hypothesis testing applet

Last revised: January 28, 2013

Hypothesis Testing Calculator

Type II Error

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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8.8 Hypothesis Tests for a Population Proportion

Learning objectives.

• Conduct and interpret hypothesis tests for a population proportion.

Some notes about conducting a hypothesis test:

• The null hypothesis [latex]H_0[/latex] is always an “equal to.”  The null hypothesis is the original claim about the population parameter.
• The alternative hypothesis [latex]H_a[/latex] is a “less than,” “greater than,” or “not equal to.”  The form of the alternative hypothesis depends on the context of the question.
• If the alternative hypothesis is a “less than”, then the test is left-tail.  The p -value is the area in the left-tail of the distribution.
• If the alternative hypothesis is a “greater than”, then the test is right-tail.  The p -value is the area in the right-tail of the distribution.
• If the alternative hypothesis is a “not equal to”, then the test is two-tail.  The p -value is the sum of the area in the two-tails of the distribution.  Each tail represents exactly half of the p -value.
• Think about the meaning of the p -value.  A data analyst (and anyone else) should have more confidence that they made the correct decision to reject the null hypothesis with a smaller p -value (for example, 0.001 as opposed to 0.04) even if using a significance level of 0.05. Similarly, for a large p -value such as 0.4, as opposed to a p -value of 0.056 (a significance level of 0.05 is less than either number), a data analyst should have more confidence that they made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.
• The significance level must be identified before collecting the sample data and conducting the test.  Generally, the significance level will be included in the question.  If no significance level is given, a common standard is to use a significance level of 5%.

Suppose the hypotheses for a hypothesis test are:

[latex]\begin{eqnarray*} H_0: & & p=20 \% \\ H_a: & & p \gt 20\% \end{eqnarray*}[/latex]

Because the alternative hypothesis is a [latex]\gt[/latex], this is a right-tail test.  The p -value is the area in the right-tail of the distribution.

[latex]\begin{eqnarray*} H_0: & & p=50 \% \\ H_a: & & p \neq  50\% \end{eqnarray*}[/latex]

Because the alternative hypothesis is a [latex]\neq[/latex], this is a two-tail test.  The p -value is the sum of the areas in the two tails of the distribution.  Each tail contains exactly half of the p -value.

[latex]\begin{eqnarray*} H_0: & & p=10\% \\ H_a: & & p \lt  10\% \end{eqnarray*}[/latex]

Because the alternative hypothesis is a [latex]\lt[/latex], this is a left-tail test.  The p -value is the area in the left-tail of the distribution.

Steps to Conduct a Hypothesis Test for a Population Proportion

• Write down the null and alternative hypotheses in terms of the population proportion [latex]p[/latex].  Include appropriate units with the values of the proportion.
• Use the form of the alternative hypothesis to determine if the test is left-tailed, right-tailed, or two-tailed.
• Collect the sample information for the test and identify the significance level.
• If [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex], use the normal distribution with [latex]\displaystyle{z=\frac{\hat{p}-p}{\sqrt{\frac{p \times (1-p)}{n}}}}[/latex].
• If one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex], use a binomial distribution.
• The results of the sample data are significant.  There is sufficient evidence to conclude that the null hypothesis [latex]H_0[/latex] is an incorrect belief and that the alternative hypothesis [latex]H_a[/latex] is most likely correct.
• The results of the sample data are not significant.  There is not sufficient evidence to conclude that the alternative hypothesis [latex]H_a[/latex] may be correct.
• Write down a concluding sentence specific to the context of the question.

USING EXCEL TO CALCULE THE P -VALUE FOR A HYPOTHESIS TEST ON A POPULATION PROPORTION

The p -value for a hypothesis test on a population proportion is the area in the tail(s) of distribution of the sample proportion.  If both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex], use the normal distribution to find the p -value.  If at least one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex], use the binomial distribution to find the p -value.

If both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p) \geq 5[/latex]:

• For x , enter the value for [latex]\hat{p}[/latex].
• For [latex]\mu[/latex] , enter the mean of the sample proportions [latex]p[/latex].  Note:  Because the test is run assuming the null hypothesis is true, the value for [latex]p[/latex] is the claim from the null hypothesis.
• For [latex]\sigma[/latex] , enter the standard error of the proportions [latex]\displaystyle{\sqrt{\frac{p \times (1-p)}{n}}}[/latex].
• For the logic operator , enter true .  Note:  Because we are calculating the area under the curve, we always enter true for the logic operator.
• Use the appropriate technique with the norm.dist function to find the area in the left-tail or the area in the right-tail.

If at least one of [latex]n \times p \lt 5[/latex] or [latex]n \times (1-p) \lt 5[/latex]:

• The p -value is found using the binomial distribution.
• For x , enter the number of successes.
• For n , enter the sample size.
• For p , enter the the value of the population proportion [latex]p[/latex] from the null hypothesis.
• For the logic operator , enter true .  Note:  Because we are calculating an at most probability, the logic operator is always true.
• For p , enter the the value of the population proportion [latex]p[/latex] in the null hypothesis.
• For the logic operator , enter true .  Note:  Because we are calculating an at least probability, the logic operator is always true.

