7) A final exam in Statistics has a mean of 73 with a standard deviation of 7.73. Assume that a random sample of 24 students is selected and the mean test score of the sample is computed. What percentage of sample means are less than 70?
To find the percentage of sample means that are less than 70, we need to calculate the z-score for the sample mean of 70 using the given information of the population mean and standard deviation.
The formula to calculate the z-score is:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
In this case:
Sample mean = 70
Population mean = 73
Standard deviation = 7.73
Sample size = 24
Substituting these values into the formula, we get:
z = (70 - 73) / (7.73 / sqrt(24))
Calculating the expression inside the square root first:
sqrt(24) ≈ 4.899
Now, we can substitute this value and calculate the z-score:
z = (70 - 73) / (7.73 / 4.899)
z = -3 / (7.73 / 4.899)
z ≈ -1.599
Now that we have the z-score, we can use a standard normal distribution table or a calculator to find the percentage of sample means that are less than -1.599.
Using a standard normal distribution table, we can look up the area to the left of -1.599, which is approximately 0.0548.
So, the percentage of sample means that are less than 70 is approximately 5.48%.