Q: A basketball player attempts 15 free throws, each with a 60% chance of success. What is the expected number of successful shots?

$$E(Y) = np = 15 \times 0.6 = 9$$

Q: A sample of 50 students has a mean score of 80 and a standard deviation of 10. What is the 95% confidence interval for the mean, using $$t^* = 2.009$$?

$$80 \pm 2.009 \times \frac{10}{\sqrt{50}}$$

Question 1

A company wants to estimate the proportion of customers who accept a new credit card offer. They randomly select 500 customers and find that 120 accepted. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{120}{500} = 0.24$$

Question 2

Which of the following scenarios is most appropriate for modeling with a Poisson distribution?

Accepted Answer

Counting the number of emails a professor receives per day

Question 3

A basketball player attempts 15 free throws, each with a 60% chance of success. What is the expected number of successful shots?

Accepted Answer

$$E(Y) = np = 15 	imes 0.6 = 9$$

Question 4

Which sampling method is most likely to introduce bias if the list has a hidden pattern?

Accepted Answer

Systematic sampling

Question 5

A survey reports a 95% confidence interval for the mean monthly balance as $$[1500, 2500]. $$Which is the correct interpretation?

Accepted Answer

We are 95% confident that the true mean monthly balance is between $$1500$$ and $$2500$$

Question 6

A professor receives 120 emails over 24 days. What is the estimated average rate $$\lambda$$ for emails per day?

Accepted Answer

$$\lambda = \frac{120}{24} = 5$$

Question 7

Which formula gives the standard error for the sample mean when the sample standard deviation $$s is $$used?

Accepted Answer

$$SE_{\bar{x}} = \frac{s}{\sqrt{n}}$$

Question 8

A binomial random variable $$Y$$ counts the number of successes in $$n$$ independent trials with probability $$p. $$Which formula gives the probability of exactly $$y$$ successes?

Accepted Answer

$$P(Y = y) = \binom{n}{y} p^y (1-p$$)^{n-y}$$

Question 9

Which of the following is NOT a reason why a Normal model might be inappropriate for a dataset?

Accepted Answer

The data are measured on a binary scale

Question 10

A sample of 50 students has a mean score of 80 and a standard deviation of 10. What is the 95% confidence interval for the mean, using $$t^* = 2.009$$?

Accepted Answer

$$80 \pm 2.009 	imes \frac{10}{\sqrt{50}}$$

Question 11

Which formula is used to convert an original value $$x to a z-$$score in the Normal model?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 12

A control chart uses 3-sigma limits. What is the approximate probability that a single point falls outside these limits if the process is in control?

Accepted Answer

$$0.0027$$

Question 13

Which sampling method is best for ensuring that key subgroups are represented proportionally in a sample?

Accepted Answer

Stratified sampling

Question 14

If you want to decrease the margin of error in a confidence interval for a mean, which action is most effective?

Accepted Answer

Increase the sample size $$n$$

Question 15

A binomial model is appropriate when:

Accepted Answer

You have a fixed number of independent trials, each with the same probability of success

Question 16

Which formula gives the 95% confidence interval for a proportion $$p$$?

Accepted Answer

$$\hat{p} \pm 1.96 \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 17

Which of the following is a common mistake when interpreting a confidence interval?

Accepted Answer

Claiming that 95% of individual data values fall within the interval

Question 18

A Normal Quantile Plot (QQ plot) is used to:

Accepted Answer

Diagnose whether a Normal model is appropriate for the data

Question 19

In a control chart, what is the tradeoff when you widen the control limits?

Accepted Answer

Fewer Type I errors but more Type II errors

Question 20

Which formula gives the probability of exactly $$x$$ events in a Poisson process with rate $$\lambda$$?

Accepted Answer

$$P(X = x) = \frac{e$$^{-\lambda}$$ \lambda^x}{x!}$$

Random Variables, Normal Probability Model, Sampling, and Confidence Intervals: Study Notes for Business Statistics

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