Marketers believe that 92% of adults own a cell phone.  A cell phone manufacturer believes that number is actually lower.  In a sample of 200 adults, 87% own a cell phone.  At the 1% significance level, determine if the proportion of adults that own a cell phone is lower than the marketers’ claim.

Hypotheses:

[latex]\begin{eqnarray*} H_0: & & p=92\% \mbox{ of adults own a cell phone} \\ H_a: & & p \lt 92\% \mbox{ of adults own a cell phone} \end{eqnarray*}[/latex]

From the question, we have [latex]n=200[/latex], [latex]\hat{p}=0.87[/latex], and [latex]\alpha=0.01[/latex].

To determine the distribution, we check [latex]n \times p[/latex] and [latex]n \times (1-p)[/latex].  For the value of [latex]p[/latex], we use the claim from the null hypothesis ([latex]p=0.92[/latex]).

[latex]\begin{eqnarray*} n \times p & = & 200 \times 0.92=184 \geq 5 \\ n \times (1-p) & = & 200 \times (1-0.92)=16 \geq 5\end{eqnarray*}[/latex]

Because both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p)  \geq 5[/latex] we use a normal distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\lt[/latex], the p -value is the area in the left tail of the distribution.

So the p -value[latex]=0.0046[/latex].

Conclusion:

Because p -value[latex]=0.0046 \lt 0.01=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis.  At the 1% significance level there is enough evidence to suggest that the proportion of adults who own a cell phone is lower than 92%.

• The null hypothesis [latex]p=92\%[/latex] is the claim that 92% of adults own a cell phone.
• The alternative hypothesis [latex]p \lt 92\%[/latex] is the claim that less than 92% of adults own a cell phone.
• The function is norm.dist because we are finding the area in the left tail of a normal distribution.
• Field 1 is the value of [latex]\hat{p}[/latex].
• Field 2 is the value of [latex]p[/latex] from the null hypothesis.  Remember, we run the test assuming the null hypothesis is true, so that means we assume [latex]p=0.92[/latex].
• Field 3 is the standard deviation for the sample proportions [latex]\displaystyle{\sqrt{\frac{p \times (1-p)}{n}}}[/latex].
• The p -value of 0.0046 tells us that under the assumption that 92% of adults own a cell phone (the null hypothesis), there is only a 0.46% chance that the proportion of adults who own a cell phone in a sample of 200 is 87% or less.  This is a small probability, and so is unlikely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.  In other words, the proportion of adults who own a cell phone is most likely less than 92%.

A consumer group claims that the proportion of households that have at least three cell phones is 30%.  A cell phone company has reason to believe that the proportion of households with at least three cell phones is much higher.  Before they start a big advertising campaign based on the proportion of households that have at least three cell phones, they want to test their claim.  Their marketing people survey 150 households with the result that 54 of the households have at least three cell phones.  At the 1% significance level, determine if the proportion of households that have at least three cell phones is less than 30%.

[latex]\begin{eqnarray*} H_0: & & p=30\% \mbox{ of household have at least 3 cell phones} \\ H_a: & & p \gt 30\% \mbox{ of household have at least 3 cell phones} \end{eqnarray*}[/latex]

From the question, we have [latex]n=150[/latex], [latex]\displaystyle{\hat{p}=\frac{54}{150}=0.36}[/latex], and [latex]\alpha=0.01[/latex].

To determine the distribution, we check [latex]n \times p[/latex] and [latex]n \times (1-p)[/latex].  For the value of [latex]p[/latex], we use the claim from the null hypothesis ([latex]p=0.3[/latex]).

[latex]\begin{eqnarray*} n \times p & = & 150 \times 0.3=45 \geq 5 \\ n \times (1-p) & = & 150 \times (1-0.3)=105 \geq 5\end{eqnarray*}[/latex]

Because both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p)  \geq  5[/latex] we use a normal distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\gt[/latex], the p -value is the area in the right tail of the distribution.

So the p -value[latex]=0.0544[/latex].

Because p -value[latex]=0.0544 \gt 0.01=\alpha[/latex], we do not reject the null hypothesis.  At the 1% significance level there is not enough evidence to suggest that the proportion of households with at least three cell phones is more than 30%.

• The null hypothesis [latex]p=30\%[/latex] is the claim that 30% of households have at least three cell phones.
• The alternative hypothesis [latex]p \gt 30\%[/latex] is the claim that more than 30% of households have at least three cell phones.
• The function is 1-norm.dist because we are finding the area in the right tail of a normal distribution.
• Field 2 is the value of [latex]p[/latex] from the null hypothesis.  Remember, we run the test assuming the null hypothesis is true, so that means we assume [latex]p=0.3[/latex].
• The p -value of 0.0544 tells us that under the assumption that 30% of households have at least three cell phones (the null hypothesis), there is a 5.44% chance that the proportion of households with at least three cell phones in a sample of 150 is 36% or more.  Compared to the 1% significance level, this is a large probability, and so is likely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis.  In other words, the claim that 30% of households have at least three cell phones is most likely correct.

