Question 1

Suppose you are testing the null hypothesis $$H_0$$: \mu = 10$$ versus the alternative H_1$: \mu 
eq 10. If $$the test statistic is $$t = -2.54$$ and the critical value at $$\alpha = 0.05 is$$ $$\pm 1.73$$, what is the correct statistical decision?

Accepted Answer

Reject $$H_0$$ because $$|t| \u003e 1.73$$

Question 2

In hypothesis testing, what does the significance level $$\alpha$$ represent?

Accepted Answer

The probability of making a Type I error

Question 3

A researcher wants to compare the means of two independent samples. Which test statistic should be used if the population variances are unknown but assumed equal?

Accepted Answer

Pooled t-test statistic

Question 4

Given a sample mean $$\bar{x} = 12$$, population mean $$\mu = 10$$, sample standard deviation $$s = 2$$, and sample size $$n = 16$$, what is the value of the t-statistic?

Accepted Answer

$$t = \frac{12 - 10}{2/\sqrt{16}}$$

Question 5

Which of the following best describes a Type II error in hypothesis testing?

Accepted Answer

Failing to reject the null hypothesis when it is false

Question 6

If the p-value for a test is 0.03 and the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 7

A medical researcher tests whether a new drug changes the average recovery time. The null hypothesis is $$H_0$$: \mu = 7$$ days. The sample mean is $$\bar{x} = 6.2$$, s = 1.5$, n = 25$. Calculate the t-statistic.

Accepted Answer

$$t = \frac{6.2 - 7}{1.5/\sqrt{25}}$$

Question 8

Which formula is used to calculate the standard error of the mean for a sample?

Accepted Answer

$$SE = \frac{s}{\sqrt{n}}$$

Question 9

In a two-sample t-test, which assumption is necessary for using the pooled variance formula?

Accepted Answer

The population variances are equal

Question 10

A scientist wants to test if the mean of a population is greater than 50. Which is the correct alternative hypothesis?

Accepted Answer

$$H_1$$: \mu \u003e 50$$

Question 11

If the calculated t-statistic is 1.2 and the critical value is 2.0 for a two-tailed test at $$\alpha = 0.05$$, what is the correct decision?

Accepted Answer

Fail to reject the null hypothesis

Question 12

Which of the following is NOT an assumption of the one-sample t-test?

Accepted Answer

The population variance is known

Question 13

A company claims their batteries last 100 hours on average. A sample of 20 batteries has a mean of 98 hours and a standard deviation of 5 hours. What is the value of the t-statistic?

Accepted Answer

$$t = \frac{98 - 100}{5/\sqrt{20}}$$

Question 14

Which of the following best describes the p-value in hypothesis testing?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value under $$H_0$$

Question 15

A researcher performs a two-sample t-test and obtains a p-value of 0.08. If $$\alpha = 0.05$$, what should the researcher conclude?

Accepted Answer

Fail to reject the null hypothesis

Question 16

Which formula is used to calculate the pooled variance for two samples with sizes $$n_1$$ and $$n_2$$, and variances $$s_1^2$ and s_2$^2$$?

Accepted Answer

$$s_p^2$ = \frac{(n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2}{n_1$ + n_2$ - 2}$$

Hypothesis Testing and Inference: Practice Problems in Statistics

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