Question 3
A researcher selects a sample of 324 participants from a normally distributed population. He determines the mean of the
population is 105 with a standard deviation of 36 . He calculates that of the sample means in his sampling distribution will fall
between a sample mean of _
(A) 101 and 109
(B) 103 and 107
© 105 and 109
(D) 103 and 109

To solve this, we can use the Central Limit Theorem, which tells us that the sample means will be normally distributed, provided the sample size is sufficiently large (which it is in this case). The expected mean of the sample means is the same as the population mean (105), and the standard error (SE) can be calculated using the formula , where is the population standard deviation and is the sample size.

So, the SE would be .

Now, for a 95% confidence interval, we look at approximately 1.96 standard deviations from the mean. Thus, we calculate:

Lower Limit:

Upper Limit:

So, about 95% of the sample means will fall between approximately 101 and 109.

The correct answer is (A) 101 and 109.

For a little extra context, did you know that the concept of sampling distributions dates back to the works of statisticians like Pierre-Simon Laplace in the 18th century? They recognized the importance of just how powerful averages can be when we gather data.In real-life situations, this statistical approach can be incredibly valuable. For instance, businesses may use sample data to gauge customer satisfaction before rolling out new initiatives, ensuring they hit the mark and resonate with their audience!