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?
To determine whether a mean of a sample is unusual, we can use the concept of the standard deviation and Z-scores.
First, let's calculate the standard deviation of the sample mean. The standard deviation of the population (σ) is 15, and since the sample size is 3, the standard deviation of the sample mean (σ_mean) can be calculated as σ / √n, where n is the sample size:
σ_mean = 15 / √3 ≈ 8.66025
Next, let's calculate the Z-score for a sample mean of 115. The Z-score measures the number of standard deviations that a sample mean is away from the population mean. The formula for calculating the Z-score is:
Z = (sample mean - population mean) / σ_mean
Z = (115 - 100) / 8.66025 ≈ 1.73205
Now, we can refer to a Z-table (or use statistical software) to determine the corresponding probability or percentile associated with a Z-score of 1.73205. The Z-table will give us the percentage of data that falls to the left of this Z-score.
Looking it up, we find that the probability corresponding to a Z-score of 1.732 is approximately 0.9582 or 95.82%.
Therefore, a mean of 115 or higher in a sample of size 3 would be considered unusual since approximately 95.82% of the sample means would fall below a mean of 115 in a normally distributed population with a mean of 100 and a standard deviation of 15.