AP Statistics : Properties of Single Sample Distributions

Study concepts, example questions & explanations for AP Statistics

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Example Questions

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Example Question #1 : Properties Of Single Sample 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

      

       

Possible Answers:

      

      

      

Correct answer:

      

Explanation:

     

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

     

    

Possible Answers:

    

    

Correct answer:

    

Explanation:

    

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

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 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 

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

  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 #2 : Sampling Distributions

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:

.35

.875

.9

.7

.65

Correct answer:

.875

Explanation:

The population standard deviation = 

The sample mean standard deviation = 

Example Question #3 : Sampling Distributions

A random variable has a average of  with a standard deviation of . What is the probability that out of the sample set the variable is less than  . The sample set is . Round your answer to three decimal places.

Possible Answers:

Correct answer:

Explanation:

There are two keys here. One, we have a large sample size since , meaning we can use the Central Limit Theorem even if points per game is not normally distributed. 

Our -score thus becomes...

where  is the specified points or less needed this season,

 is the average points per game of the previous season,

 is the standard deivation of the previous season,

and  is the number of games.

 

Example Question #1 : Sampling 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 : Properties Of Single Sample Distributions

Which of the following is a sampling distribution?

Possible Answers:

The average height of a sample of college students.

The average height of all college students.

The height of a particular college student.

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 : Sampling Distributions

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:

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

College students are getting shorter.

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

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

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.

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