A teacher believes that 70% of students in the class will want to go on a field trip to the local zoo.  The students in the class believe the proportion is much higher and ask the teacher to verify her claim.  The teacher samples 50 students and 39 reply that they would want to go to the zoo.  At the 5% significance level, determine if the proportion of students who want to go on the field trip is higher than 70%.

[latex]\begin{eqnarray*} H_0: & & p = 70\% \mbox{ of students want to go on the field trip}  \\ H_a: & & p \gt 70\% \mbox{ of students want to go on the field trip}   \end{eqnarray*}[/latex]

From the question, we have [latex]n=50[/latex], [latex]\displaystyle{\hat{p}=\frac{39}{50}=0.78}[/latex], and [latex]\alpha=0.05[/latex].

[latex]\begin{eqnarray*} n \times p & = & 50 \times 0.7=35 \geq 5 \\ n \times (1-p) & = & 50 \times (1-0.7)=15 \geq 5\end{eqnarray*}[/latex]

Because both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p)  \geq 5[/latex] we use a normal distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\gt[/latex], the p -value is the area in the right tail of the distribution.

So the p -value[latex]=0.1085[/latex].

Because p -value[latex]=0.1085 \gt 0.05=\alpha[/latex], we do not reject the null hypothesis.  At the 5% significance level there is not enough evidence to suggest that the proportion of students who want to go on the field trip is higher than 70%.

• The null hypothesis [latex]p=70\%[/latex] is the claim that 70% of the students want to go on the field trip.
• The alternative hypothesis [latex]p \gt 70\%[/latex] is the claim that more than 70% of students want to go on the field trip.
• The p -value of 0.1085 tells us that under the assumption that 70% of students want to go on the field trip (the null hypothesis), there is a 10.85% chance that the proportion of students who want to go on the field trip in a sample of 50 students is 78% or more.  Compared to the 5% significance level, this is a large probability, and so is likely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis.  In other words, the teacher’s claim that 70% of students want to go on the field trip is most likely correct.

Joan believes that 50% of first-time brides in the United States are younger than their grooms.  She performs a hypothesis test to determine if the percentage is the same or different from 50%.  Joan samples 100 first-time brides and 56 reply that they are younger than their grooms.  Use a 5% significance level.

[latex]\begin{eqnarray*} H_0: & & p=50\% \mbox{ of first-time brides are younger than the groom} \\ H_a: & & p \neq 50\% \mbox{ of first-time brides are younger than the groom} \end{eqnarray*}[/latex]

From the question, we have [latex]n=100[/latex], [latex]\displaystyle{\hat{p}=\frac{56}{100}=0.56}[/latex], and [latex]\alpha=0.05[/latex].

To determine the distribution, we check [latex]n \times p[/latex] and [latex]n \times (1-p)[/latex].  For the value of [latex]p[/latex], we use the claim from the null hypothesis ([latex]p=0.5[/latex]).

[latex]\begin{eqnarray*} n \times p & = & 100 \times 0.5=50 \geq 5 \\ n \times (1-p) & = & 100 \times (1-0.5)=50 \geq 5\end{eqnarray*}[/latex]

Because both [latex]n \times p \geq 5[/latex] and [latex]n \times (1-p)  \geq 5[/latex] we use a normal distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\neq[/latex], the p -value is the sum of area in the tails of the distribution.

Because there is only one sample, we only have information relating to one of the two tails, either the left or the right.  We need to know if the sample relates to the left or right tail because that will determine how we calculate out the area of that tail using the normal distribution.  In this case, the sample proportion [latex]\hat{p}=0.56[/latex] is greater than the value of the population proportion in the null hypothesis [latex]p=0.5[/latex] ([latex]\hat{p}=0.56>0.5=p[/latex]), so the sample information relates to the right-tail of the normal distribution.  This means that we will calculate out the area in the right tail using 1-norm.dist .  However, this is a two-tailed test where the p -value is the sum of the area in the two tails and the area in the right-tail is only one half of the p -value.  The area in the left tail equals the area in the right tail and the p -value is the sum of these two areas.

So the area in the right tail is 0.1151 and  [latex]\frac{1}{2}[/latex]( p -value)[latex]=0.1151[/latex].  This is also the area in the left tail, so

p -value[latex]=0.1151+0.1151=0.2302[/latex]

Because p -value[latex]=0.2302 \gt 0.05=\alpha[/latex], we do not reject the null hypothesis.  At the 5% significance level there is not enough evidence to suggest that the proportion of first-time brides that are younger than the groom is different from 50%.

• The null hypothesis [latex]p=50\%[/latex] is the claim that the proportion of first-time brides that are younger than the groom is 50%.
• The alternative hypothesis [latex]p \neq 50\%[/latex] is the claim that the proportion of first-time brides that are younger than the groom is different from 50%.
• We use norm.dist([latex]\hat{p}[/latex],[latex]p[/latex],[latex]\mbox{sqrt}(p*(1-p)/n)[/latex],true) to find the area in the left tail.  The area in the right tail equals the area in the left tail, so we can find the p -value by adding the output from this function to itself.
• We use 1-norm.dist([latex]\hat{p}[/latex],[latex]p[/latex],[latex]\mbox{sqrt}(p*(1-p)/n)[/latex],true) to find the area in the right tail.  The area in the left tail equals the area in the right tail, so we can find the p -value by adding the output from this function to itself.
• The p -value of 0.2302  is a large probability compared to the 5% significance level, and so is likely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis.  In other words, the claim that the proportion of first-time brides who are younger than the groom is most likely correct.

