Q: If a sample of size $$n = 15$$ yields a sample variance $$s^2 = 4.0$, and the chi-square critical values for a 95% confidence interval are 6.571$ and 27.488$, what is the 95% confidence interval for the population variance?

$$\left( \dfrac{14 \times 4.0}{27.488}, \dfrac{14 \times 4.0}{6.571} \right)$$

Question 1

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

Accepted Answer

The chi-square distribution is positively skewed and becomes more symmetric as degrees of freedom increase.

Question 2

Suppose you want to construct a 95% confidence interval for the variance of a normally distributed population. Which distribution should you use?

Accepted Answer

Chi-square distribution

Question 3

Given a sample size of $$n = 25$$, what are the degrees of freedom used when constructing a confidence interval for the population variance?

Accepted Answer

$$n - 1$$

Question 4

Which of the following is the correct formula for the confidence interval for the population variance $$\$$sigma^2$$ using the chi-square distribution?

Accepted Answer

$$\left( \dfrac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}, \dfrac{(n-1)s^2$}{\$$chi^2$$_{1-\alpha/2}} \right)$$

Question 5

A researcher collects a sample of $$n = 18$$ and wants to find the critical values for a 95% confidence interval for the variance. What degrees of freedom should be used to look up the chi-square critical values?

Accepted Answer

17

Question 6

If the sample standard deviation is $$s = 2.5$$ and the sample size is $$n = 10$$, what is the point estimate for the population variance?

Accepted Answer

$$2.5^2 = 6.25$

Question 7

Which of the following best describes the effect of increasing the sample size on the chi-square distribution?

Accepted Answer

The distribution becomes more symmetric and approaches a normal distribution.

Question 8

A pharmaceutical company weighs 30 samples of a drug and finds a sample standard deviation of $$1.20 mg. $$Assuming normality, which formula would you use to construct a 99% confidence interval for the population standard deviation?

Accepted Answer

$$\left( \sqrt{\dfrac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}}, \sqrt{\dfrac{(n-1)s^2$}{\$$chi^2$$_{1-\alpha/2}}} \right)$$

Question 9

Which of the following is NOT a property of the chi-square distribution?

Accepted Answer

The chi-square distribution is symmetric for all degrees of freedom.

Question 10

A 99% confidence interval for the variance of a normally distributed population is calculated as $$(0.80, 3.18). $$What does this mean in context?

Accepted Answer

If the estimation process is repeated many times, 99% of the intervals constructed will contain the true population variance.

Question 11

Why are there two critical values when constructing a confidence interval for variance using the chi-square distribution?

Accepted Answer

Because the confidence interval is based on both the left and right tails of the distribution.

Question 12

Which of the following real-world scenarios would be most appropriate for constructing a confidence interval for a population variance?

Accepted Answer

Estimating the variability in the amount of active ingredient in allergy medicine tablets.

Question 13

If a sample of size $$n = 15$$ yields a sample variance $$s^2 = 4.0$, and the chi-square critical values for a 95% confidence interval are 6.571$ and 27.488$, what is the 95% confidence interval for the population variance?

Accepted Answer

$$\left( \dfrac{14 	imes 4.0}{27.488}, \dfrac{14 	imes 4.0}{6.571} ight)$$

Question 14

Which of the following best explains why the chi-square distribution is used for variance and not for means?

Accepted Answer

Because the distribution of $$\dfrac{(n-1)s^2$}{\$$sigma^2$$}$$ follows a chi-square distribution under normality.

Question 15

A researcher wants to estimate the standard deviation of a population with 95% confidence. Which of the following steps is NOT required?

Accepted Answer

Use the formula for the confidence interval for the mean.

Confidence Intervals for Variance and Standard Deviation (Chapter 6.4 Study Notes)

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