Wiki Page, Math 13 - Perez Sandbox : Module 22 - Distribution of Sample Means (4 of 4)

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2026-01-17T21:15:36-08:00

Now we will use the sampling distribution as a probability model to lay the ground work for inference about a population mean.

As before, we can always use a simulation; however, we can also use a normal model when conditions are met: the sample size is large (n>30) or if the variable is normally distributed in the population.

As we have done before, we convert the statistic (in this case the sample mean) to a z-score and use a standard normal curve to find probabilities and identify unusual sample means.

Normal Model Applet Useful Again

We will again use the same Normal Distribution Calculator.

EXAMPLE Surprising heights for individual basketball players

Suppose we have a population of adult male basketball players and we know their heights: the mean height is cm and the standard deviation of their heights is cm. The heights are normally distributed, which is often the case with body measurements.

Would it be surprising to find a randomly chosen player from this population with a height of 195 cm?

We can answer this question by computing the probability that a randomly chosen player from this population has height greater than 195 cm, P(X 195).

1. Heights are normally distributed in the population, so we can use the Standard Normal Curve to estimate the probability.

2. Convert the interval X 195 to an interval of z-scores.

Recall that the z-score of an X-value is the number of standard deviations that value is away from the mean. The formula is

So the z-score of X = 195 is

This tells us that the interval of X-values “X > 195” corresponds to the interval of Z-values “Z 0.69".

3. Find P(Z 0.69) by finding the area associated with the interval Z 0.69.

We used the applet given above to find the area to the right of Z = 0.69.

So we have found that P(X 195) = P(Z 0.69) = 0.2451.

Conclusion: This probability is fairly large (almost 25%). We conclude that it would be not be surprising to find a randomly chosen individual basketball player from this population with a height of 195 cm.

EXAMPLE Surprising heights for samples of basketball players

As before, suppose the heights of individual players are normally distributed with cm and cm.

Would it be surprising to find a randomly chosen team of 25 players with a mean height of 195 cm?

This time the probability model is a sampling distribution. Here’s what we know about this sampling distribution when n=25 players:

The distribution of sample means is normal, even though our sample size is less than 30, because we know the distribution of individual heights is normal. (If the individual heights were not normally distributed, we would need a larger sample size before using a normal model for the sampling distribution.)

The mean of the sampling distribution is 195 cm, the same as the mean of the individual heights.

The standard deviation of the sampling distribution is . As before, we also call the standard deviation of the sampling distribution the standard error.

Now we can answer the question by computing the probability that a randomly chosen player from this population has height greater than 195 cm, P(X > 195).

1. Sample means are normally distributed so we can use the Standard Normal Curve to estimate the probability.

2. Convert the interval to an interval of z-scores.

Note that the z-score is the number of standard errors the sample mean is from . So the z-score calculation for the sampling distribution uses mean and standard error .

In symbols this is .
For this problem

Therefore, the interval corresponds to the interval .

3. Find by finding the area associated with the interval .

So we have found that .

Conclusion: This probability is very low (much, much less than 1%). We conclude that it would be very surprising to find a random sample of 25 players from this population of basketball players with a mean height of 195 cm.

Comment

It’s interesting to notice that the height cutoff we used in these two examples is the same (195 cm). When considering the individual, we concluded that finding a randomly chosen individual with height of 195 cm would not be surprising. However, when we considered the team, we concluded that it would be very surprising to find a random sample of 25 players with a mean height of 195 cm. This makes sense because as sample size grows, variability shrinks (here we considered a sample of size 1 versus a sample of size 25).

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