Watch this video: Hypothesis Testing for Proportions: z -test by ExcelIsFun [7:27] 

An online retailer believes that 93% of the visitors to its website will make a purchase.   A researcher in the marketing department thinks the actual percent is lower than claimed.  The researcher examines a sample of 50 visits to the website and finds that 45 of the visits resulted in a purchase.  At the 1% significance level, determine if the proportion of visits to the website that result in a purchase is lower than claimed.

[latex]\begin{eqnarray*} H_0: & & p=93\% \mbox{ of visitors make a purchase} \\ H_a: & & p \lt 93\% \mbox{ of visitors make a purchase} \end{eqnarray*}[/latex]

From the question, we have [latex]n=50[/latex], [latex]x=45[/latex], and [latex]\alpha=0.01[/latex].

To determine the distribution, we check [latex]n \times p[/latex] and [latex]n \times (1-p)[/latex].  For the value of [latex]p[/latex], we use the claim from the null hypothesis ([latex]p=0.93[/latex]).

[latex]\begin{eqnarray*} n \times p & = & 50 \times 0.93=46.5 \geq 5 \\ n \times (1-p) & = & 50 \times (1-0.93)=3.5 \lt 5\end{eqnarray*}[/latex]

Because [latex]n \times (1-p)  \lt 5[/latex] we use a binomial distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\lt[/latex], the p -value is the probability of getting at most 45 successes in 50 trials.

So the p -value[latex]=0.2710[/latex].

Because p -value[latex]=0.2710 \gt 0.01=\alpha[/latex], we do not reject the null hypothesis.  At the 1% significance level there is not enough evidence to suggest that the proportion of visitors who make a purchase is lower than 93%.

• The null hypothesis [latex]p=93\%[/latex] is the claim that 93% of visitors to the website make a purchase.
• The alternative hypothesis [latex]p \lt 93\%[/latex] is the claim that less than 93% of visitors to the website make a purchase.
• The function is binom.dist because we are finding the probability of at most 45 successes.
• Field 1 is the number of successes [latex]x[/latex].
• Field 2 is the sample size [latex]n[/latex].
• Field 3 is the probability of success [latex]p[/latex].  This is the claim about the population proportion made in the null hypothesis, so that means we assume [latex]p=0.93[/latex].
• The p -value of 0.2710 tells us that under the assumption that 93% of visitors make a purchase (the null hypothesis), there is a 27.10% chance that the number of visitors in a sample of 50 who make a purchase is 45 or less.  This is a large probability compared to the significance level, and so is likely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely correct, and so the conclusion of the test is to not reject the null hypothesis.  In other words, the proportion of visitors to the website who make a purchase adults is most likely 93%.

A drug company claims that only 4% of people who take their new drug experience any side effects from the drug.  A researcher believes that the percent is higher than drug company’s claim.  The researcher takes a sample of 80 people who take the drug and finds that 10% of the people in the sample experience side effects from the drug.  At the 5% significance level, determine if the proportion of people who experience side effects from taking the drug is higher than claimed.

[latex]\begin{eqnarray*} H_0: & & p=4\% \mbox{ of people experience side effects} \\ H_a: & & p \gt 4\% \mbox{ of people experience side effects} \end{eqnarray*}[/latex]

From the question, we have [latex]n=80[/latex], [latex]\hat{p}=0.1[/latex], and [latex]\alpha=0.05[/latex].

To determine the distribution, we check [latex]n \times p[/latex] and [latex]n \times (1-p)[/latex].  For the value of [latex]p[/latex], we use the claim from the null hypothesis ([latex]p=0.04[/latex]).

[latex]\begin{eqnarray*} n \times p & = & 80 \times 0.04=3.2 \lt 5\end{eqnarray*}[/latex]

Because [latex]n \times p  \lt 5[/latex] we use a binomial distribution to calculate the p -value.  Because the alternative hypothesis is a [latex]\gt[/latex], the p -value is the probability of getting at least 8 successes in 80 trials.  (Note:  In the sample of size 80, 10% have the characteristic of interest, so this means that [latex]80 \times 0.1=8[/latex] people in the sample have the characteristic of interest.)

So the p -value[latex]=0.0147[/latex].

Because p -value[latex]=0.0147 \lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis.  At the 5% significance level there is enough evidence to suggest that the proportion of people who experience side effects from taking the drug is higher than 4%.

