Q: A company is analyzing the number of defective items in batches of 100, where the probability of a defect is 0.05. What is the mean and standard deviation of the number of defective items?

Mean: $$\mu = np = 100 \times 0.05 = 5$$; Standard deviation: $$\sigma = \sqrt{npq} = \sqrt{100 \times 0.05 \times 0.95}$$

Question 1

A company wants to estimate the average time employees spend on training per month. They collect a random sample of 50 employees, and the sample mean is 12 hours with a sample standard deviation of 3 hours. What is the 95% confidence interval for the population mean training time?

Accepted Answer

12 \pm z_{0.025} \frac{3}{\sqrt{50}}

Question 2

A manufacturer claims that 80% of its products pass quality inspection. In a random sample of 100 products, 75 pass inspection. What is the appropriate test statistic to test the manufacturer's claim at the 5% significance level?

Accepted Answer

z = \frac{\hat{p} - $$p_0$$}{\sqrt{$$p_0(1$$-$$p_0)/n$$}}

Question 3

Which formula would you use in Excel to calculate the probability that at most 3 out of 10 customers make a purchase, assuming the probability of purchase is 0.2?

Accepted Answer

=BINOM.DIST(3,10,0.2,TRUE)

Question 4

A business analyst uses the central limit theorem to justify using the normal distribution for sample means. Which statement best describes the central limit theorem?

Accepted Answer

When the sample size $$n is $$large, the distribution of sample means $$\bar{x} is $$approximately normal with mean $$\mu$$ and standard deviation $$\frac{\sigma}{\sqrt{n}}$$

Question 5

A retail store wants to predict sales based on advertising spending using linear regression. What is the formula for the 95% confidence interval for the mean response at a specific advertising amount $$x_p$$?

Accepted Answer

$$\hat{y} \pm t_{\alpha/2} s \sqrt{\frac{1}{n} + \frac{(x_p - \bar{x})^2$}{SS_{xx}}}$$

Question 6

A bank wants to estimate the proportion of customers who use online banking. In a sample of 200 customers, 120 use online banking. What is the 95% confidence interval for the proportion?

Accepted Answer

$$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 7

A business wants to test if the average delivery time is less than 2 days. They collect a sample of 30 deliveries, with a sample mean of 1.8 days and sample standard deviation of 0.5 days. Which test statistic should they use?

Accepted Answer

$$t = \frac{\bar{x} - \mu_0$}{s/\sqrt{n}}$$

Question 8

A company is analyzing the number of defective items in batches of 100, where the probability of a defect is 0.05. What is the mean and standard deviation of the number of defective items?

Accepted Answer

Mean: $$\mu = np = 100 	imes 0.05 = 5$$; Standard deviation: $$\sigma = \sqrt{npq} = \sqrt{100 	imes 0.05 	imes 0.95}$$

Question 9

A marketing manager wants to find the probability that exactly 4 out of 20 customers respond to a campaign, given the response rate is 0.1. Which Excel formula should they use?

Accepted Answer

=BINOM.DIST(4,20,0.1,FALSE)

Question 10

A business wants to predict the sales for a new product using linear regression. What is the formula for the 95% prediction interval for the response at a specific value $$x_p$$?

Accepted Answer

$$\hat{y} \pm t_{\alpha/2} s \sqrt{1 + \frac{1}{n} + \frac{(x_p - \bar{x})^2$}{SS_{xx}}}$$

Question 11

A business wants to calculate the sample mean and sample standard deviation of sales data in Excel. Which formulas should they use?

Accepted Answer

=average() for mean and =stdev() for standard deviation

Question 12

A business wants to find the critical value $$z_{\alpha}$$ for a 90% confidence interval using Excel. Which formula should they use?

Accepted Answer

=norm.s.inv(1-0.10)

Question 13

A business wants to test if the proportion of customers who prefer online shopping is different from 0.5. Which formula should they use for the test statistic?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{p_0$(1-p_0$)/n}}$$

Question 14

A business wants to calculate the correlation coefficient between sales and advertising spending in Excel. Which formula should they use?

Accepted Answer

=correl(x data, y data)

Question 15

A business wants to calculate $$SS_{xx}$$ for a set of advertising data in Excel. Which formula should they use?

Accepted Answer

=devsq(x data)

Question 16

A business wants to fit a linear regression model in Excel. Which tool should they use?

Accepted Answer

Data analysis toolpak =\u003e Regression

Question 17

A business wants to find the probability that a normally distributed variable with mean 100 and standard deviation 15 is less than 120 in Excel. Which formula should they use?

Accepted Answer

=norm.dist(120,100,15,true)

Question 18

A business wants to create a histogram of sales data in Excel. Which tool should they use?

Accepted Answer

Data analysis toolpak =\u003e Histogram

Business Statistics: Math and Excel Formula Sheet Study Guide

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