Q: In a $$\$$chi^2$$ test for independence, which formula is used to calculate the expected count for a cell in a contingency table?

$$\text{Expected} = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$$

Question 1

A researcher wants to compare the average lifespan of a species of tree from two different locations. She collects two independent samples: 35 trees from Location 1 and 40 trees from Location 2. Which statistical test is most appropriate to determine if there is a significant difference in mean lifespan between the two locations?

Accepted Answer

A two-sample t-test for independent means

Question 2

Given the following sample means and standard deviations for tree lifespans: Location 1: $$\bar{x}_1 = 42.0$$, $$s_1 = 5.8$, n_1$ = 35$$; Location 2: $$\bar{x}_2 = 45.1$$, $$s_2 = 6.2$, n_2$ = 40. $$What is the formula for the test statistic to compare the means?

Accepted Answer

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}}$$

Question 3

In a study of nutrition in children aged 5-10, only 16 out of 61 children met the recommended intake of daily fruits and vegetables. Which formula would you use to construct a 95% confidence interval for the population proportion?

Accepted Answer

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

Question 4

A sample of 61 children is surveyed, and 16 meet the recommended intake of fruits and vegetables. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{16}{61}$$

Question 5

A survey of adults asks for their preferred social media platform, sorted by age. Which statistical test is appropriate to determine if there is an association between age group and preferred platform?

Accepted Answer

Chi-square test for independence

Question 6

In a $$\$$chi^2$$ test for independence, which formula is used to calculate the expected count for a cell in a contingency table?

Accepted Answer

$$	ext{Expected} = \frac{	ext{Row Total} 	imes 	ext{Column Total}}{	ext{Grand Total}}$$

Question 7

An eight-sided die is rolled 80 times, and the observed counts for each outcome are given. Which test should be used to determine if the die is fair?

Accepted Answer

Chi-square goodness-of-fit test

Question 8

For a chi-square goodness-of-fit test with $$k$$ categories and $$n$$ total observations, what is the formula for the test statistic?

Accepted Answer

$$\$$chi^2$$ = \sum_{i=1}^{k} \frac{(O_i - E_i)^2$}{E_i}$$

Question 9

A researcher performs a two-sample t-test and obtains a p-value of $$2.18 	imes 10$$^{-5}$$. What is the correct interpretation if $$\alpha = 0.05$$?

Accepted Answer

There is strong evidence to reject the null hypothesis

Question 10

In a chi-square test for independence, the null hypothesis states:

Accepted Answer

The two categorical variables are independent

Question 11

A confidence interval for a population proportion is calculated as $$0.262 \pm 0.116. $$What is the correct interpretation?

Accepted Answer

We are 95% confident that the true proportion is between 0.146 and 0.378

Question 12

If the p-value for a chi-square test for independence is greater than $$\alpha$$, what should you conclude?

Accepted Answer

Fail to reject the null hypothesis; the variables are independent

Question 13

A researcher rolls an eight-sided die 80 times and observes the following counts: 8, 12, 11, 10, 9, 13, 7, 10. What is the expected count for each outcome if the die is fair?

Accepted Answer

Expected count = $$\frac{80}{8} = 10$$

Question 14

Which of the following is NOT an assumption for the two-sample t-test for independent means?

Accepted Answer

The samples are paired

Question 15

A chi-square test for independence is performed on survey data about age and preferred social media platform. The p-value is 0.001. What does this indicate?

Accepted Answer

There is a significant association between age and preferred platform

Question 16

In a two-sample t-test, the null hypothesis is:

Accepted Answer

$$H_0$$: \$$mu_1$$ = \$$mu_2$$

Question 17

A researcher wants to test if the proportion of children meeting recommended fruit and vegetable intake is less than 0.5. Which is the correct null and alternative hypothesis?

Accepted Answer

$$H_0$$: p = 0.5$$, H_a$$: p \u003c 0.5$$

Question 18

Which of the following real-world scenarios is best analyzed using a chi-square goodness-of-fit test?

Accepted Answer

Testing if a die is fair by comparing observed and expected frequencies

Exam 1 Practice Problems: Key Concepts in Inferential Statistics

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