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?
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 related to the Z score.
98.099
To find the percentage of sample means that are less than 70, we can use the Central Limit Theorem. According to the Central Limit Theorem, if the sample size is large enough (typically greater than 30) and the population follows a normal distribution, then the distribution of sample means will be approximately normal as well, regardless of the shape of the population distribution.
In this case, we have a sample size of 24, which is not very large but can still be considered as an approximation to normality. Since the population standard deviation is known, we can use the z-score formula to calculate the z-score corresponding to a sample mean of 70.
The formula for the z-score is:
z = (x - μ) / (σ / sqrt(n))
Where:
- x is the sample mean (70 in this case)
- μ is the population mean (73 in this case)
- σ is the population standard deviation (7.73 in this case)
- n is the sample size (24 in this case)
Plugging in the values into the formula:
z = (70 - 73) / (7.73 / sqrt(24))
z = -3 / (7.73 / √24)
Next, we can use a standard normal distribution table or a calculator to find the area under the curve to the left of this z-score. This will give us the percentage of sample means that are less than 70.
Using a standard normal distribution table or calculator, the area under the curve to the left of z = -1 is approximately 0.1587. Therefore, the percentage of sample means less than 70 is approximately 0.1587 * 100% = 15.87%.