Question

In one region of the country, the mean length of stay in hospitals is 5.5 days with standard deviation 2.6 days. Because many patients stay in the hospital for considerably more days, the distribution of length of stay is strongly skewed to the right. Consider random samples of size 100 taken from the distribution with the mean length of stay, x, recorded for each sample. Which of the following is the best description of the sampling distribution of x ?Strongly skewed to the right with mean 5.5 days and standard deviation 2.6 daysStrongly skewed to the right with mean 5.5 days and standard deviation 0.26 dayStrongly skewed to the right with mean 5.5 days and standard deviation 0.026 dayApproximately normal with mean 5.5 days and standard deviation 2.6 daysApproximately normal with mean 5.5 days and standard deviation 0.26 day

Answer 1

The best description of the sampling distribution of x would be approximately normal with mean 5.5 days and standard deviation 0.26 day.

According to the central limit theorem, when a sufficiently large sample is taken from a population with any distribution, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution. In this case, a sample size of 100 is large enough to assume that the sampling distribution of the sample mean will be approximately normal.

The standard deviation of the sampling distribution of the sample mean can be estimated using the formula: standard deviation of sample mean = standard deviation of population / square root of sample size. Substituting the given values, we get:

standard deviation of sample mean = 2.6 / sqrt(100) = 0.26

Therefore, the best description of the sampling distribution of x would be approximately normal with mean 5.5 days and standard deviation 0.26 day.