Q: A car battery manufacturer claims a standard deviation of 1 year. A sample of 5 batteries has a sample variance of 0.815. What is the value of the chi-squared test statistic?

$$\$$chi^2$$ = \frac{4 \times 0.815}{1}$$

Question 1

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

As the sample size increases, the distribution of the sample mean approaches a normal distribution, regardless of the population's distribution, provided the population has finite variance.

Question 2

Suppose you roll a fair six-sided die $$n$$ times and sum the results. As $$n$$ increases, what happens to the shape of the probability distribution of the sum?

Accepted Answer

It becomes more bell-shaped and approaches a normal distribution.

Question 3

Given a random sample $$X_1$$, $$X_2$$, ..., X_n$$ from a population with mean $$\mu$$ and variance $$\$$sigma^2$$, what is the formula for the sample variance $$S^2$$?

Accepted Answer

$$S^2$$ = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X}$$)^2$$

Question 4

A machine fills soft-drink cans with a mean of 12.1 fl oz and a standard deviation of 0.05 fl oz. If a random sample of 10 cans is selected, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{0.05}{\sqrt{10}}$$

Question 5

Which distribution should be used to construct a confidence interval for the mean when the population variance is unknown and the sample size is 15?

Accepted Answer

Student-$$t$$ distribution with $$n-1$$ degrees of freedom

Question 6

Given two independent samples with means $$\overline{X}_1$$ and $$\overline{X}_2$$, population variances $$\$$sigma_1$$^2$$ and $$\$$sigma_2$$^2$$, and sample sizes $$n_1$$ and $$n_2$$, what is the variance of the sampling distribution of $$\overline{X}_1 - \overline{X}_2$$?

Accepted Answer

$$\frac{\$$sigma_1$$^2}{n_1$} + \frac{\$$sigma_2$$^2}{n_2$}$$

Question 7

A sample of 8 cereal bars has a mean saturated fat content of 0.475 grams and a sample standard deviation of 0.183 grams. The claimed population mean is 0.5 grams. What is the value of the $$t$$ statistic for testing the claim?

Accepted Answer

$$t = \frac{0.475 - 0.5}{0.183/\sqrt{8}}$$

Question 8

Which of the following is the correct formula for the chi-squared statistic used to test the variance of a normal population?

Accepted Answer

$$\$$chi^2$$ = \frac{(n-1)S^2$}{\$$sigma^2$$}$$

Question 9

A car battery manufacturer claims a standard deviation of 1 year. A sample of 5 batteries has a sample variance of 0.815. What is the value of the chi-squared test statistic?

Accepted Answer

$$\$$chi^2$$ = \frac{4 	imes 0.815}{1}$$

Question 10

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

Accepted Answer

F-distribution

Question 11

Suppose you have two independent samples with sample variances $$S_1^2 = 15750$ and S_2$^2 = 10920$$ and sample sizes $$n_1 = 5$ and n_2$ = 6. $$What is the value of the F statistic for testing equality of variances?

Accepted Answer

$$F = \frac{15750}{10920}$$

Question 12

Which of the following statements about the t-distribution is TRUE?

Accepted Answer

As the sample size increases, the t-distribution approaches the normal distribution.

Question 13

A machine produces resistors with mean resistance $$\mu = 40$$ ohms and $$\sigma = 2$$ ohms. For a sample of $$n = 36$$, what is the probability that the sample mean exceeds 40.5 ohms?

Accepted Answer

0.0668

Question 14

Which of the following is NOT a property of the F-distribution?

Accepted Answer

It is symmetric about zero.

Question 15

If $$n \geq 30$$, which distribution is generally appropriate for the sample mean, regardless of the population's shape?

Accepted Answer

Normal distribution

Question 16

Which of the following best describes a sampling distribution?

Accepted Answer

The probability distribution of a statistic based on all possible samples from a population.

Fundamental Sampling Distributions and Their Applications in Introductory Statistics

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