AP Stats Conditions for a Mean

Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 515 and standard deviation 114. You choose an SRS of 100 students and average their SAT Math scores. Suppose that you do this many, many times. Which of the following are the mean and standard deviation of the sampling distribution of x̄ ?

• Mean = 515; SD = 114

• Mean = 515; SD = 114/√100

• Mean = 515/100; SD = 114/√100

• Mean = 515/100; SD = 114/100

• Mean = 515/100; SD = 114/√100

The number of hours a light bulb burns before failing varies from bulb to bulb. The population distribution of burnout times is strongly skewed to the right. The central limit theorem says that

• As we look at more and more bulbs, their average burnout time gets ever closer to the mean for all bulbs of this type.

• The average burnout time of a large number of bulbs has a sampling distribution with the same shape (strongly skewed) as the population distribution.

• The average burnout time of a large number of bulbs has a sampling distribution with similar shape but not as extreme (skewed but not as strongly) as the population distribution.

• The average burnout time of a large number of bulbs has a sampling distribution that is close to Normal.

• The average burnout time of a large number of bulbs has a sampling distribution that is exactly Normal.

A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was x̄ = 810 grams. This sample mean is an unbiased estimator of the mean weight in the population of all ELBW babies, which means that

• In all possible samples of size 219 from the population, the mean of the values of will equal 810.

• In all possible samples of size 219 from this population, the mean of the values of will equal μ

• As we take larger and larger samples from this population, x̄ will get closer and closer to μ .

• In all possible samples of size 219 from the population, the values of will have a distribution that is close to Normal.

• The person measuring the children’s weights does so without any error.

Which of the following statements about the sampling distribution of the sample mean is incorrect?

• The standard deviation of the sampling distribution will decrease as the sample size increases

• The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.

• The sample mean is an unbiased estimator of the population mean.

• The sampling distribution shows how the sample mean will vary in repeated samples.

• The sampling distribution shows how the sample was distributed around the sample mean.

Why is it important to check the 10% condition before calculating probabilities involving x̄?

• To reduce the variability of the sampling distribution of x̄

• To ensure that the distribution of x̄ is approximately Normal

• To ensure that we can generalize the results to a larger population

• To ensure that x̄ will be an unbiased estimator of μ.

• To ensure that the observations in the sample are close to independent.

You have an SRS of 23 observations from a large population. The distribution of sample values is roughly symmetric with no outliers. What critical value would you use to obtain a 98% confidence interval for the mean of the population?

• 2.177

• 2.183

• 2.326

• 2.500

• 2.508

One reason for using a t distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for a population mean is that

• z can be used only for large samples

• z requires that you know the population standard deviation

• z requires that you can regard your data as an SRS from the population

• z requires that the sample size is at most 10% of the population size

• a z critical value will lead to a wider interval than a t critical value

A researcher wants to find a confidence interval estimate of the mean time it takes to complete an order over the phone at a call center for a large retail company. He has selected a random sample of 18 orders and recorded the time to complete the order. The sample mean time was 4.27 minutes and the sample standard deviation was 0.78 minutes. Assuming the conditions needed for inference are met, which of the following is the appropriate 98% confidence interval?

• 4.27 ± 2.326 ⋅ (0.78/√18)

• 4.27 ± 2.054 ⋅ (0.78/√18)

• 4.27 ± 2.552 ⋅ (0.78/√18)

• 4.27 ± 2.567 ⋅ (0.78/√18)

• 4.27 ± 2.567 ⋅ (0.78/√17)

A 98% confidence interval for a population mean is found to be 127 ± 18. Which of the following is the correct interpretation of this confidence level?

• The probability that the true mean is contained in this interval is 0.98

• The probability that the sample mean is contained in this interval is 0.98

• The probability that another interval will give a true mean of 127 is 0.98

• If many different random samples of the same size are selected, about 98% of the confidence intervals constructed using this method will contain the value of the population mean

• If many different random samples of the same size are selected, about 98% of the confidence intervals constructed using this method will contain the sample mean of 127

Which of the following would not decrease the width of a confidence interval?

I. Increasing the sample size

II. Decreasing the degrees of freedom

III. Decreasing the confidence level

• I only

• II only

• III only

• I and II only

• II and III only