Final answer:
The mean of the sampling distribution of X is 178.5 centimeters with a standard deviation of 1.28 centimeters. The expected number of sample means between 176.4 and 179.6 centimeters can be calculated using z-scores and the area under the normal curve. The expected number of sample means below 176.0 centimeters can also be calculated using z-scores and the area under the normal curve.
Step-by-step explanation:
(a) The mean of the sampling distribution of X is equal to the mean of the population, which is 178.5 centimeters. The standard deviation of the sampling distribution of X, also known as the standard error, can be calculated using the formula:
Standard Error = Population Standard Deviation / Square Root of Sample Size
Therefore, the standard deviation of the sampling distribution of X is 6.4 / square root of 25 = 1.28 centimeters.
(b) To determine the expected number of sample means that fall between 176.4 and 179.6 centimeters inclusive, we need to find the z-scores corresponding to these values. The formula to calculate the z-score is:
z = (X - Mean) / Standard Deviation
Using the z-score table or a calculator, we can find that the z-score for 176.4 centimeters is -1.23 and the z-score for 179.6 centimeters is 0.93. Then, we can find the area under the normal curve between these z-scores, which gives us the probability of a sample mean falling between those values. Lastly, we multiply this probability by the total number of samples (400) to find the expected number of sample means.
(c) To determine the expected number of sample means falling below 176.0 centimeters, we need to find the z-score for this value (-2.14) and calculate the area under the normal curve to the left of this z-score. We then multiply this probability by the total number of samples (400) to find the expected number of sample means.