Question: A simple random sample of size n = 38 is obtained from a population with \(\mu =\) 61 and \(\sigma =14\)

A. Can we use the normal model to calculate probabilities regarding the sample mean? Why or why not? Explain this in sentence form.

B. Assuming the normal model can be used, determine the probability that the sample mean is less than 55. In other words, find \(\Pr \left( \bar{X}<55 \right)\)

C. Assuming the normal model can be used, determine the probability that the sample mean is greater than or equal to 68.6. In other words, find \(\Pr \left( \bar{X}\ge 68.6 \right)\).

D. Assuming that the normal model can be used, describe the sampling distribution of \(\bar{X}\). Include the values for the mean of the sampling distribution \(\bar{X}\) ( \({{\mu }_{{\bar{X}}}}\) ), the standard deviation of the sampling distribution \(\left( {{\sigma }_{{\bar{X}}}} \right)\) and a description of the overall distribution.