Q: A medical researcher wants a margin of error less than $$0.03$$ for a 95% confidence interval for a population proportion. What is the minimum sample size needed if the estimated proportion is $$0.5$$? (Use $$z^* = 1.96.)$$

$$n = \left( \dfrac{1.96}{0.03} \$$right)^2$$ \times 0.5 \times 0.5 = 1067$$

Q: How do you calculate the expected count for a cell in a contingency table?

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

Question 1

A researcher wants to estimate the proportion of university students who prefer online classes. She surveys 200 students and finds that 120 prefer online classes. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \dfrac{120}{200} = 0.60$$

Question 2

Which of the following best describes the sampling distribution of the sample proportion $$\hat{p}$$ when the sample size is large?

Accepted Answer

It is approximately normal with mean $$p$$ and standard deviation $$\sqrt{\dfrac{p(1-p)}{n}}.$$

Question 3

A 95% confidence interval for a population proportion is calculated as $$0.55 \pm 0.07. $$What is the margin of error?

Accepted Answer

$$0.07$$

Question 4

Which formula is used to calculate the margin of error for a 1-proportion $$z-$$interval?

Accepted Answer

$$z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 5

A medical researcher wants a margin of error less than $$0.03$$ for a 95% confidence interval for a population proportion. What is the minimum sample size needed if the estimated proportion is $$0.5$$? (Use $$z^* = 1.96.)$$

Accepted Answer

$$n = \left( \dfrac{1.96}{0.03} \$$right)^2$$ 	imes 0.5 	imes 0.5 = 1067$$

Question 6

In hypothesis testing for one population proportion, what is the null hypothesis $$H_0$$ typically stated as?

Accepted Answer

$$H_0$$: p = $$p_0$$

Question 7

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

Accepted Answer

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

Question 8

Which of the following statements about the chi-square distribution is TRUE?

Accepted Answer

It is right-skewed and its shape depends on the degrees of freedom.

Question 9

A chi-square goodness of fit test is used to:

Accepted Answer

Determine if observed categorical data fit a specified distribution.

Question 10

Given observed counts $$O_i$$ and expected counts $$E_i$$, what is the formula for the chi-square test statistic?

Accepted Answer

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

Question 11

A researcher performs a chi-square goodness of fit test with 4 categories. What are the degrees of freedom for this test?

Accepted Answer

$$df = 3$$

Question 12

In a chi-square test for independence, a contingency table has 3 rows and 4 columns. What is the degrees of freedom?

Accepted Answer

$$df = (3-1)(4-1) = 6$$

Question 13

How do you calculate the expected count for a cell in a contingency table?

Accepted Answer

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

Question 14

Which of the following is a correct interpretation of a p-value in the context of a chi-square test?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

Question 15

A hospital wants to test if the distribution of blood types among its patients matches the national distribution. Which statistical test should be used?

Accepted Answer

Chi-square goodness of fit test

Question 16

A study investigates whether there is an association between exercise frequency (low, medium, high) and cholesterol level (normal, high) in adults. Which test is appropriate?

Accepted Answer

Chi-square test for independence

Question 17

If the calculated chi-square statistic is greater than the critical value from the chi-square table at a given significance level, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 18

Which of the following is NOT an assumption for using the chi-square test for independence?

Accepted Answer

The data are paired measurements.

Question 19

A researcher calculates a p-value of 0.03 in a chi-square test for independence at $$\alpha = 0.05. $$What should the researcher conclude?

Accepted Answer

There is sufficient evidence to suggest an association between the variables.

Statistical Inference for Proportions and Chi-Square Procedures

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