Among a random sample of 500 college students, the mean number of hours worked per week at non-college related jobs is 14.6. This mean lies 0.4 standard deviations below the mean of the sampling distribution. If a second sample of 500 students is selected, what is the probability that for the second sample, the mean number of hours worked will be less than 14.6?
A. 0.5
B. 0.6179
C. 0.6554
D. 0.3446
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
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.
o.5
b)0.6179
o.6554
0.3446
To answer this question, we will use the concept of the sampling distribution of the sample mean. The sampling distribution represents the distribution of sample means that we would expect to obtain if we were to take multiple samples from the same population.
In this case, we are given that the mean of the sampling distribution is 14.6 and that it lies 0.4 standard deviations below the mean of the sampling distribution. This information allows us to determine the standard deviation of the sampling distribution.
The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:
Standard error = population standard deviation / square root of sample size
Since the standard deviation of the population is not given, we cannot calculate the exact standard error. However, we can use the information given to estimate the standard error.
We know that the mean of the sampling distribution (14.6) is 0.4 standard deviations below the mean of the sampling distribution. This means that the difference between the mean of the sampling distribution and the mean of the population is 0.4 times the standard deviation of the sampling distribution.
So, if we define the standard deviation of the sampling distribution as s, we can write the equation:
0.4s = 14.6 - mean of the population
Since the mean of the population is not given, we cannot solve this equation exactly. However, we can use the fact that the mean of the sampling distribution is equal to the mean of the population to simplify the equation:
0.4s = 0
This equation tells us that the standard deviation of the sampling distribution is 0. By definition, this means that every sample from the population will have the same mean as the population mean. In other words, the mean of the second sample will also be 14.6.
To find the probability that the mean number of hours worked for the second sample is less than 14.6, we need to calculate the area to the left of the mean (14.6) under the normal distribution curve.
Since the second sample mean will be exactly equal to 14.6, the probability that it is less than 14.6 is 0.
Therefore, the correct answer is A. 0.