Our researcher friend wishes to know how accurate the sample mean is likely to be if she samples 25 people. To simulate this, in the applet select the following options: **Normal, N=25, Show sample data** (no other ‘show’ options).

Click on

**Draw a sample**, and enter the value shown for

**Last mean**in the first space below. Then click on

**Draw a sample**nine more times and record each mean below.

How many of your 10 sample means fell outside of the range 470 to 530?

Click on **Show obtained means** to see all ten means displayed. Notice how tightly clustered the obtained means are compared to the individual scores.

Click **Draw 100 samples** to see the means for 100 different samples, each of size *N*=25.

Click **Show sampling distribution of the mean** to see how closely the distribution of 100 observed sample means matches the actual distribution of possible means of size *N*=25.

In the box below describe how this sampling distribution of the mean (for *N*=25) compares to the sampling distribution of the mean for *N*=100. Be sure to consider the variability of sample means (i.e., the standard error of the sampling distribution of the means). [Hint: Select **Show sampling distribution of the mean** and compare the sampling distributions for *N*=100 and *N*=25.]

**Q3. What is the probability that a randomly selected sample of N=25 American adults has a mean Life Satisfaction score within 30 points of the population mean?**

First, estimate the answer by examining your ten sample means, the displays of 100 sample means with *N*=25 for each mean, and the sampling distribution of the mean. What is your rough estimate based on these observations?

Now solve for the exact answer. You may use a table of probabilities for the standardized normal distribution or use the *p*–*z* converter to convert from a *z*-score to probability.

#### Show A Hint

The possible means are normally distributed with a mean of 500. If we can find the standard deviation of this distribution, we can find the z score corresponding to 530, and then use the z table or p-z converter to find the probability of observing a sample mean between 500 and 530, and between 500 and 470.

#### Show Another Hint

According to the CLT, the standard error of the mean is equal to the population standard deviation, 100, divided by the square root of the sample size, 25. Thus, the standard error = 100/ (square root of 25) = 100/5 = 20. Using this standard error of 20, you can now convert 470 and 530 (scores that deviate from the mean by 30) into z scores and find their corresponding probabilities.

The exact answer is about 87%. A detailed solution is in the answer section, but try it on your own before consulting the solution.