Q: A sample of 25 light bulbs is taken from a large shipment. The sample mean lifetime is 1200 hours, and the sample standard deviation is 100 hours. What is the 95% confidence interval for the population mean lifetime, assuming the population is approximately normal?

$$1200 \pm t_{0.025,24} \times \frac{100}{\sqrt{25}}$$

Question 1

A company claims that their new battery lasts an average of 100 hours. A consumer group tests 40 batteries and finds a sample mean of 98 hours with a sample standard deviation of 5 hours. Assuming battery life is approximately normal, which of the following is the correct test statistic for testing the company's claim at the 0.05 significance level?

Accepted Answer

$$z = \frac{98 - 100}{5/\sqrt{40}}$$

Question 2

Which of the following statements about the Central Limit Theorem (CLT) is TRUE?

Accepted Answer

For any population, as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution.

Question 3

A random variable $$X$$ counts the number of emails received per hour at a help desk. If the average rate is 6 emails per hour, what is the probability of receiving exactly 4 emails in an hour? (Use the Poisson formula: $$P(X = k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!})$$

Accepted Answer

$$P(X=4) = \frac{6^4$ e$$^{-6}$$}{4!}$$

Question 4

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 5

A sample of 25 light bulbs is taken from a large shipment. The sample mean lifetime is 1200 hours, and the sample standard deviation is 100 hours. What is the 95% confidence interval for the population mean lifetime, assuming the population is approximately normal?

Accepted Answer

$$1200 \pm t_{0.025,24} 	imes \frac{100}{\sqrt{25}}$$

Question 6

Which of the following best describes a Type II error in hypothesis testing?

Accepted Answer

Failing to reject a null hypothesis that is actually false.

Question 7

A quality control manager wants to estimate the proportion of defective items in a shipment. She samples 200 items and finds 12 are defective. What is the standard error of the sample proportion?

Accepted Answer

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

Question 8

Which of the following is TRUE about the standard normal random variable $$Z$$?

Accepted Answer

$$Z$$ has a mean of 0 and a standard deviation of 1.

Question 9

A researcher wants to construct a 99% confidence interval for a population proportion $$p$$ but has no prior estimate for $$p. $$What value should she use for $$p in $$the margin of error formula?

Accepted Answer

$$0.5$$

Question 10

In a hypothesis test, if the p-value is less than the significance level $$\alpha$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis and accept the alternative hypothesis.

Question 11

A hypergeometric random variable is most appropriate when:

Accepted Answer

Sampling is done without replacement from a finite population.

Question 12

Which of the following is the correct formula for the variance of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p(x_i)$$?

Accepted Answer

$$\sum (x_i - \mu)^2$ p(x_i)$$

Question 13

A normal random variable $$X$$ has mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the $$z-$$score for $$X = 65$$?

Accepted Answer

$$z = \frac{65 - 50}{10}$$

Question 14

Which of the following statements about confidence intervals is FALSE?

Accepted Answer

A confidence interval guarantees the parameter is within the interval for the current sample.

Question 15

If a sample of size $$n = 100 is $$taken from a population with standard deviation $$\sigma = 20$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{20}{\sqrt{100}}$$

Question 16

A two-tailed hypothesis test is being conducted at $$\alpha = 0.05. $$What is the area in each tail of the standard normal distribution?

Accepted Answer

$$0.025$$

Question 17

Which of the following is an unbiased estimator for the population mean $$\mu$$?

Accepted Answer

The sample mean $$\bar{x}$$

Question 18

A manufacturer wants to ensure that the margin of error for a 95% confidence interval for a population proportion is no more than 0.03. What sample size $$n$$ should be used if no prior estimate for $$p is $$available? (Use $$z_{0.025} = 1.96$$ and $$p = 0.5)$$

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.5 	imes 0.5}{0.03^2$} ight)$$

Question 19

Which of the following best describes the role of alpha ($$\alpha) in $$hypothesis testing?

Accepted Answer

It is the probability of making a Type I error.

Question 20

A sample of 10 students' test scores is used to construct a confidence interval for the mean score. Which distribution should be used to find the critical value?

Accepted Answer

The $$t$$ distribution with 9 degrees of freedom

Study Notes: Random Variables, Sampling Distributions, Confidence Intervals, and Hypothesis Testing

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