# AP Statistics : How to find sampling distribution of a sample mean

## Example Questions

### 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?

.875

.9

.7

.65

.35

.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?

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.

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 #4 : How To Find Sampling Distribution Of A Sample Mean

Which of the following is a sampling distribution?

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

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.

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 #5 : 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:

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.

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

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 #6 : How To Find Sampling Distribution Of A Sample Mean

The standard deviation of a sampling distribution is called:

Sampling deviation

Standard error

John McEnroe

Sample variance

Standard error

Explanation:

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

### Example Question #7 : How To Find Sampling Distribution Of A Sample Mean

Suppose that the mean height of college students is 70 inches with a standard deviation of 5 inches. If a random sample of 60 college students is taken, what is the probability that the sample average height for this sample will be more than 71 inches?

Explanation:

First check to see if the Central Limit Theorem applies. Since n > 30, it does. Next we need to calculate the standard error. To do that we divide the population standard deviation by the square-root of n, which gives us a standard error of 0.646. Next, we calculate a z-score using our z-score formula:

Plugging in gives us:

Finally, we look up our z-score in our z-score table to get a p-value.

The table gives us a p-value of,

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

A random variable has an 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.

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.