Question 1

Which of the following is a key assumption when conducting a hypothesis test for a population standard deviation using the chi-square distribution?

Accepted Answer

The population must be normally distributed.

Question 2

A quality-control engineer wants to test if the standard deviation of the weights of a product is less than 0.75 grams after recalibration. Which of the following is the correct set of hypotheses for this left-tailed test?

Accepted Answer

$$H_0$$: \sigma = 0.75$$ grams, H_1$: \sigma \u003c 0.75$$ grams

Question 3

Given a sample of size $$n$$ from a normally distributed population, which formula is used to calculate the test statistic for a hypothesis test about the population variance?

Accepted Answer

$$\$$chi^2$$ = \dfrac{(n-1)s^2$}{\$$sigma^2$$}$$

Question 4

Which of the following statements about the chi-square distribution is TRUE?

Accepted Answer

As the degrees of freedom increase, the distribution becomes more symmetric.

Question 5

A researcher collects a simple random sample of $$n=15$$ from a normally distributed population and calculates a sample variance $$s^2. $$What are the degrees of freedom for the corresponding chi-square test statistic?

Accepted Answer

$$14$$

Question 6

In a right-tailed hypothesis test for a population variance, when should the null hypothesis be rejected using the classical approach?

Accepted Answer

If $$\$$chi^2$$_0 \u003e \$$chi^2$$_{\alpha}$$

Question 7

Suppose a sample of $$n=20$$ yields a test statistic $$\$$chi^2$$_0 = 35$$ for a two-tailed test at $$\alpha = 0.05. $$The critical values are $$\$$chi^2$$_{0.025,19} = 8.907$$ and $$\$$chi^2$$_{0.975,19} = 32.852. $$What is the correct decision?

Accepted Answer

Reject the null hypothesis because $$\$$chi^2$$_0 \u003e \$$chi^2$$_{0.025,19}.$$

Question 8

A medical device manufacturer wants to ensure the consistency of a new blood pressure monitor. They test $$n=10$$ devices and find $$s=2.1$$ mmHg. If the claimed population standard deviation is $$\$$sigma_0$$=2.5$$ mmHg, what is the value of the test statistic?

Accepted Answer

$$\$$chi^2$$_0 = \dfrac{(10-1)(2.1)^2$}{(2.5)^2$} = 6.3648$$

Question 9

Which of the following would invalidate the use of the chi-square test for a population variance?

Accepted Answer

The population is not normally distributed.

Question 10

In the context of hypothesis testing for a population standard deviation, what does the P-value represent?

Accepted Answer

The probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

Question 11

A researcher is testing $$H_0$$: \sigma = 5$$ versus H_1$: \sigma \u003e 5 at$$ $$\alpha = 0.01. $$The sample size is $$n=12$$ and the sample standard deviation is $$s=6. $$What is the test statistic?

Accepted Answer

$$\$$chi^2$$_0 = \dfrac{(12-1)(6)^2$}{(5)^2$} = 15.84$$

Question 12

If the P-value for a left-tailed test is 0.12 and the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Do not reject the null hypothesis.

Question 13

Which of the following best describes the effect of increasing the degrees of freedom on the shape of the chi-square distribution?

Accepted Answer

The distribution becomes more symmetric.

Question 14

A company claims the variance of their product's weight is $$0.25$$ grams$$^2. A $$sample of $$n=9$$ products yields $$s^2 = 0.36$ grams^2$. What is the value of the test statistic?

Accepted Answer

$$\$$chi^2$$_0 = \dfrac{(9-1)(0.36)}{0.25} = 11.52$$

Question 15

In a hypothesis test for a population variance, which of the following is NOT a possible structure for the alternative hypothesis?

Accepted Answer

$$H_1$$: \sigma = \$$sigma_0$$

Question 16

A left-tailed test for a population variance is being conducted. The test statistic is $$\$$chi^2$$_0 = 2.5$$ and the critical value is $$\$$chi^2$$_{1-\alpha, n-1} = 3.0. $$What is the correct decision?

Accepted Answer

Reject the null hypothesis.

Question 17

Which of the following real-world scenarios is most appropriate for a chi-square test of a population standard deviation?

Accepted Answer

A pharmaceutical company wants to test if the variability in the amount of active ingredient in pills is less than a specified value after a new manufacturing process.

Question 18

If a sample variance is much larger than the hypothesized population variance, what will happen to the value of the chi-square test statistic?

Accepted Answer

It will be much greater than 1.

Question 19

A researcher uses the P-value approach for a right-tailed test and obtains a P-value of 0.03 with $$\alpha = 0.05. $$What should the researcher conclude?

Accepted Answer

Reject the null hypothesis.

Hypothesis Tests for a Population Standard Deviation

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