Question

Old Faithful, in Yellowstone National Park, has a mean time between eruptions of 92 minutes. with a standard deviation of 18 minutes. You are going to collect a sample of 45 time intervals between eruptions and will compute the sample mean.

a. What is the mean of the sample mean?
b. What is the standard deviation of the sample mean?
c. This distribution of interval times is not known to be normal. What conditions must be satisfied for the sampling distribution of χ to be assumed to be approximately normally distributed? Are those met?
d. Assuming the conditions are met, what interval contains the sample mean for 95% of all samples of size 45 ? Show work to receive credit.
e. Assuming the conditions are met, what is the probability that the sample mean is less than 95.86 minutes? Show work to receive credit.
f. Describe the effect on both the mean and standard deviation of the sampling distribution of χ if you increase the sample size. For example, from 45 to 150 ?

Answer 1

The mean of the sample mean is equal to the population mean. The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size. The conditions for the sampling distribution of the sample mean to be approximately normally distributed are: the sample size should be large enough, the population distribution should be approximately normal or the sample size should be large enough for the Central Limit Theorem to apply, and the samples should be independent of each other.

Step-by-step explanation:

a. The mean of the sample mean is equal to the population mean, which is 92 minutes.

b. The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size. In this case, it is 18 minutes divided by the square root of 45.

c. The conditions that must be satisfied for the sampling distribution of the sample mean to be approximately normally distributed are: the sample size should be large enough (typically n > 30), the population distribution should be approximately normal or the sample size should be large enough for the Central Limit Theorem to apply, and the samples should be independent of each other. Since the sample size in this case is 45, it meets the condition for the Central Limit Theorem to apply.

d. To find the interval that contains the sample mean for 95% of all samples of size 45, we need to find the z-score corresponding to a 95% confidence level, which is 1.96. We then multiply the standard deviation of the sample mean by this z-score and add it to the sample mean to get the upper bound of the interval, and subtract it from the sample mean to get the lower bound of the interval.

e. To find the probability that the sample mean is less than 95.86 minutes, we need to calculate the z-score corresponding to this sample mean, and then find the corresponding cumulative probability in the standard normal distribution table.

f. If you increase the sample size from 45 to 150, the mean of the sampling distribution of the sample mean will remain the same, but the standard deviation will decrease. The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size, so as the sample size increases, the standard deviation of the sample mean decreases.