• The null hypothesis [latex]p=4\%[/latex] is the claim that 4% of the people experience side effects from taking the drug.
• The alternative hypothesis [latex]p \gt 4\%[/latex] is the claim that more than 4% of the people experience side effects from taking the drug.
• The function is 1-binom.dist because we are finding the probability of at least 8 successes.
• Field 1 is [latex]x-1[/latex] where [latex]x[/latex] is the number of successes.  In this case, we are using the compliment rule to change the probability of at least 8 successes into 1 minus the probability of at most 7 successes.
• Field 3 is the probability of success [latex]p[/latex].  This is the claim about the population proportion made in the null hypothesis, so that means we assume [latex]p=0.04[/latex].
• The p -value of 0.0147 tells us that under the assumption that 4% of people experience side effects (the null hypothesis), there is a 1.47% chance that the number of people in a sample of 80 who experience side effects is 8 or more.  This is a small probability compared to the significance level, and so is unlikely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.  In other words, the proportion of people who experience side effects is most likely greater than 4%.

Concept Review

The hypothesis test for a population proportion is a well-established process:

• Find the p -value (the area in the corresponding tail) for the test using the appropriate distribution (normal or binomial).
• Compare the p -value to the significance level and state the outcome of the test.

Attribution

“ 9.6   Hypothesis Testing of a Single Mean and Single Proportion “ in Introductory Statistics by OpenStax  is licensed under a  Creative Commons Attribution 4.0 International License.

Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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• Hypothesis Testing
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• Confidence Interval Creation
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Instructor’s Guide: Hypothesis Testing Tutorial

How the Hypothesis Testing tutorial fits into the typical statistics course: WISE tutorials are modularized to allow instructors to pick or choose modules that best fit their course needs. Each module is a self-contained lesson that does not depend on any of the other modules, although some specific prerequisite information may be required.

The Hypothesis Testing (HT) Tutorial assumes that students have some familiarity with basic statistics, such as means, standard deviation, and variance, and are able to calculate standard errors of the mean and z -scores. Students should also have an understanding of the normal distribution, sampling distributions, and the Central Limit Theorem.

When to use the HT tutorial? Instructors can go over hypothesis testing after sampling distributions and the Central Limit are discussed, and before introducing power and effect sizes. Completing the WISE tutorials on the sampling distribution of the mean and on the Central Limit Theorem prior to the HT tutorial may enhance students’ experience with using this tutorial. After this tutorial, students may go on to the WISE Power Tutorial .

Suggestions for Using the HT Tutorial

• Class demonstration/Lecture aid
• Lab assignment
• Homework assignment
• Review assignment

There are many ways in which the HT Module can be inserted into your lesson plan. Your choices may depend on students’ level of computer literacy, computer resources available at your school, and class time restrictions. Here are a few suggestions:

1. Pre-lecture Assignment

Assign the module as homework to introduce hypothesis testing to students. This will allow you to use more class time for in-depth discussions and activities instead of a full lecture. Students may download and print the Tutorial Worksheet (which contains the multiple choice questions in the tutorial as well as spaces for written explanations of their responses) that they may complete and submit.

2. Live Demonstration

As part of either a lecture or guided lab assignment, the Power applet itself may be used by the instructor to demonstrate visually different aspects of the sampling distribution, z -scores, and statistical significance. Some instructors may choose to step through parts or all of the tutorial in a demonstration mode. This demonstration may serve as a stimulus for classroom discussion and/or introduction to an assignment for students.

You may wish to use the Sampling Distribution Applet to demonstrate how sample size affects the variability of sample means. Power Applet to illustrate concepts that will lead to discussion of statistical power.

Access: SDM Applet | SDM Applet Instructions | Power Applet | Power Applet Instructions

Demonstration guides for: Sampling Distribution of the Mean Applet | Power Applet

3. Post-lecture Assignment

After your presentation on hypothesis testing, the module can be used to demonstrate lecture points and give students practice using the concepts. This applet allows students to gain a perspective on the concepts that complement a lecture or other presentations. The more perspectives students are exposed to in the course of instruction, the more likely they are to understand and retain the material.  Students may download and print the Tutorial Worksheet that they may submit to you.

For more information, see the Introduction to the tutorial .

4. Tutorial Worksheet

The main portion of the module is designed to give students feedback without evaluating their performance. The multiple-choice questions provide feedback and explanations on both correct and incorrect responses. However, no record is kept of student answers. You may have your students download and print the Tutorial Worksheet so that they may produce a written record of their responses, calculations, and explanations. They may submit this to you for evaluation.

5. Supplemental Activities

• Final Quiz on Hypothesis Testing – Paper quiz on an application of z -test similar to the tutorial that also examines a two-tailed test. No answers to the questions are posted online.
• Online Quiz on p -values – The follow-up online quiz assesses understanding of how to interpret hypothesis test results. There are two versions of this quiz. Each version contains seven true-false questions that ask for short-answer essays, designed to examine conceptual understanding of the topic. Feedback and explanations are provided at the end of the quiz.

WISE modules are designed as self-contained lessons that students can use with little, if any, guidance. If you are concerned that students may not feel comfortable using web pages and applets, you may consider using the module as part of an in-class activity. Most students complete the module in 40 – 50 minutes.

Your Feedback

We hope this tutorial is helpful for you and your students, and we welcome your feedback on this tutorial and other aspects of the WISE site. Please send your comments to [email protected] .

Questions, comments, difficulties? See our technical support page or contact us: [email protected] .

