Q: A retail company collects a sample of $$n=64$$ transactions with $$\bar{x}=82.50$$ and $$s=16.00. $$What is the 95% confidence interval for the true mean transaction amount?

$$82.50 \pm 1.96 \times \frac{16.00}{8}$$

Question 1

A random sample of 35 Ontario secondary school students had a mean travel time to school of $$\bar{x} = 17$$ minutes with a sample standard deviation $$s = 9.66$$ minutes. Test at the 5% significance level whether the mean travel time is less than 20 minutes. What is the correct null and alternative hypothesis for this test?

Accepted Answer

$$H_0$$: \mu = 20$$; H_a$$: \mu \u003c 20$$

Question 2

Given the sample data from the Ontario school travel time study ($$n=35$$, $$\bar{x}=17$$, $$s=9.66$$), what is the value of the test statistic for testing $$H_0$$: \mu = 20$$ vs H_a$$: \mu \u003c 20$$?

Accepted Answer

$$t = \frac{17 - 20}{9.66/\sqrt{35}} = -1.80$$

Question 3

For the travel time hypothesis test, what is the critical value for a one-tailed t-test at the 5% significance level with 34 degrees of freedom?

Accepted Answer

$$t_{0.05, 34} = -1.690$$

Question 4

Based on the calculated test statistic $$t = -1.80$$ and the critical value $$t_{0.05, 34} = -1.690$$, what is the correct conclusion for the travel time hypothesis test?

Accepted Answer

Reject $$H_0$$; there is statistically significant evidence that the mean travel time is less than 20 minutes.

Question 5

A coffee shop chain has a population mean of $$\mu = 5.75$$ and population standard deviation $$\sigma = 1.20. $$For a random sample of $$n = 30$$ customers, what is the standard error of the sample mean?

Accepted Answer

$$SE = \frac{1.20}{\sqrt{30}} = 0.219$$

Question 6

For the coffee shop sample ($$n=30$$, $$\mu=5.75$$, $$\sigma=1.20$$), what is the z-score for a sample mean of $$5.00$$?

Accepted Answer

$$z = \frac{5.00 - 5.75}{1.20/\sqrt{30}} = -3.42$$

Question 7

Using the coffee shop data, what is the probability that the sample mean is between $$5.00$$ and $$6.00$$?

Accepted Answer

Approximately $$0.803 ($$using $$P(-3.42 \u003c Z \u003c 1.14))$$

Question 8

A retail company collects a sample of $$n=64$$ transactions with $$\bar{x}=82.50$$ and $$s=16.00. $$What is the 95% confidence interval for the true mean transaction amount?

Accepted Answer

$$82.50 \pm 1.96 	imes \frac{16.00}{8}$$

Question 9

Calculate the margin of error for the 95% confidence interval in the retail company example ($$n=64$$, $$s=16.00).$$

Accepted Answer

$$1.96 	imes 2 = 3.92$$

Question 10

Interpret the 95% confidence interval $$[78.58, 86.42]$$ for the retail company’s average transaction amount.

Accepted Answer

We are 95% confident that the true mean transaction amount is between $$78.58$$ and $$86.42.$$

Question 11

Which of the following is a correct interpretation of a p-value in the context of the travel time hypothesis test?

Accepted Answer

The probability of observing a sample mean as extreme as $$17$$ minutes or less, assuming the true mean is $$20$$ minutes.

Question 12

If the population distribution is slightly skewed, which statistical principle justifies using the normal distribution to approximate the sampling distribution of the sample mean for $$n=30$$?

Accepted Answer

Central Limit Theorem

Question 13

Suppose the retail company wants to reduce the margin of error of its confidence interval by half. By what factor should the sample size be increased?

Accepted Answer

The sample size should be quadrupled.

Question 14

Which of the following is NOT an assumption for constructing a confidence interval for the mean using the t-distribution?

Accepted Answer

The population standard deviation is known.

Question 15

A coffee shop manager claims the average amount spent per customer is at least $$6.00. Based on the population mean 5.75$ and $$\sigma=1.20$$, what is the probability that a random sample of 30$ customers has a sample mean of at least 6.00$?

Accepted Answer

Approximately $$0.127 ($$using $$P(Z \u003e 1.14))$$

Question 16

If the sample size increases while the sample standard deviation remains the same, what happens to the width of the confidence interval for the mean?

Accepted Answer

It decreases.

Question 17

Which of the following best describes the effect of increasing the confidence level from 95% to 99% on the confidence interval for the mean?

Accepted Answer

The confidence interval becomes wider.

STAT 101 Assignment 8 – Step-by-Step Statistics Guidance

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