Performing Hypothesis Tests: Proportions: Videos & Practice Problems

A research team surveys $100$ employees and asks each, “How do you feel about your current stress level compared to your colleagues?” The responses are:

• $47$ employees said “more stressed”  
• $35$ employees said “less stressed”
• $18$ employees said “about the same.”

Using the sign test, examine the null hypothesis that the proportion of employees who feel more stressed equals the proportion who feel less stressed. Use $\alpha =0.05$.  

A health study asks participants, “Do you exercise in the morning or evening?” Responses are 'morning', 'evening', and 'never'. Use a sign test to test if the proportion exercising in the morning equals the proportion exercising in the evening. Assign $+$ to 'morning', $-$ to 'evening', and $0$ to 'never'. Suppose there are $12$ 'morning', $17$ 'evening', and $6$ 'never' responses. Use $\alpha =0.05$. What is the $p$-value?

A company advertises that at least $70\%$ of its customers are satisfied with their service. In a survey of $150$ customers, $99$ say they are satisfied. Which of the following is the correct null hypothesis for testing the company's claim?

In a study of users’ adoption of a new app feature, researchers claim that the true adoption rate $p$ is greater than $30\%$. They collect a sample of $n=100$ users and find a sample proportion $\hat{p}=0.35$. At $\alpha =0.05$, can they use the normal approximation, and do the data support the claim?

A researcher claims that at least $55\%$ of adults prefer online shopping. In a sample of $120$ adults, $60$ say they prefer online shopping. Can a normal sampling distribution be used? If so, test the claim at $\alpha =0.05$.

In a sample of $200$ employees, a firm claims that the true proportion of staff who prefer remote work is not equal to $0.40$. The sample shows $\hat{p}=0.48$. At $\alpha =0.05$, determine whether you can use the normal approximation, and if so, test the claim.

A researcher claims that the proportion of adults who exercise regularly is greater than $0.55$. In a random sample of $n=180$ adults, $\hat{p}=0.50$. Can a normal sampling distribution be used? If so, test the claim at $\alpha =0.05$.

A survey analyst claims that at least $60\%$ of consumers support stricter labeling laws for organic products. A recent random sample of $180$ consumers shows that $102$ of them support stricter labeling laws. Using a significance level of $\alpha =0.10$, is there enough evidence to reject the analyst’s claim?

A campus survey claims that at least $60\%$ of full-time students commute to campus daily. In a random sample of $120$ students, $66$ reported that they commute daily. At the $0.10$ significance level, is there enough evidence to reject the survey’s claim?

A corporate trainer claims that more than half of employees in large organizations believe that workplace communication has improved since switching to hybrid work models. In a random sample of $300$ employees, $162$ agree with this statement. At the $0.01$ significance level, is there enough evidence to support the trainer’s claim?

A workforce analyst claims that more than half of remote employees believe their time management skills improved while working from home. In a random sample of $280$ remote employees, $158$ say their time management skills improved. At the $0.01$ significance level, is there enough evidence to support the analyst’s claim?

An environmental researcher claims that $25\%$ of households regularly participate in local recycling programs. In a random sample of $500$ households, $135$ say they participate regularly. At the $0.05$ significance level, is there enough evidence to reject the researcher’s claim?

A sustainability officer claims that at least $70\%$ of office employees try to reduce paper usage to help the environment. In a random sample of $100$ employees, $62$ report that they actively try to reduce paper usage. At the $0.05$ significance level, is there enough evidence to reject the officer’s claim?

A sustainability researcher claims that less than $30\%$ of household appliances sold in a particular region are certified as energy efficient. A random sample of $35$ appliances is selected, and $8$ of them are found to be energy efficient. At the $0.05$ significance level, can you conclude that less than $30\%$ of appliances are energy efficient, as the researcher claims?

A college asserts that at least $75\%$ of its students participate in extracurricular activities. Out of a random sample of $80$ students, $62$ are involved in such activities. At $\alpha =0.05$, is there sufficient evidence to reject the college's claim?