Conceptual Learning with Interactive Applets

Interactive applets for undergraduate mathematics and statistics

Conceptual Learning with Interactive Applets is a project to build high-quality web-based applets and supporting resources for enhancing conceptual understanding in undergraduate mathematics and statistics. Our applets are built using GeoGebra .

The project is based in the University of Melbourne , School of Mathematics and Statistics , and funded by a University of Melbourne Learning & Teaching Initiatives grant.

Select a subject to view applets for that subject, or browse the full collection below.

This applet demonstrates the concept of coordinate vectors in $\mathbb R^2$ A basis $\mathcal B$ of a vector space $V$ is a linearly independent spanning set. One useful feature of a basis is that it gives rise to a way of writing coordinates on $V$. Any vector $\mathbf v \in V$ can be written uniquely […]

This applet demonstrates the Gram-Schmidt algorithm performed in $\mathbb R^3$. The Gram-Schmidt algorithm converts a basis of an inner product space into an orthonormal basis. It does this by building up the orthonormal basis one vector at a time. For each vector in turn, we remove any component that is parallel to the vectors which […]

This applet shows the geometric effect of a linear transformation $T: \mathbb R^2 \to \mathbb R^2$. You can type a matrix $M$ on the right hand side of the applet, and then click Play to see how the vertices of a triangle are transformed when multiplying by $M$. Other resources: Geogebratube page for this applet

This applet shows the geometric effect of a linear transformation $T$ in $\mathbb R^3$. You can type a 3×3 matrix $M$ on the right hand side of the applet, and then click Play to see how the vertices of a cube are transformed when multiplying by $M$. Can you see if the corresponding transformations are […]

This applet helps visualise the surface generated by cylindrical coordinates using r,θ and z. Click and drag on the sliders on the left to adjust the ranges for r,θ and z. Geogebratube page for this applet

This applet visualises surfaces generated by spherical coordinates using r,θ and φ. Click and drag on the sliders on the left to change the values for r,θ and φ. Click and drag on the graph to change/rotate the view. Geogebratube page for this applet

This applet shows a solution of the heat equation, a partial differential equation from MAST20029 Engineering Mathematics.

This applet visualises the span of two vectors in R3 using linear combinations.

This applet shows a line in R2 and the vector form of its equation.

This applet shows a plane in R3 and the vector form of its equation.

This applet shows a line in R3 and the vector form of its equation. 

This applet shows how the determinant is unaffected by the elementary row operation of addition of a scalar multiple of a row to another row.

This applet shows the row, column, and solution spaces of a 3×3 matrix M.

This applet shows how the column space, solution space, rank and nullity of a matrix M change as you append additional columns. Initially the matrix M has a single column. You can add extra columns to M by editing the text boxes on the right of the applet, and clicking the ‘Append column’ button. The […]

This applet explores a geometric interpretation of the parameter t in the parameterisation of the standard hyperbola using cosh and sinh.

This applet explores the velocity vector of a parametric curve, and its relationship to the chord r(t+h)-r(t) and the difference quotient.

This applet illustrates the connection between a confidence interval, a formal hypothesis test, and the p-value of a hypothesis test.

This applet gives a visualisation of the concept of statistical power, and helps illustrate the relationship between power, sample size, standard deviation and difference between the means.

This applet illustrates partitioning of variability into explained (fitted) and unexplained (residual) variability.

This applet illustrates the partitioning of variability into explained and unexplained variability, in the context of ANOVA.

This applet illustrates the effect of a linear transformation in R2 on the unit circle/unit disk, and the geometric meaning of eigenvectors, eigenvalues and determinant.

This applet displays the distribution for the order statististics of a sample of size n from an arbitrary population distribution.

This applet shows the maximum likelihood estimator and (log) likelihood function for several statistical models.

This applet illustrates the ε-δ definitions of the limit and continuity of a function. It can be used to investigate (non-)convergence or (dis)continuity of real functions, including the Dirichlet everywhere discontinuous function and variants.

This applet illustrates the definition of derivative as the limit of the gradient of a chord.

This applet illustrates the ε-M definition of convergence of a sequence.

This applet illustrates upper and lower Riemann sums and refinement of partitions.

This applet shows the relationship between terms of a sequence and the partial sums of a series. It also allows exploration of some important sequences & series including geometric and harmonic sequences.

This applet explores the normal approximation to the binomial distribution.

This applet shows the construction of the inverse of a function, and can be used to explore whether the inverse is a function.

This applet plots and traces a parametric curve, given as a vector function in R2.

This applet plots two parametric curves simultaneously. It can be used to explore whether two particles collide.

This applet explores the relationship between the pmf or density and the cumulative distribution function of a range of discrete and continuous probability distributions.

This applet explores a logistic population growth model with no harvesting. The phase plot is shown alongside the plot of p vs t.

This applet explores a logistic population growth model with constant harvesting.

This applet shows the relationship between a plot of an estimated empirical CDF and a normal probability plot.

This applet explores QQ-plots for a range of distributions.

This applet simulates a spring acting under gravity, subject to drag and an external driving force.

This applet illustrates the concept of independent identically distributed random variables.

This applet aims to demonstrate visually the projection of a vector u onto a vector v.

