A normal population has a mean of 62 and a standard deviation of 5. You select a sample of 40.
Compute the probability that the sample mean is Between 61 and 63?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability.
To compute the probability that the sample mean is between 61 and 63, we need to use the concept of the sampling distribution of the sample mean.
The sampling distribution of the sample mean follows a normal distribution if the population from which the samples are drawn is normally distributed or if the sample size is large enough. According to the Central Limit Theorem, if the population is not normally distributed but the sample size is large (typically n ≥ 30), the sampling distribution of the sample mean will still be approximately normal.
In this case, we are given that the population mean (μ) is 62 and the standard deviation (σ) is 5. The sample size (n) is 40.
To calculate the probability that the sample mean is between 61 and 63, we first need to calculate the standard deviation of the sampling distribution of the sample mean (also known as the standard error). The formula for the standard error is:
Standard Error (SE) = σ / √n
Where σ is the population standard deviation and n is the sample size.
Substituting the given values, we have:
SE = 5 / √40 ≈ 0.7906
Next, to find the z-scores corresponding to the sample mean values of 61 and 63, we will use the formula:
z = (x - μ) / SE
For x = 61:
z1 = (61 - 62) / 0.7906 ≈ -1.26
For x = 63:
z2 = (63 - 62) / 0.7906 ≈ 1.26
Now, we can use a standard normal distribution table or a calculator to find the probability corresponding to these z-scores.
Using a standard normal distribution table, we find that the cumulative probability for a z-score of -1.26 is approximately 0.1038, and the cumulative probability for a z-score of 1.26 is approximately 0.8962.
To find the probability that the sample mean is between 61 and 63, we subtract the cumulative probability corresponding to the lower z-score from the cumulative probability corresponding to the higher z-score:
Probability = 0.8962 - 0.1038 ≈ 0.7924
Therefore, the probability that the sample mean is between 61 and 63 is approximately 0.7924, or 79.24%.