STAT 5.7-8

For which of the following conditions is it not appropriate to assume that the sampling distribution of the sample mean is approximately normal?

• A random sample of 8 taken from a normally distributed population

• A random sample of 50 taken from a normally distributed population

• A random sample of 10 taken from a population distribution that is skewed to the right

• A random sample of 75 taken from a population distribution that is skewed to the left

• A random sample of 100 taken from a population that is uniform

A fair six-sided die, with sides numbered 1 through 6, will be rolled a total of 15 times. Let X¯1 represent the average of the first ten rolls, and let X¯2 represent the average of the remaining five rolls. What is the mean P(X¯1−X¯2) of the sampling distribution of the difference in sample means X¯1−X¯2 ?

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At a certain high school, the distribution of backpack weight is approximately normal with mean 19.7 pounds and standard deviation 3.1 pounds. A random sample of 5 backpacks will be selected, and the weight, in pounds, of each backpack will be recorded.

For samples of size 5, which of the following is the best interpretation of P(X¯>22)≈0.05?

• The probability that each of the 5 backpacks selected will have a weight above 22 pounds is approximately 0.05.

• The probability that each of the 5 backpacks selected will have a weight above 19.7 pounds is approximately 0.05.

• The probability that the population mean is greater than 22 pounds is approximately 0.05.

• For all samples of size 5, approximately 5% of the sample will have a probability greater than 22 pounds.

• For all samples of size 5, the probability that the sample mean will be greater than 22 pounds is approximately 0.05.

Researchers working for a certain airline are investigating the weight of carry-on bags. The researchers will use the mean weight of a random sample of 800 carry-on bags to estimate the mean weight of all carry-on bags for the airline. Which of the following best describes the effect on the bias and the variance of the estimator if the researchers increase the sample size to 1,300 ?

• The bias will decrease and the variance will remain the same.

• The bias will increase and the variance will remain the same.

• The bias will remain the same and the variance will decrease.

• The bias will remain the same and the variance will increase.

• The bias will decrease and the variance will decrease.

At a large university, the division of computing services surveyed a random sample of 45 biology majors and 55 business majors from populations of over 1,000 biology and 1,000 business majors. The sampled students were asked how many hours they spend per week using any university computer lab. Let X¯1 represent the average hours per week spent in any university computer lab by the 45 biology majors, and let X¯2 represent the average hours per week spent in any university computer lab by the 55 business majors.

Which of the following is the best explanation for why the sampling distribution of X¯1−X¯2 can be modeled with a normal distribution?

• The two sample standard deviations are assumed to be equal.

• The sample sizes are both sufficiently large.

• The distributions of the population are normal.

• The population standard deviations are assumed equal.

• There are at least 30 students in each of the two populations.

Suppose the variance in trunk diameter of the giant sequoia tree species is 15.7m2, while the variance in trunk diameter of the California redwood tree species is 10.6m2. Let X¯1 represent the average trunk diameter of four randomly sampled giant sequoia trees, and let X¯2 represent the average trunk diameter of three randomly sampled California redwood trees. If the random sampling is done with replacement, what is the standard deviation P(X¯1−X¯2) of the sampling distribution of the difference in sample means X¯1−X¯2 ?

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The distribution of prices for a certain car model is approximately normal with mean $21,800 and standard deviation $400. A random sample of 4 cars of the model will be selected.

What is the correct unit of measure for the mean of the sampling distribution of X¯ ?

• Dollars

• Models

• Cars

• Samples

• There are no units associated with the mean of the sampling distribution.

A manufacturer of cell phone batteries claims that the average number of recharge cycles for its batteries is 400. A consumer group will obtain a random sample of 100 of the manufacturer’s batteries and will calculate the mean number of recharge cycles. Which of the following statements is justified by the central limit theorem?

• The distribution of the number of recharge cycles for the sample is approximately normal because the population mean of 400 is greater than 30.

• The distribution of the number of recharge cycles for the sample is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the number of recharge cycles for the population is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the sample means of the number of recharge cycles is approximately normal because the sample size of 100 is greater than 30.

• The distribution of the sample means of the number of recharge cycles is approximately normal because the population mean of 400 is greater than 30.

There were 5,317 previously owned homes sold in a western city in the year 2000. The distribution of the sales prices of these homes was strongly right-skewed, with a mean of $206,274 and a standard deviation of $37,881. If all possible simple random samples of size 100 are drawn from this population and the mean is computed for each of these samples, which of the following describes the sampling distribution of the sample mean?

• Approximately normal with mean $206,274 and standard deviation $3,788

• Approximately normal with mean $206,274 and standard deviation $37,881

• Strongly right-skewed with mean $206,274 and standard deviation $3,788

• Approximately normal with mean $206,274 and standard deviation $520

• Strongly right-skewed with mean $206,274 and standard deviation $37,881

At a certain restaurant, the distribution of wait times between ordering a meal and receiving the meal has mean 11.4 minutes and standard deviation 2.6 minutes. The restaurant manager wants to find the probability that the mean wait time will be greater than 12.0 minutes for a random sample of 84 customers. Assuming the wait times among customers are independent, which of the following describes the sampling distribution of the sample mean wait time for random samples of size 84 ?

• Approximately normal with mean 11.4 minutes and standard deviation 2.6 minutes

• Binomial with mean 84(0.41) minutes and standard deviation 4.51 minutes

• Approximately normal with mean 12.0 minutes and standard deviation 2.6 minutes

• Approximately normal with mean 11.4 minutes and standard deviation 0.2837 minute

• Binomial with mean 84(0.5) minutes and standard deviation 4.58 minutes