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Answer :
To find the mean and standard error of the mean for the sampling distribution, let's start by breaking down the information given and applying some relevant statistical concepts.
Mean of the Sampling Distribution:
The mean of the sampling distribution of the sample mean is the same as the mean of the population.
Given that the mean of the population is [tex]\mu = 157[/tex] lb, the mean of the sampling distribution is also [tex]\mu_{\bar{x}} = 157[/tex] lb.
Standard Error of the Mean (SEM):
The standard error of the mean is calculated by dividing the standard deviation of the population by the square root of the sample size [tex]n[/tex].
We have the standard deviation of the population [tex]\sigma = 23[/tex] lb and a sample size of [tex]n = 8[/tex].
The formula for the standard error of the mean (SEM) is:
[tex]SEM = \frac{\sigma}{\sqrt{n}} = \frac{23}{\sqrt{8}}[/tex]
Calculating this gives us:
[tex]SEM = \frac{23}{\sqrt{8}} \approx \frac{23}{2.828} \approx 8.13[/tex] lb
Conclusion:
- The mean of the sampling distribution is [tex]157[/tex] lb.
- The standard error of the mean is approximately [tex]8.13[/tex] lb.
The standard error provides a measure of how much the sample mean would vary from sample to sample, showing that with smaller samples, the variability or 'error' tends to be larger than with larger samples. This calculation is essential when we're trying to understand how representative a sample mean might be of the actual population mean.
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