Question 1

A study compares the effectiveness of a nicotine patch versus a placebo in helping people quit smoking. Out of 20 subjects using the placebo, 11 quit, and out of 23 using the patch, 17 quit. At $$\alpha = 0.05$$, what is the correct null hypothesis for testing if the proportions are different?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$

Question 2

Given the same nicotine patch study, what is the formula for the test statistic to compare the two proportions?

Accepted Answer

$$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1$} + \frac{1}{n_2$}\right)}}$$

Question 3

In a two-proportion z-test, what does it mean if the p-value is less than $$\alpha$$?

Accepted Answer

Reject $$H_0$$; there is enough evidence to suggest a difference in proportions.

Question 4

A 90% confidence interval for the difference in proportions between two groups is $$(-0.13, 0.05). $$What does this interval suggest?

Accepted Answer

There is not enough evidence to conclude a difference in proportions.

Question 5

Which formula is used to calculate the margin of error for a confidence interval for the difference in proportions?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 6

A company rolls a fair 6-sided die 60 times and records the observed frequencies. Which test should be used to determine if the die is fair?

Accepted Answer

Chi-square goodness of fit test

Question 7

What is the formula for the chi-square test statistic in a goodness of fit test?

Accepted Answer

$$\$$chi^2$$ = \sum \frac{(O - E)^2$}{E}$$

Question 8

If the p-value from a chi-square test is 0.047 and $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$; there is enough evidence to conclude the observed frequencies do not match the claimed distribution.

Question 9

A gym owner wants to test if the number of members is equal across all age groups. What are the degrees of freedom for the chi-square test if there are 5 age groups?

Accepted Answer

$$df = 4$$

Question 10

Which of the following is a requirement for using the chi-square goodness of fit test?

Accepted Answer

Each expected frequency must be at least 5.

Question 11

A study tests if the proportion of car ownership is the same for different age groups using a chi-square statistic. What type of test is this?

Accepted Answer

Test of homogeneity

Question 12

In a hypothesis test for two proportions, which of the following is a pooled proportion formula?

Accepted Answer

$$\hat{p} = \frac{x_1$ + x_2$}{n_1$ + n_2$}$$

Question 13

A company surveys 200 customers about their preferred flavor of a new drink (4 flavors). What is the expected frequency for each flavor if preferences are equally distributed?

Accepted Answer

50

Question 14

Which of the following statements is true regarding confidence intervals for the difference in proportions?

Accepted Answer

If the interval includes 0, we fail to reject $$H_0.$$

Question 15

A study on seatbelt effectiveness finds 50 injuries out of 2400 people not wearing seatbelts and 15 injuries out of 1600 people wearing seatbelts. What is the alternative hypothesis to test if not wearing a seatbelt results in a greater proportion of injuries?

Accepted Answer

$$H_a: p_1$ \u003e p_2$

Question 16

Which calculator function is used to construct a confidence interval for the difference between two proportions?

Accepted Answer

2-PropZInt

Question 17

If a p-value is 0.44 and $$\alpha = 0.05 in a $$goodness of fit test, what is the correct conclusion?

Accepted Answer

Fail to reject $$H_0$$; there is not enough evidence to conclude the observed frequencies do not match the claimed distribution.

Inference for Proportions and Goodness-of-Fit Tests in Business Statistics

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