a random sample of 100 college students is taken from the student body of a large university. assume that a population mean of 20 hours and a population standard deviation of 15 hours describes the weekly study estimates for the entire student body. About 68 percent of the sample means in this sampling distribution should be between?
68% are approximately within mean ± 1 SD
Z = ±1 = (score - mean)/SEm
SEm = SD/√(n-1) = approx. SD/10
Insert values in above Z equation to get scores.
consider a normal distribution with mean 27 and standard deviation 6. What is the probability a value selected at random from this distribution is great than 27?
To determine the range within which 68 percent of the sample means lie, we need to use the concept of the standard error.
The standard error measures the average amount of variation we would expect in the sample means due to random sampling. It is calculated by dividing the population standard deviation by the square root of the sample size, also known as the sample's standard deviation (denoted as σ).
In this case, the population standard deviation is 15 hours, and the sample size is 100 (n = 100). Therefore, the standard error (SE) is calculated as follows:
SE = σ / √n
= 15 / √100
= 15 / 10
= 1.5
Now, since we are considering a normal distribution, we can use the 68-95-99.7 rule, also known as the empirical rule, which states that approximately:
- 68 percent of the sample means will fall within 1 standard error (SE) of the population mean.
- 95 percent of the sample means will fall within 2 SEs of the population mean.
- 99.7 percent of the sample means will fall within 3 SEs of the population mean.
Hence, about 68 percent of the sample means in this sampling distribution should fall within 1.5 hours of the population mean of 20 hours.
To determine the range, we can calculate the lower and upper limits by adding and subtracting 1.5 hours from the population mean:
Lower limit = 20 - 1.5 = 18.5 hours
Upper limit = 20 + 1.5 = 21.5 hours
Therefore, about 68 percent of the sample means should be between 18.5 and 21.5 hours.