Sampling Distribution Center and Variability

Do you favor a law banning lawn mowing after 8 pm? Suppose that 35% of all citizens favor the law. If we took a very large number of SRSs of size n = 250 from this population, the sampling distribution of the sample proportion [p hat] would have which of the following characteristics?

• Mean = 0.35

  SD = 0.0009

  Shape: unknown

• Mean = 0.35

  SD = 0.0009

  Shape: Approx. Normal

• Mean = 0.35

  SD = 0.022

  Shape: unknown

• Mean = 0.35

  SD = 0.022

  Shape: Approx. Normal

Which of the following best describes a sampling distribution?

• ...a random sample of values of a sample statistic found in less than 10% of simulated samples of size n from the population

• ...values taken by a statistic in all possible samples of the same size from the same population

• ...parameter values in all possible samples of size n

• ...all values in a single random sample selected from the population

• ...is approximately normal with mean 60 and standard deviation 10

• is approximately normal with mean 60 and standard deviation 2

• has an unknown shape with mean 60 and standard deviation 1.414

• ...has an unknown shape with mean 60 and standard deviation 2

At a high school, 29% of students are freshmen. Let (p-hat) = the proportion of students that are freshmen in a random sample of 25 students from the school. Which of the following statements about the mean and standard deviation of the sampling distribution of (p-hat) is true?

Do you favor a law banning lawn mowing after 8 pm? If 35% of all citizens favor the law, and we take a very large number of SRSs of an increased size n = 600 rather than n = 150 from this population, how would it affect the estimate?

• reduce the bias of the estimate

• increase the bias of the estimate

• reduce the variability of the estimate

• increase the variability of the estimate

The figures below show approximate sampling distributions of 4 different statistics intended to estimate the same parameter. The true value of the parameter is 12. Which statistic has the smallest bias and the smallest variability?

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The basketball team has four players who can play the center position. Their heights (in inches) are 78, 80, 77, and 80.

List all 6 possible SRSs of size n = 2, calculate the mean height for each sample.

True of False: The sampling distribution of the sample proportion ˆp describes the distribution of values taken by the sample proportion ˆp in all possible samples of the same size from the same population.

• True

• False

This describes _______________: The mean of the sampling distribution of pˆ is equal to the population proportion p. This makes sense because the sample proportion pˆ is an unbiased estimator of p.

• Shape

• Outliers

• Center

• Variability

This describes ________________: The value of σpˆ depends on both n and p. For a specific sample size, the standard deviation σpˆ is larger for values of p close to 0.5 and smaller for values of p close to 0 or 1. For a specific value of p, the standard deviation σpˆ gets smaller as n gets larger. Specifically, multiplying the sample size by 4 cuts the standard deviation in half.

• Shape

• Outliers

• Center

• Variability