Q: A botanist measures the nitrogen content in 15 leaves and wants to test if the mean is different from 3.0%. Which hypotheses should be used?

$$ H_0$: \mu = 3.0, \quad H_a: \mu \neq 3.0 $$

Question 1

Which of the following scenarios is most appropriate for a one-sample t-test for the mean?

Accepted Answer

Testing whether the mean caffeine content of a new energy drink is 80 mg per can.

Question 2

What is the formula for the test statistic in a one-sample t-test for the mean?

Accepted Answer

$$ t = \frac{\bar{x} - \mu_0$}{s / \sqrt{n}} $$

Question 3

Which assumption is NOT required for the one-sample t-test for the mean?

Accepted Answer

The population variance is known.

Question 4

A botanist measures the nitrogen content in 15 leaves and wants to test if the mean is different from 3.0%. Which hypotheses should be used?

Accepted Answer

$$ H_0$: \mu = 3.0, \quad H_a: \mu 
eq 3.0 $$

Question 5

Which formula gives the confidence interval for the population mean in a one-sample t-test?

Accepted Answer

$$ \bar{x} \pm t^* \frac{s}{\sqrt{n}} $$

Question 6

Which test is used to compare the variances of two independent samples?

Accepted Answer

F-test

Question 7

What is the formula for the F-test statistic when comparing two sample variances?

Accepted Answer

$$ F = \frac{s_1$^2}{s_2$^2} $$

Question 8

Which of the following is TRUE about Welch's t-test?

Accepted Answer

It is used when variances are unequal between groups.

Question 9

A researcher wants to compare the effectiveness of two advertising campaigns using independent samples. Which test should be used if the variances are unequal?

Accepted Answer

Welch's (nonpooled) t-test

Question 10

What is the formula for the test statistic in a paired samples (matched-pairs) t-test?

Accepted Answer

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

Question 11

Which assumption is required for the paired samples t-test?

Accepted Answer

Differences between pairs are approximately normally distributed.

Question 12

In a manufacturing study, the sample variance of spring length is $$0.06 cm$^2$$ with $$n = 24$$ springs. What is the test statistic for the chi-square test for variance if the hypothesized variance is $$0.04 cm$^2$$?

Accepted Answer

$$ \$$chi^2$$ = \frac{(n-1)s^2$}{\$$sigma_0$$^2} = \frac{23 	imes 0.06}{0.04} $$

Question 13

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

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 14

A medical researcher tests blood pressure before and after treatment in the same subjects. Which test should be used?

Accepted Answer

Paired samples t-test

Question 15

Which formula gives the confidence interval for the ratio of two population variances using the F-test?

Accepted Answer

$$ \left( \frac{s_1$^2}{s_2$^2} \cdot \frac{1}{F_{1-\alpha/2, n_1$-1, n_2$-1}}, \frac{s_1$^2}{s_2$^2} \cdot \frac{1}{F_{\alpha/2, n_1$-1, n_2$-1}} \right) $$

Question 16

Which of the following is a key difference between Student's (pooled) t-test and Welch's t-test?

Accepted Answer

Student's t-test assumes equal variances, while Welch's t-test does not.

Question 17

In a paired samples t-test, what does $$\bar{d}$$ represent?

Accepted Answer

The mean of the differences between paired observations.

Question 18

Which test would you use to determine if the variance of a manufacturing process is greater than a specified value?

Accepted Answer

Chi-square test for variance

Question 19

A scientist wants to compare the mean enzyme activity between two species, but suspects the variances are not equal. Which test should be used?

Accepted Answer

Welch's (nonpooled) t-test

Question 20

Which of the following is NOT a valid assumption for the F-test for equality of variances?

Accepted Answer

Sample sizes must be equal.

Methods for Quantitative Response Variables – One and Two Groups: Statistical Tests and Applications

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