Q: A sample of 100 delinquent accounts has a mean overdue amount of $$\233.28$ and a standard deviation of $$\$$90.34. $$What is the 95% confidence interval for the mean overdue amount?

$$\ 233.28$$ \pm 1.96 \times \frac{90.34}{\sqrt{100}} = (215.57, 250.99) \$

Question 1

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

Accepted Answer

The sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution, as long as the sample size is sufficiently large.

Question 2

Given a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the standard error of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\$$ \frac{20}{\sqrt{25}} = 4 \$

Question 3

A pharmaceutical company finds that the amount of active ingredient in its tablets is normally distributed with $$\mu = 500 mg $$and $$\sigma = 5 mg. $$What is the probability that a randomly selected tablet contains less than 490 mg?

Accepted Answer

$$\ P(Z$$ \u003c -2) \approx 0.0228 \$

Question 4

Which formula would you use to construct a 95% confidence interval for the population mean when the population standard deviation $$\sigma is $$known?

Accepted Answer

$$\$$ \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \$

Question 5

A sample of 100 delinquent accounts has a mean overdue amount of $$\233.28$ and a standard deviation of $$\$$90.34. $$What is the 95% confidence interval for the mean overdue amount?

Accepted Answer

$$\ 233.28$$ \pm 1.96 	imes \frac{90.34}{\sqrt{100}} = (215.57, 250.99) \$

Question 6

If the population standard deviation $$\sigma is $$unknown and the sample size is small ($$n \u003c 30$$), which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Student's t-distribution

Question 7

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals would contain the true population mean.

Question 8

A researcher draws 1,000 samples of size $$n=5$$ from a uniform population. What happens to the distribution of sample means as $$n$$ increases to 100?

Accepted Answer

The distribution of sample means becomes more normal and less spread out.

Question 9

Which of the following is the correct formula for the standard error of the mean?

Accepted Answer

$$\$$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \$

Question 10

A sample of 40 volunteers taking a sleep aid has a sample mean sleep onset latency of 5.935 minutes. If the population mean is 15 minutes and $$\sigma = 10$$ minutes, what is the probability of observing a sample mean below 6 minutes?

Accepted Answer

$$\ P$$(\bar{X} \u003c 6) \approx 0 \$

Question 11

Which of the following is NOT a condition required for a valid large-sample confidence interval for $$\mu$$?

Accepted Answer

The population must be normally distributed.

Question 12

If you want to estimate the mean overpayment per Medicare claim using a sample of 52 claims, which formula would you use for a 95% confidence interval if $$\sigma is $$unknown?

Accepted Answer

$$\$$ \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}} \$

Question 13

Which of the following best explains why the t-distribution is used instead of the normal distribution when $$\sigma is $$unknown and $$n is $$small?

Accepted Answer

The t-distribution accounts for the extra uncertainty in estimating $$\sigma$$ with $$s.$$

Question 14

What is the effect on the width of a confidence interval for the mean if the sample size increases, all else held constant?

Accepted Answer

The confidence interval becomes narrower.

Question 15

A 99% confidence interval for a mean will be _______ than a 95% confidence interval, assuming all other factors are the same.

Accepted Answer

wider

Question 16

Which of the following is the correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we repeated the sampling process many times, about 95% of the constructed intervals would contain the true mean.

Sampling Distributions and Confidence Intervals: The Central Limit Theorem and Estimation

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