This applet illustrates how the distribution of the sample mean converges towards normality as sample size increases.

This applet calculates the zygote and adult allele and genotype frequencies according to the Fisher-Haldane-Wright model of population genetics, and plots the results.

This applet iterates a difference equation (also known as recurrence relation) and displays the resulting sequence both graphically and numerically.

This worksheet performs iteration and produces cobweb diagrams for a first-order difference equation (AKA recurrence relation, discrete dynamical system).

This applet shows a linear approximation to a non-linear difference equation close to an equilibrium, using cobwebbing. It can be used to investigate the accuracy of a linear approximation, or to motivate the linear stability criterion for equilibria of a first-order difference equation.

Guess the correlation of a sample of bivariate data drawn from a linear or non-linear population.

Repeatedly sample from a bivariate population, and construct a histogram of sample regression line slope.

This applet illustrates a solution of the wave equation, from the MAST20029 Engineering Mathematics lecture notes.

This applet displays the direction field and solutions for an ordinary differential equation (ODE), and calculates approximate solutions using Euler’s method.

This applet investigates the continuity of a 2-branch piecewise-defined function.

This applet investigates the continuity of a piecewise-defined function.

This applet displays the direction field and solutions for an ordinary differential equation (ODE).

This applet explores how the rate of change of a composite function y = f(g(x)) depends on the rates of change of both f and g.

