AP Statistics : Sampling Distributions

Study concepts, example questions & explanations for AP Statistics

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Example Questions

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Example Question #1 : Sampling Distributions

A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?

Possible Answers:

Correct answer:

Explanation:

This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.

Example Question #1 : Sampling Distributions

      

       

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Example Question #1 : Sampling Distributions

     

    

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Example Question #1 : Properties Of Single Sample Distributions

The president of a country is trying to estimate the average income of his citizens. He randomly samples residents and collects information about their salaries. A  percent confidence interval computed from this data for the mean income per citizen is  Which of the following provides the best interpretation of this confidence interval?

Possible Answers:

There is a  percent probability that all the citizens of the country have an income  between  and 

There is a  percent probability that the mean of another sample with the same size will fall between  and 

There is a   percent probability that the mean income per citizen in the school is between  and  

If he was to take another sample of the same size and compute a  percent confidence interval, we would have a  percent chance of getting the interval 

  percent of the citizens of the country have an income that is between  and 

Correct answer:

There is a   percent probability that the mean income per citizen in the school is between  and  

Explanation:

A confidence interval is a statement about the mean of the population the sample is drawn from so there is a  percent probability that a  percent confidence interval contains the true mean of the population. 

Example Question #1 : How To Find Sampling Distribution Of A Sample Mean

Assume you have taken 100 samples of size 64 each from a population. The population variance is 49.

What is the standard deviation of each (and every) sample mean?

Possible Answers:

.875

.7

.35

.65

.9

Correct answer:

.875

Explanation:

The population standard deviation = 

The sample mean standard deviation = 

Example Question #1 : Properties Of Single Sample Distributions

Reaction times in a population of people have a standard deviation of  milliseconds. How large must a sample size be for the standard deviation of the sample mean reaction time to be no larger than  milliseconds?

Possible Answers:

Correct answer:

Explanation:

Use the fact that .

Alternately, you can use the fact that the variance of the sample mean varies inversely by the square root of the sample size, so to reduce the variance by a factor of 10, the sample size needs to be 100.

Example Question #1 : How To Find Sampling Distribution Of A Sample Mean

A machine puts an average of  grams of jelly beans in bags, with a standard deviation of  grams.  bags are randomly chosen, what is the probability that the mean amount per bag in the sampled bags is less than  grams. 

Possible Answers:

Correct answer:

Explanation:

A sample size of  bags means that the central limit theorem is applicable and the distribution can be assumed to be normal. The sample mean would be   and  

Therefore, 

Example Question #1 : How To Find Sampling Distribution Of A Sample Mean

Which of the following is a sampling distribution?

Possible Answers:

The average height of a sample of college students.

The height of a particular college student.

The average height of all college students.

The distribution of average height statistics that could happen from all possible samples of college students.

Correct answer:

The distribution of average height statistics that could happen from all possible samples of college students.

Explanation:

The correct answer is the distribution of average height statistics that could happen from all possible samples of college students. Remember that a sampling distribution isn't just a statistic you get form taking a sample, and isn't just a piece of data you get from doing sampling. Instead, a sampling distribution is a distribution of sample statistics you could get from all of the possible samples you might take from a given population.

Example Question #1 : How To Find Sampling Distribution Of A Sample Mean

If a sampling distribution for samples of college students measured for average height has a mean of 70 inches and a standard deviation of 5 inches, we can infer that:

Possible Answers:

Roughly 68% of college students are between 65 and 75 inches tall.

College students are getting shorter.

Roughly 68% of random samples of college students will have a sample mean of between 65 and 75 inches.

Any particular random sample of college students will have a mean of 70 inches and a standard deviation of 5 inches.

Correct answer:

Roughly 68% of random samples of college students will have a sample mean of between 65 and 75 inches.

Explanation:

We can infer that roughly 68% of random samples of college students will have a sample mean of between 65 and 75 inches. Anytime we try to make an inference from a sampling distribution, we have to keep in mind that the sampling distribution is a distribution of samples and not a distribution about the thing we're trying to measure itself (in this case the height of college students). Also, remember that the empirical rules tells us that roughly 68% of the distribution will fall within one standard deviation of the mean.

Example Question #1 : How To Find Sampling Distribution Of A Sample Mean

The standard deviation of a sampling distribution is called:

Possible Answers:

Standard error

Sample variance

Sampling deviation

John McEnroe

Correct answer:

Standard error

Explanation:

The standard error (SE) is the standard deviation of the sampling distribution.

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