Assume that a population is normally distributed with a mean of 100 and a standard deviation of 20. Would a sample mean of 115 or more be considered unusual? Why or why not?
To determine if a sample mean of 115 or more is considered unusual, we can use the concept of z-scores.
A z-score measures the number of standard deviations a particular value is from the mean of a distribution. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- x is the sample mean (115 in this case)
- μ is the population mean (100 in this case)
- σ is the population standard deviation (20 in this case)
Let's calculate the z-score for a sample mean of 115:
z = (115 - 100) / 20
z ≈ 0.75
Now, we need to determine if this z-score is considered unusual. Generally, a z-score greater than 2 or less than -2 is considered unusual. It means that the value is more than 2 standard deviations away from the mean.
In this case, the z-score of 0.75 is less than 2, which suggests that a sample mean of 115 or more is not considered unusual in this population. It means that the sample mean of 115 falls within a reasonable range of values given the population mean and standard deviation.
So, based on the given population data, a sample mean of 115 or more would not be considered unusual.