STAT 5.5 sampling distribution

A bag contains chips of which 27.5 percent are blue. A random sample of 5 chips will be selected one at a time and with replacement. What are the mean and standard deviation of the sampling distribution of the sample proportion of blue chips for samples of size 5 ?

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A bag contains chips of which 27.5 percent are blue. A random sample of 5 chips will be selected one at a time and with replacement. What are the mean and standard deviation of the sampling distribution of the sample proportion of blue chips for samples of size 5 ?

• The mean is 0.275, and the standard deviation is 0.250

• The mean is 0.275, and the standard deviation is 0.156

• The mean is 0.275, and the standard deviation is 0.200.

• The mean is 0.275, and the standard deviation is 0.350

• The mean is 0.275, and the standard deviation is 0.1.00

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A manufacturer of cell phone screens has found that 5 percent of all screens produced have defects. Let pd represent the population proportion of all cell phone screens with a screen defect, therefore pd=0.05. For the sampling distribution of the sample proportion of cell phone screens from this manufacturer with a screen defect for sample size 400, mupd=0.05. Which of the following is the best interpretation of mu pd=0.05 ?

• For a randomly selected cell phone screen from this population, the mean number of screen defects for the selected screen will be equal to 0.05.

• For every sample of size 400 from this population, the proportion of cell phone screens with a screen defect will be 0.05.

• For all samples of size 400 from this population, the mean number of screen defects for the samples is 0.05.

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05.

• For all samples of size 400 from this population, the standard deviation of all resulting sample proportions of cell phone screens with a screen defect is 0.05.

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A manufacturer of cell phone screens has found that 5 percent of all screens produced have defects. Let pd represent the population proportion of all cell phone screens with a screen defect, therefore pd=0.05. For the sampling distribution of the sample proportion of cell phone screens from this manufacturer with a screen defect for sample size 400, mupd=0.05. What is the standard deviation of of a sample of 400 with mean pd=0.05?

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05 and the standard deviation is 0.005

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05 and the standard deviation is 0.033

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05 and the standard deviation is 0.022

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05 and the standard deviation is 0.011

• For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05 and the standard deviation is 0.017

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A national charity contacted 100 randomly selected people by phone, and 7 percent of those contacted made a donation to the charity. The population proportion of those who make a donation when contacted by phone is known to be mu=0.05. For samples of size 100, which of the following best interprets the mean of the sampling distribution of the sample proportion of people who make a donation when contacted by phone?

• For all random samples of 100 people contacted by phone, the sample proportion of those who make a donation will be 0.05.

• For all random samples of 100 people contacted by phone, the sample proportion of those who make a donation will be 0.07.

• The mean of all sample proportions of those who make a donation from all random samples of 100 people contacted by phone is 0.05.

• The mean of all sample proportions of those who make a donation from all random samples of 100 people contacted by phone is 0.0

• The probability that the mean of the sampling distribution of sample proportions is greater than 0.07 is 0.05.

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Which of the following pairs of sample size n and population proportion p would produce the greatest standard deviation for the sampling distribution of a sample proportion p̂?

• n = 1,000 and p close to 0

• n = 1,000 and p close to 1

• n = 1,000 and p close to 1/2

• n = 100 and p close to 0

• n = 100 and p close to 1/2

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According to data from the United States Elections Project, only 36 percent of eligible voters voted in the 2014 elections. For random samples of size 40, which of the following best describes the sampling distribution of ps, the sample proportion of people who voted in the 2014 elections?

• The sampling distribution is skewed to the left, with mean 0.36 and standard deviation 0.076.

• The sampling distribution is skewed to the right, with mean 0.64 and standard deviation 0.006.

• The sampling distribution is approximately normal, with mean 0.36 and standard deviation 0.076.

• The sampling distribution is approximately normal, with mean 0.36 and standard deviation 0.006.

• The sampling distribution is approximately normal, with mean 0.64 and standard deviation 0.076.

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The director of a marketing department wants to estimate the proportion of people who purchase a certain product online. The director originally planned to obtain a random sample of 2,500 people who purchased the product. However, because of budget concerns, the sample size will be reduced to 1,500 people. Which of the following describes the effect of reducing the number of people in the sample?

• The variance of the sample will increase.

• The variance of the population will decrease.

• The variance of the sampling distribution of the estimator will increase.

• The variance of the sampling distribution of the estimator will decrease.

• The variance of the sampling distribution of the estimator will remain the same.

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For the population of people in the United States who are at least 18 years of age, approximately 61 percent voted in a recent election. A random sample of 200 people and a random sample of 100 people will be selected from the population.

For which sample size is it more likely that greater than 70 percent of the sample voted in the recent election?

• They are equally likely, because the mean of the sampling distribution is the same for both sample sizes.

• It is more likely for a sample size of 200 people, because there is more variability in the sampling distribution for larger sample sizes.

• It is more likely for a sample size of ⁢200 people, because there is less variability in the sampling distribution for larger sample sizes.

• It is more likely for a sample of size 100 people, because there is more variability in the sampling distribution for smaller sample sizes.

• It is more likely for a sample size of 100 people, because there is less variability in the sampling distribution for smaller sample sizes.

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On any given day, the proportion of workers at a factory who are more than 5 minutes late to work is 0.11. A random sample of 20 workers will be selected. Which of the following is the best interpretation of the mean of the sampling distribution of the sample proportion of workers in the sample who are more than 5 minutes late to work for samples of size 20 ?

• For all samples of size 20, the mean of all possible sample proportions is equal to 0.11.

• For a randomly selected worker, the probability the worker will be more than 5 minutes late to work is 0.11.

• For a random sample of 20 workers, the proportion of workers who are more than 20 minutes late to work will be 0.11.

• For a random sample of 20 workers, the probability that 2.2 workers will be more than 5 minutes late to work is very high.

• For repeated samples of size 20, the proportion of workers who are more than 5 minutes late to work varies in each sample by no more than 0.11.

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