Chi-Square Distribution $X \sim \chi^2_{(\nu)}$ $\nu=$ $x=$ This applet computes probabilities and percentiles for the chi-square distribution: $$X \sim \chi^2_{(\nu)}$$ Directions: Enter the degrees of freedom in the $\nu$ box. To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue (left) box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink (right) box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability. To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink (right) box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your keyboard. The percentile $x$ will appear in the blue (left) box. On the graph, the $x$ value appears in blue while the probability is shaded in pink . Probability density function $$f(x)=\frac{1}{2^{\nu/2}\Gamma\left(\frac{\nu}{2}\right)} x^{\nu/2-1} e^{-x/2}$$ where $x > 0$ and $\nu > 0$ $\mu=E(X)=\nu$ $\sigma^2=Var(X)=2\nu$ $\sigma=SD(X)=\sqrt{2\nu}$ Caption Contest Dollar Street Lifetime Achievement SPARKS Challenge Stats + Stories Lesson Plan Contest Presenter Guide Statistics Jobs Getting Started Chance News Fun Collection Resource Library Suggest Resource ** Hypothesis Test for a Mean Report a problem with this resource You must Login or Register to post comments.   MathCS.org - Java Java Programs Android Tutorial Java by Definition Asset Survey System Discussion Board Hypothesis Testing This applet illustrates the process and theoretical background of testing the hypothesis of a population mean having a specific value against different alternatives. You should also check out the closely related Central Limit Theorem applet. Start the HypoTester applet Carefully move through the four steps involved when testing a hypothesis to understand how the applet works. Now answer the following questions: Using the default values, would you feel comfortable rejecting the null hypothesis, and hence accepting the alternative hypothesis ? If so, what is the probability that you are making a mistake when you reject the null hypothesis? Use the default values, but now choose the " two-sided alternative " in Step 1. Do you still reject the null hypothesis? What is the probability now of making a mistake when rejecting the null hypothesis? What is the relation of this probability to the probability in the previous question? Is that relation always true between one- and two-sided alternatives? Use the default values, but now choose the " left-sided alternative " in Step 1. How does the result of the test change? Would you still reject the null hypothesis? Why or why not? Reset all values to their default values by clicking on [Reset]. Now experiment with different standard deviations, keeping all other values the same. How does the result of the test change when you change the standard deviation? Explain. What does this all have to do with the Central Limit Theorem (you could start the HypoCentral applet to see an illustration of the Central Limit Theorem) Rossman/Chance Applet Collection Updated 2021 applets.   - categorical - quantitative - quantitative (click here for help on running java on , )     -intervals for different population shapes     Probability Probability Calculator Statistical Inference    (multiple groups)   (multiple groups)    Multiple Variables (js) (js) (js) (js) (js) Click to access old applets page Teachers: Click to set up memory quiz for your class 214 IMAGES Hypothesis testing Hypothesis testing using the binomial distribution (2.05a) SOLVED:Use the applet Hypothesis Testing (for Proportions) (refer to Use the applet Hypothesis Test for a Mean to investigate the Use the applet Hypothesis Test for a Mean to investigate the WISE Applets COMMENTS Interactive applets This is a list of the most popular applets. For more go to StatCrunch.. Confidence Intervals/Hypothesis Testing. Confidence intervals for a mean ; Confidence intervals for a proportion Hypothesis tests for a mean The hypothesis test is based on the T statistic. The resulting statistic from the test drops into the plot. Red values represent tests where the null hypothesis is rejected at the specified level of significance. The default significance level of 0.05 used for the tests can be changed by adjusting the Level input within the applet. Hypothesis tests for a proportion The hypothesis test is based on the Z statistic. The resulting statistic from the test drops into the plot. Red values are tests where the null hypothesis is rejected at the specified level of significance. Change the default significance level (set at 0.05) by adjusting the Level in the applet. 5 tests and 1000 tests add the hypothesis results ... Chapter 1 . P-Value of a Test of Significance This applet illustrates the P-value of a test of significance. Here we're testing a hypothesis about the mean of a normal distribution whose standard deviation we know, but the concepts are essentially the same for any other type of significance test. The normal curve shows the sampling distribution of the sample mean when your null hypothesis ... Two Applets for Teaching Hypothesis Testing Nevertheless, students have good intuition about what makes a thing "too unlikely to be true". This applet guides students to make a decision based on a given probability without introducing formal concepts of hypothesis testing. The applet consists of two "views", called "test view" (. Figure 1. ) and "investigate view" (. Power of a hypothesis test This applet gives a visualisation of the concept of statistical power, and helps illustrate the relationship between power, sample size, standard deviation and ... it has no effect on the hypothesis test.) Other resources: Online tutorial using this applet. Alternate version of the applet with large sample sizes. Geogebratube page for this ... Hypothesis tests for a mean Hypothesis tests for a mean. The applet below allows one to visually investigate hypothesis tests for a mean. Specify the sample size, n, the shape of the distribution (Normal or Right skewed), the true population mean (Mean), the true population standard deviation (Std. Dev.), the null value for the mean (Null mean) and the alternative for the test (Alternative). Hypothesis Testing The Four Steps in Hypothesis Testing. STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha. STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. If the conditions are met, summarize the data using a test statistic. WISE Applets This applet converts probability values to z values and vice versa. The simple version converts only right-tail p and z values. The graphic version allows the user to input left-tail p, raw scores, and the mean and standard deviation of the group of interest. Hypothesis Testing Applet Power of a Hypothesis Test Applet This probability is knows as the power of the test, and it depends on the true value of . (Clearly, a test would have more power for an extreme value of than for a that is very close to .) To use this applet, you must specify the hypothesized mean , the true mean , and the value of , and select the appropriate alternative hypothesis. Confidence intervals, hypothesis testing and p-values This applet illustrates the connection between a confidence interval, a formal hypothesis test, and the p-value of a hypothesis test. Conceptual Learning with Interactive Applets. Menu. Confidence intervals, hypothesis testing and p-values Hypothesis Testing Calculator with Steps Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ... 8.8 Hypothesis Tests for a Population Proportion Suppose the hypotheses for a hypothesis test are: H 0: p = 50% H a: p ≠ 50% H 0: p = 50 % H a: p ≠ 50 %. Because the alternative hypothesis is a ≠ ≠, this is a two-tail test. The p -value is the sum of the areas in the two tails of the distribution. Each tail contains exactly half of the p -value. 10 2A Hypothesis Test for a Proportion Using Coin Flipping Applet Introduces hypothesis test for a population proportion using simulation with a coin-flipping applet. Utilizes the P-value approach. Based on Section 10.2A ... WISE » Instructor's Guide: Hypothesis Testing Tutorial This applet allows students to gain a perspective on the concepts that complement a lecture or other presentations. The more perspectives students are exposed to in the course of instruction, the more likely they are to understand and retain the material. ... Final Quiz on Hypothesis Testing - Paper quiz on an application of z-test similar to ... Hypothesis Testing The applet restricts the case to where the null and alternative distributions are both normal and only differ by a shift in the mean. Specifically, where 0 is fixed at 10 and 2 is fixed at 1.0. The hypothesis of interest will be the right-tailed hypothesis: Power is calculated using the following formula assuming 2 is known: Conceptual Learning with Interactive Applets Power of a hypothesis test. This applet gives a visualisation of the concept of statistical power, and helps illustrate the relationship between power, sample size, standard deviation and difference between the means. 12 Jul 2021 Applets. Partitioning of variability in regression. Two Applets for Teaching Hypothesis Testing This applet guides students to make a decision based on a given probability without introducing formal concepts of hypothesis testing. The applet consists of two "views", called "test view" (Figure 1) and "investigate view" (Figure 2). In the first, the user carries out an experiment to determine if a displayed coin is fair; in the second he ... Hypothesis Testing Applet Open in new window Open in current window ... Chi-Square Distribution Applet/Calculator ©2021 Matt Bognar Department of Statistics and Actuarial Science University of Iowa ** Hypothesis Test for a Mean This Java applet tutorial prompts the user to input the components of a hypothesis test for the mean. Hints are provided whenever the user enters an incorrect value. Once the steps are completed and the user has chosen the correct conclusion for accepting or rejecting the null hypothesis, a statement summarizing the conclusion is displayed. The applet is supported by an explanation of the ... MathCS.org Click for help! Start the HypoTester applet. Carefully move through the four steps involved when testing a hypothesis to understand how the applet works. Now answer the following questions: Using the default values, would you feel comfortable rejecting the null hypothesis, and hence accepting the alternative hypothesis ? Rossman/Chance Applet Collection One proportion inference. Goodness of Fit. Analyzing Two-way Tables. Matched Pairs. Randomization test for quantitative response (multiple groups) two means. Randomization test for categorical response (multiple groups) Dolphin Study applet. assignment essay homework phd report resume review speech thesis