Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
What do you mean by unusual? P < .10? P < .05? P < .01?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion for that Z score.
To determine whether a sample mean of 115 or more is unusual, we need to calculate the z-score associated with this value and see if it falls within the range of typical values.
We can find the z-score using the formula:
z = (x - μ) / (σ / √n)
Where:
x = sample mean (115)
μ = population mean (100)
σ = population standard deviation (15)
n = sample size (3)
Substituting the values into the formula:
z = (115 - 100) / (15 / √3)
z = 15 / (15 / √3)
z = √3
The z-score for a sample mean of 115 or more is √3.
To determine if this z-score is unusual, we can compare it to the standard normal distribution (sometimes called the Z-distribution). In the standard normal distribution, most values fall within a range of -3 to 3.
Since the z-score (√3) falls within this range, we can conclude that a sample mean of 115 or more is not unusual in this case. It is within the typical range of possibilities for samples of size 3 from a normal distribution with a mean of 100 and a standard deviation of 15.