Q: A company tests the lifetime of a battery and collects a random sample of $$n = 12$$ batteries. The sample mean is $$\bar{x} = 306.25$$ minutes and the sample standard deviation is $$s = 29.31$$ minutes. What is the 95% confidence interval for the mean battery life? (Use $$t_{11, 0.025} = 2.201)$$

$$306.25 \pm 2.201 \times \frac{29.31}{\sqrt{12}}$$

Question 1

Which of the following best describes the accuracy and precision of the sample mean as an estimator of the population mean $$\mu$$?

Accepted Answer

The sample mean is accurate if, on average, it equals the population mean, and it is precise if its variability is low across samples.

Question 2

Suppose a population has a mean $$\mu = 45$$ and variance $$\$$sigma^2$$ = 150. $$What is the variance of the sample mean for a random sample of $$n = 30$$?

Accepted Answer

$$\frac{150}{30} = 5$$

Question 3

A researcher collects a random sample of size $$n = 100$$ from a population with mean $$\mu = 20$$ and standard deviation $$\sigma = 35. $$What is the standard deviation of the sample mean?

Accepted Answer

$$\frac{35}{\sqrt{100}} = 3.5$$

Question 4

Which of the following is NOT a condition required for the Central Limit Theorem to apply to the sample mean?

Accepted Answer

The population must be exactly Normal, regardless of sample size.

Question 5

A company tests the lifetime of a battery and collects a random sample of $$n = 12$$ batteries. The sample mean is $$\bar{x} = 306.25$$ minutes and the sample standard deviation is $$s = 29.31$$ minutes. What is the 95% confidence interval for the mean battery life? (Use $$t_{11, 0.025} = 2.201)$$

Accepted Answer

$$306.25 \pm 2.201 	imes \frac{29.31}{\sqrt{12}}$$

Question 6

Which of the following statements about the $$t-$$distribution is TRUE?

Accepted Answer

The $$t-$$distribution has thicker tails than the Normal distribution, especially for small sample sizes.

Question 7

A 95% confidence interval for a population mean is calculated as $$\bar{x} \pm t^* \frac{s}{\sqrt{n}}. $$What does the '95%' refer to?

Accepted Answer

The proportion of all possible samples for which the method produces an interval containing the population mean.

Question 8

A hospital manager wants to test if the mean years of experience among nursing staff has increased from 14.3 years (2010) to a higher value in 2022. Which is the correct null and alternative hypothesis for this test?

Accepted Answer

$$H_0$$: \mu = 14.3$$, H_A$$: \mu \u003e 14.3$$

Question 9

Suppose you are constructing a confidence interval for a mean using a sample of size $$n = 31. $$What is the degrees of freedom for the $$t-$$distribution?

Accepted Answer

$$30$$

Question 10

Which of the following best describes the effect of increasing the sample size on the margin of error in a confidence interval for the mean?

Accepted Answer

The margin of error decreases as sample size increases.

Question 11

A random sample of 34 colleges has a mean tuition of $$\5218$ with a standard deviation of $$\$$893. $$What is the standard error of the mean?

Accepted Answer

$$\frac{893}{\sqrt{34}} \approx 153.1$$

Question 12

Which of the following is the correct formula for the one-sample $$t-$$statistic used in hypothesis testing for a population mean?

Accepted Answer

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

Question 13

A study finds that the mean number of cigarettes smoked per day by Canadian teenagers is 6, with a standard deviation of 5. In a random sample of 36 teenagers, what is the probability that the sample mean is above 7? (Assume Normality.)

Accepted Answer

About 0.23

Question 14

Which of the following best describes the difference between the $$z-$$statistic and the $$t-$$statistic?

Accepted Answer

The $$z-$$statistic is used when the population standard deviation is known; the $$t-$$statistic is used when it is unknown.

Question 15

A 99% confidence interval for a mean is wider than a 95% confidence interval because:

Accepted Answer

The multiplier $$t^* is $$larger for 99% confidence.

Question 16

Which of the following is a correct interpretation of a p-value in hypothesis testing?

Accepted Answer

The probability of observing data as extreme or more extreme than the sample, assuming the null hypothesis is true.

Question 17

A researcher wants to estimate the mean mass of baby rhinos born in 2011. A random sample of 10 rhinos has a mean of 54 kg and a standard deviation of 14 kg. What is the 95% confidence interval for the mean mass? (Use $$t_{9, 0.025} = 2.262)$$

Accepted Answer

$$54 \pm 2.262 	imes \frac{14}{\sqrt{10}}$$

Question 18

Which of the following is a reason why the sample mean is considered an unbiased estimator of the population mean?

Accepted Answer

Because the expected value of the sample mean equals the population mean.

Chapter 9

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