Q: A researcher wants to test whether the proportion of customers who prefer online shopping is different from 60%. Which of the following is the correct null and alternative hypothesis for a two-sided test?

$$H_0$$: p = 0.6$$ \\ H_A: p \neq 0.6$

Q: A medical researcher tests $$H_0$$: p = 0.2$$ against H_A$$: p \u003e 0.2$$ using a sample of 400 patients. The sample proportion is $$\hat{p} = 0.25$$. Calculate the z-statistic for this test.

$$z = \frac{0.25 - 0.2}{\sqrt{\frac{0.2 \times 0.8}{400}}} = 2.5$$

Q: A store manager samples 200 purchases and finds 140 are under $$10. Test H_0$: p = 2/3 vs$$ $$H_A: p \u003e 2/3$. What is the z-statistic?

$$z = \frac{0.7 - 0.6667}{\sqrt{\frac{0.6667 \times 0.3333}{200}}} \approx 0.46$$

Question 1

A researcher wants to test whether the proportion of customers who prefer online shopping is different from 60%. Which of the following is the correct null and alternative hypothesis for a two-sided test?

Accepted Answer

$$H_0$$: p = 0.6$$ \ H_A: p 
eq 0.6$

Question 2

In a hypothesis test for a proportion, which formula is used to calculate the standard deviation of the sampling distribution under the null hypothesis?

Accepted Answer

$$SD(\hat{p}) = \sqrt{\frac{p_0$(1-p_0$)}{n}}$$

Question 3

A company claims that 70% of its deliveries arrive on time. In a random sample of 150 deliveries, 99 were on time. What is the value of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{99}{150} = 0.66$$

Question 4

Which of the following best describes a Type I error in the context of hypothesis testing?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 5

A medical researcher tests $$H_0$$: p = 0.2$$ against H_A$$: p \u003e 0.2$$ using a sample of 400 patients. The sample proportion is $$\hat{p} = 0.25$$. Calculate the z-statistic for this test.

Accepted Answer

$$z = \frac{0.25 - 0.2}{\sqrt{\frac{0.2 	imes 0.8}{400}}} = 2.5$$

Question 6

Which of the following statements about P-values is TRUE?

Accepted Answer

A small P-value indicates that the observed data would be unlikely if the null hypothesis were true.

Question 7

A store manager samples 200 purchases and finds 140 are under $$10. Test H_0$: p = 2/3 vs$$ $$H_A: p \u003e 2/3$. What is the z-statistic?

Accepted Answer

$$z = \frac{0.7 - 0.6667}{\sqrt{\frac{0.6667 	imes 0.3333}{200}}} \approx 0.46$$

Question 8

If the significance level $$\alpha is $$decreased from 0.05 to 0.01, what is the effect on Type I and Type II errors?

Accepted Answer

Type I error probability decreases, but Type II error probability increases.

Question 9

Which of the following actions will increase the power of a hypothesis test?

Accepted Answer

Increasing the sample size.

Question 10

A 95% confidence interval for a proportion is (0.6365, 0.7635). Is there evidence at the 5% significance level that the true proportion is greater than 2/3?

Accepted Answer

No, because 2/3 (0.6667) is contained within the confidence interval.

Question 11

Which of the following best describes the power of a statistical test?

Accepted Answer

The probability of correctly rejecting a false null hypothesis.

Question 12

A jury trial is used as an analogy for hypothesis testing. In this analogy, what does 'failing to reject the null hypothesis' correspond to?

Accepted Answer

Declaring the defendant 'not guilty'.

Question 13

Which of the following is NOT a condition that must be checked before performing a one-proportion z-test?

Accepted Answer

The population standard deviation is known.

Question 14

Suppose you conduct a hypothesis test and obtain a P-value of 0.03. If your significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 15

Which of the following is a correct interpretation of a 95% confidence interval for a population proportion?

Accepted Answer

We are 95% confident that the true population proportion lies within the interval.

Question 16

Which of the following is a reason to report the effect size along with the P-value in hypothesis testing?

Accepted Answer

To distinguish between statistical significance and practical importance.

Question 17

If the sample size is quadrupled, what happens to the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

It is halved.

Question 18

Which of the following is true about the relationship between confidence intervals and two-sided hypothesis tests?

Accepted Answer

A confidence interval with confidence level $$C\%$$ corresponds to a two-sided hypothesis test with $$\alpha = 1 - \frac{C}{100}.$$

Testing Hypotheses about Proportions: A Comprehensive Study Guide

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