Q: A sample of 100 light bulbs has a mean lifetime of 1200 hours with a standard deviation of 100 hours. What is the 95% confidence interval for the population mean?

$$1200 \pm 1.96 \times \frac{100}{\sqrt{100}} = (1180.4, 1219.6)$$

Q: A random sample of 25 batteries has a mean life of 40 hours and a standard deviation of 5 hours. What is the margin of error for a 90% confidence interval for the mean?

$$E = z_{0.05} \times \frac{5}{\sqrt{25}} = 1.645 \times 1 = 1.645$$

Q: A hospital administrator wants to estimate the mean length of stay for patients. She wants to be 99% confident that the estimate is within 2 days. If the standard deviation is 6 days, what is the minimum sample size required?

$$n = \left(\frac{z_{0.005} \times \sigma}{E}\$$right)^2$$ = \left(\frac{2.576 \times 6}{2}\$$right)^2$$ = 119.3 \approx 120$$

Q: Which of the following is the correct formula for the standard error of the mean when the population standard deviation is known?

$$\text{Standard Error} = \frac{\sigma}{\sqrt{n}}$$

Question 1

Suppose the heights of adult men in a city are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. What is the probability that a randomly selected man is taller than 73 inches?

Accepted Answer

$$P(X \u003e 73) = P\left(Z \u003e \frac{73-70}{3}\right) = P(Z \u003e 1) = 0.1587$$

Question 2

A standardized test has scores that are normally distributed with a mean of 500 and a standard deviation of 100. What is the z-score for a student who scored 650?

Accepted Answer

$$z = \frac{650 - 500}{100} = 1.5$$

Question 3

If the area to the left of $$z is 0.975$$, what is the value of $$z$$?

Accepted Answer

$$z = 1.96$$

Question 4

A sample of 100 light bulbs has a mean lifetime of 1200 hours with a standard deviation of 100 hours. What is the 95% confidence interval for the population mean?

Accepted Answer

$$1200 \pm 1.96 	imes \frac{100}{\sqrt{100}} = (1180.4, 1219.6)$$

Question 5

In a survey of 800 adults, 240 said they exercise regularly. What is the point estimate for the proportion of adults who exercise regularly?

Accepted Answer

$$\hat{p} = \frac{240}{800} = 0.30$$

Question 6

A random sample of 36 students has a mean test score of 82 with a population standard deviation of 12. What is the probability that the sample mean is greater than 85?

Accepted Answer

$$P\left(Z \u003e \frac{85-82}{12/\sqrt{36}}\right) = P(Z \u003e 1.5) = 0.0668$$

Question 7

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

Accepted Answer

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

Question 8

A 95% confidence interval for a population mean is calculated as $$(50.2, 54.8). $$Which of the following statements is correct?

Accepted Answer

We are 95% confident that the population mean lies between 50.2 and 54.8.

Question 9

A random sample of 25 batteries has a mean life of 40 hours and a standard deviation of 5 hours. What is the margin of error for a 90% confidence interval for the mean?

Accepted Answer

$$E = z_{0.05} 	imes \frac{5}{\sqrt{25}} = 1.645 	imes 1 = 1.645$$

Question 10

If the area between $$z = -a$$ and $$z = a is 0.90$$, what is the value of $$a$$?

Accepted Answer

$$a = 1.96$$

Question 11

A hospital administrator wants to estimate the mean length of stay for patients. She wants to be 99% confident that the estimate is within 2 days. If the standard deviation is 6 days, what is the minimum sample size required?

Accepted Answer

$$n = \left(\frac{z_{0.005} 	imes \sigma}{E}\$$right)^2$$ = \left(\frac{2.576 	imes 6}{2}\$$right)^2$$ = 119.3 \approx 120$$

Question 12

Which of the following is the correct formula for the standard error of the mean when the population standard deviation is known?

Accepted Answer

$$	ext{Standard Error} = \frac{\sigma}{\sqrt{n}}$$

Question 13

A sample of 150 students has a mean SAT score of 1025 with a standard deviation of 220. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{220}{\sqrt{150}} = 17.97$$

Question 14

A 90% confidence interval for a population proportion is $$(0.45, 0.55). $$What is the margin of error?

Accepted Answer

$$E = \frac{0.55 - 0.45}{2} = 0.05$$

Question 15

Which of the following best describes the meaning of a p-value in hypothesis testing?

Accepted Answer

The probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

Question 16

A researcher wants to test if the mean weight of a population is different from 150 lbs. Which of the following is the correct set of hypotheses?

Accepted Answer

$$H_0$$: \mu = 150$$, H_a$$: \mu 
eq 150$$

Question 17

If a 95% confidence interval for the mean difference between two treatments is $$(-2.1, 0.5)$$, what can you conclude?

Accepted Answer

There is not enough evidence to conclude a significant difference between the treatments at the 0.05 level.

Question 18

A random sample of 1000 U.S. adults found that 120 visited an emergency room last year. What is the 95% confidence interval for the proportion of U.S. adults who visited an emergency room last year?

Accepted Answer

$$(0.102, 0.138)$$

Normal Distributions, Confidence Intervals, and Sampling Distributions: Study Notes

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