Q: A sample of 40 invoices has a mean value of $$500 and a population standard deviation of 80. $$What is the 90% confidence interval for the mean invoice value? (Use $$z^* = 1.645$$ for 90% confidence.)

$$ 500 \pm 1.645 \times \frac{80}{\sqrt{40}} $$

Question 1

Which of the following best describes the purpose of the Z table in business statistics?

Accepted Answer

To find the probability associated with a given Z-score in a standard normal distribution.

Question 2

A company wants to estimate the average time customers spend on their website. They collect a sample of 50 customers and calculate a sample mean of 8 minutes with a standard deviation of 2 minutes. Which formula should they use to construct a 95% confidence interval for the population mean?

Accepted Answer

$$ \bar{x} \pm z^* \frac{s}{\sqrt{n}} $$

Question 3

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

Accepted Answer

t-distribution

Question 4

A business analyst is testing whether the average delivery time for a product is less than 3 days. Which of the following represents the correct null and alternative hypotheses?

Accepted Answer

$$ H_0$: \mu = 3, \quad H_a: \mu \u003c 3 $$

Question 5

A sample of 25 business students has a mean test score of 78 and a sample standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$ \frac{10}{\sqrt{25}} = 2 $$

Question 6

Which of the following statements about the Central Limit Theorem is correct?

Accepted Answer

As sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's distribution.

Question 7

A manager wants to test if a new training program increases employee productivity. She collects productivity data before and after the program for 15 employees. Which statistical test should she use?

Accepted Answer

Paired t-test

Question 8

If the p-value for a hypothesis test is 0.03 and the significance level is 0.05, what should the business analyst conclude?

Accepted Answer

Reject the null hypothesis.

Question 9

A sample of 40 invoices has a mean value of $$500 and a population standard deviation of 80. $$What is the 90% confidence interval for the mean invoice value? (Use $$z^* = 1.645$$ for 90% confidence.)

Accepted Answer

$$ 500 \pm 1.645 	imes \frac{80}{\sqrt{40}} $$

Question 10

Which of the following is NOT an assumption for using the t-distribution to estimate a population mean?

Accepted Answer

The population standard deviation is known.

Question 11

A business wants to know if the average amount spent by customers has changed from last year’s mean of $120. Which type of hypothesis test should be used?

Accepted Answer

Two-tailed test

Question 12

A sample of 16 sales transactions has a mean of $$200 and a standard deviation of 40. $$What is the value of the t-statistic for testing $$H_0$$: \mu = 180$$?

Accepted Answer

$$ t = \frac{200 - 180}{40/\sqrt{16}} = 2 $$

Question 13

Which of the following best explains why larger sample sizes lead to narrower confidence intervals?

Accepted Answer

Because the standard error decreases as sample size increases.

Question 14

A business analyst is using a t-table to find the critical value for a 95% confidence interval with 12 degrees of freedom. What value should she look up?

Accepted Answer

The value in the t-table corresponding to 12 degrees of freedom and a two-tailed probability of 0.05.

Question 15

A company claims that their new process reduces the average production time to less than 5 hours. A sample of 36 production runs has a mean of 4.8 hours and a standard deviation of 0.6 hours. What is the test statistic for testing $$H_0$$: \mu = 5$$?

Accepted Answer

$$ z = \frac{4.8 - 5}{0.6/\sqrt{36}} = -2 $$

Question 16

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

Accepted Answer

We are 95% confident that the interval contains the true population mean.

Question 17

A business wants to compare the mean sales of two different stores. Which test should be used if the population variances are unknown but assumed equal?

Accepted Answer

Two-sample t-test (pooled variance)

Question 18

If a 99% confidence interval for the mean is wider than a 95% confidence interval for the same data, what is the reason?

Accepted Answer

A higher confidence level requires a larger margin of error.

Question 19

A business collects a random sample of 100 customer ratings (on a 1-10 scale) and finds a mean of 7.2 with a standard deviation of 1.5. What is the probability that a randomly selected customer rates the business above 8? (Assume normality.)

Accepted Answer

$$ P(Z \u003e \frac{8 - 7.2}{1.5}) = P(Z \u003e 0.53) $$

Question 20

Which of the following is a real-world example of when to use a one-sample t-test?

Accepted Answer

A restaurant wants to test if the average wait time is different from the advertised 10 minutes.

Quiz 3 Study Guide: Sampling Distributions and Estimation in Business Statistics

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