Q: Given a sample proportion $$\hat{p} = 0.4$$, sample size $$n = 250$$, and $$z_{0.025} = 1.96$$, what is the 95% confidence interval for the population proportion $$p$$?

$$0.4 \pm 1.96 \times \sqrt{\frac{0.4 \times 0.6}{250}}$$

Q: A company wants to ensure that the margin of error for estimating the proportion of defective products is less than 0.03 at a 95% confidence level. If the estimated proportion is $$\hat{p} = 0.1$$, what is the minimum sample size required? (Use $$z_{0.025} = 1.96)$$

$$n = \left( \frac{1.96^2$ \times 0.1 \times 0.9}{0.03^2$} \right)$$

Q: In a chi-square test for independence, how do you calculate the expected frequency for a cell in a contingency table?

$$E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$$

Question 1

A pharmaceutical company wants to compare the effectiveness of two different drugs in lowering blood pressure. They randomly assign 30 patients to Drug A and 30 patients to Drug B. The population variances are unknown and believed to be unequal. Which statistical test should be used to compare the mean reduction in blood pressure between the two drugs?

Accepted Answer

Non-Pooled t-Test

Question 2

A researcher wants to estimate the proportion of adults in a city who support a new public transportation initiative. She surveys 400 adults and finds that 220 support the initiative. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{220}{400} = 0.55$$

Question 3

Which of the following formulas correctly represents the margin of error for a 1-proportion z-interval at confidence level $$1-\alpha$$?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 4

A study is conducted to determine if a new teaching method improves test scores. The same group of students is tested before and after the method is applied. Which hypothesis test is most appropriate?

Accepted Answer

Paired t-Test

Question 5

A hospital administrator wants to know if the proportion of patients who recover within a week is different from 0.6. She samples 100 patients and finds 70 recover within a week. What is the null hypothesis $$H_0$$ for a one-proportion z-test?

Accepted Answer

$$H_0$$: p = 0.6$$

Question 6

Which of the following best describes the degrees of freedom for a chi-square goodness of fit test with $$k$$ categories?

Accepted Answer

$$df = k - 1$$

Question 7

A researcher wants to determine if there is an association between gender (male/female) and preference for a new product (like/dislike). Which statistical test should be used?

Accepted Answer

Chi-Square Independence Test

Question 8

Given a sample proportion $$\hat{p} = 0.4$$, sample size $$n = 250$$, and $$z_{0.025} = 1.96$$, what is the 95% confidence interval for the population proportion $$p$$?

Accepted Answer

$$0.4 \pm 1.96 	imes \sqrt{\frac{0.4 	imes 0.6}{250}}$$

Question 9

A company wants to ensure that the margin of error for estimating the proportion of defective products is less than 0.03 at a 95% confidence level. If the estimated proportion is $$\hat{p} = 0.1$$, what is the minimum sample size required? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.1 	imes 0.9}{0.03^2$} ight)$$

Question 10

In a chi-square test for independence, how do you calculate the expected frequency for a cell in a contingency table?

Accepted Answer

$$E = \frac{(	ext{row total}) 	imes (	ext{column total})}{	ext{grand total}}$$

Question 11

A nutritionist wants to test if the mean calorie intake differs between two independent groups: vegetarians and non-vegetarians. The sample sizes are small, and the population variances are assumed equal. Which test should be used?

Accepted Answer

Pooled t-Test

Question 12

Which of the following is the correct formula for the test statistic in a chi-square goodness of fit test?

Accepted Answer

$$\$$chi^2$$ = \sum \frac{(O_i - E_i)^2$}{E_i}$$

Question 13

A clinical trial compares the mean recovery times of two treatments using independent samples. The sample sizes are large, and the population variances are unknown and unequal. Which test is most appropriate?

Accepted Answer

Non-Pooled t-Test

Question 14

Which of the following statements about the sampling distribution of the sample proportion $$\hat{p} is $$correct?

Accepted Answer

It is approximately normal if $$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 15

A researcher performs a chi-square test for independence and obtains a p-value of 0.03. At a significance level of $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence of an association.

Question 16

Which of the following is NOT an assumption of the pooled t-test for two population means?

Accepted Answer

The two samples are paired observations.

Question 17

A scientist wants to test if the observed frequencies of four types of bacteria in a sample differ from the expected frequencies. Which test should be used?

Accepted Answer

Chi-Square Goodness of Fit Test

Question 18

In a paired t-test, which of the following is the correct formula for the test statistic?

Accepted Answer

$$t = \frac{\bar{d}}{s_d / \sqrt{n}}$$

Question 19

A city council wants to know if the distribution of preferred modes of transportation (car, bus, bike, walk) among residents matches the national distribution. Which hypothesis test should be used?

Accepted Answer

Chi-Square Goodness of Fit Test

Question 20

Which of the following is the correct formula for the test statistic in a one-proportion z-test?

Accepted Answer

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

Study Notes: Hypothesis Testing for Two Means, Inferences for Proportions, and Chi-Square Procedures

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