Answer:
There is a probability P=0.86 that the sample mean lies within $100 of the population mean.
This is estimated knowing the sample standard deviation, that depends on the sample size and the standard deviation of the population. If the sample is taken randomly and its representative of the population, the sample mean is expected to be around the population mean, following a sample distribution which width depends on the sample standard deviation.
Step-by-step explanation:
In this question we know the standard deviation of the population, and we need to know the deviation of a sample of n=400.
The standard deviation for a sample is calculated as:
[tex]\sigma_M=\frac{\sigma}{\sqrt{N}}=\frac{1350}{\sqrt{400}}=\frac{1350}{20}= 67.5[/tex]
Assuming the sampling distribution is normal, we can calculate the z-value and estimate the probability:
[tex]z=\frac{M-\mu}{\sigma_M}= \frac{100}{67.5}= 1.48[/tex]
[tex]P(-1.48<X<1.48)=0.86[/tex]
There is a probability P=0.86 that the sample mean lies within $100 of the population mean.
