Q: A manufacturer claims the mean diameter of bolts is 30mm. You collect a sample and want to test this claim. Which is the correct formulation of the null and alternative hypotheses?

$H_0: \mu = 30$, $H_1: \mu \neq 30$

Question 1

Which of the following statements correctly describes the null hypothesis ($H_0$) in hypothesis testing?

Accepted Answer

It represents the current belief and is always about a population parameter.

Question 2

A manufacturer claims the mean diameter of bolts is 30mm. You collect a sample and want to test this claim. Which is the correct formulation of the null and alternative hypotheses?

Accepted Answer

$H_0: \mu = 30$, $H_1: \mu 
eq 30$

Question 3

In hypothesis testing, what does a Type I error represent?

Accepted Answer

Rejecting a true null hypothesis.

Question 4

If the probability of a Type I error ($\alpha$) increases, what happens to the probability of a Type II error ($\beta$), all else equal?

Accepted Answer

$\beta$ decreases

Question 5

Which formula is used to calculate the Z test statistic for a one-sample mean test when the population standard deviation is known?

Accepted Answer

$Z_{STAT} = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$

Question 6

A sample of 100 bolts has a mean diameter of 29.84mm. The population standard deviation is 0.8mm. Test the claim that the true mean is 30mm at $\alpha = 0.05$. What is the value of the test statistic?

Accepted Answer

$Z_{STAT} = -2.0$

Question 7

In a two-tailed Z test with $\alpha = 0.05$, what are the critical values for the test statistic?

Accepted Answer

$\pm 1.96$

Question 8

If the calculated test statistic falls in the non-rejection region, what is the correct decision?

Accepted Answer

Do not reject the null hypothesis.

Question 9

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

Accepted Answer

The probability of obtaining a test statistic as extreme or more extreme than the observed value, given $H_0$ is true.

Question 10

A researcher performs a hypothesis test and obtains a p-value of 0.0456 with $\alpha = 0.05$. What should the researcher conclude?

Accepted Answer

Reject $H_0$ because the p-value is less than $\alpha$.

Question 11

Which of the following factors will decrease the probability of a Type II error ($\beta$), all else equal?

Accepted Answer

Increasing the level of significance ($\alpha$)

Question 12

In the context of hypothesis testing, what does the 'power' of a statistical test refer to?

Accepted Answer

The probability of rejecting $H_0$ when it is false.

Question 13

A company wants to test if the mean monthly cell phone bill in a city is $\mu = $42. Which of the following is a correct alternative hypothesis?

Accepted Answer

$H_1: \mu 
eq 42$

Question 14

Which step in the critical value approach to hypothesis testing involves determining the critical values that divide the rejection and nonrejection regions?

Accepted Answer

Step 4

Question 15

In a two-tailed hypothesis test, what is the rejection region for $\alpha = 0.05$?

Accepted Answer

$Z < -1.96$ or $Z > 1.96$

Question 16

Which of the following is NOT a correct property of the alternative hypothesis ($H_1$)?

Accepted Answer

It is always about a sample statistic.

Question 17

A sample mean is very close to the population mean stated in $H_0$. What is the most appropriate decision?

Accepted Answer

Do not reject $H_0$

Question 18

Which of the following is the correct interpretation of the confidence coefficient ($1-\alpha$)?

Accepted Answer

The probability of not rejecting $H_0$ when it is true.

Question 19

Which of the following is a correct step in the p-value approach to hypothesis testing?

Accepted Answer

Compute the p-value after collecting the sample data.

Fundamentals of Hypothesis Testing: One-Sample Tests (Chapter 9 Study Notes)

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