A poll suggests that over $75\%$ of city residents support expanding public transportation. In a random sample of $60$ residents, $78\%$ expressed support. At $\alpha =0.05$, is there sufficient evidence to back the poll's claim?

Consider a hypothesis test for the proportion of a population, where the test statistic obtained is $z=-1.45$. The claim being tested is that the population proportion is greater than $0.4$.

i. Determine whether the test is two-tailed, right-tailed, or left-tailed.

ii. Calculate the $p$-value for this test.

iii. At a significance level of $\alpha =0.05$, should we reject the null hypothesis or fail to reject it?

In a study, a researcher tests the hypothesis that the population proportion is less than $0.6$. The obtained test statistic is $z=-2.10$.

i. Determine whether the test is two-tailed, right-tailed, or left-tailed.

ii. Calculate the $p$-value for this test.

iii. At a significance level of $\alpha =0.05$, should we reject the null hypothesis or fail to reject it?

A researcher wants to test the claim that the population proportion is greater than $0.45$. The sample proportion is $0.50$, based on a sample of size $50$. Using a significance level of $\alpha =0.05$, should the researcher reject the null hypothesis or fail to reject it? Find the critical $z$-value and determine the conclusion of the hypothesis test.

A company claims that at least $60\%$ of its customers are satisfied with its product. To test this claim, a sample of $80$ customers is surveyed, and it is found that $55\%$ of the sample reports being satisfied. Using a significance level of $\alpha =0.05$, test the claim by calculating the critical $z$-value, and determine whether to reject or fail to reject the null hypothesis.

A researcher claims that more than $40\%$ of adults prefer drinking coffee over tea. To test this claim, a random sample of $100$ adults is surveyed, and $47\%$ report preferring coffee. Using a significance level of $\alpha =0.01$, determine the critical value $z_{\text{crit}}$, and decide whether to reject or fail to reject the null hypothesis.

A survey claims that over $60\%$ of college students prefer digital textbooks to printed ones. A hypothesis test is conducted, and results in a $p$-value of $0.002$ at a significance level of $\alpha =0.05$. Based on this information, which of the following is the correct conclusion?

A vaccine is tested on $950$ volunteers, and $38$ of them report mild fever as a side effect. The manufacturer claims that at least $7$ percent of vaccinated individuals experience mild fever. Use a $0.05$ significance level to test the claim. Answer the following:

a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?

In a poll of $980$ randomly chosen employment discrimination cases, $610$ were dismissed before reaching trial. At the $0.01$ significance level, test the claim that more than half of employment discrimination cases are dismissed before trial. What are the implications of this finding for employers?

A recent survey of $4200$ Canadian adults found that $630$ do not own a smartphone. In $2012$, $35\%$ of Canadian adults did not own a smartphone. At the $0.05$ significance level, test the claim that the current percentage of Canadian adults who do not own a smartphone is now less than $35\%$. If a difference is found, is it substantial?

A hospital in Texas records $512$ girls out of $1020$ births in a given year. At the $0.05$ significance level, test the claim that $50\%$ of newborns are girls. Do the results support the belief that $50\%$ of newborns are girls? 

A researcher claims that $40\%$ of adults exercise at least three times a week. In a survey of $500$ adults, $215$ said they exercise at least three times a week. At the $0.01$ significance level, test the researcher's claim.

A survey of $1800$ college students found that $38\%$ said they prefer online classes over in-person classes. Test the claim, at the $0.01$ significance level, that fewer than half of college students prefer online classes.

A researcher claims that the proportion of defective items produced by a machine is $0.12$. In a random sample of $20$ items, $5$ are found to be defective. Using the exact binomial method, what is the P-value for testing the alternative hypothesis $H_a:p>0.12$? Use a significance level of $0.05$.

To test the claim that the proportion of twos is $0.1$ at the $0.05$ significance level using the P-value method, consider this scenario: in a survey of $800$ randomly chosen credit card numbers, $68$ of the last digits were twos, while if the digits were truly random, the proportion of twos should be $0.1$.