1. A statistics department has found that the final exam grades for its introductory course are well approximated by a normal distibution with mean (mu) = 74.8% and standard deviation (sigma) = 8.2%.
a. Is it unusual for a student to score below a 70% on this final exam? (Justify) (be sure to include all notation)
b. Is it unusual for a sample of 6 students to have an average score below a 70% of the final exam? (Justify and mention what theorem is being utilized here) (be sure to show all notation)
c. Is it unusual for a sample of 14 students to have an average score below a 70% on this final exam? (Justify and mention what theorem is being utilized here) (be sure to show all notation)
d. Suppose 6 students took the final exam. The average score was calculated to be 70%. What inference would a statistician make?
e. Suppose 14 students took the final exam. The average score was calculated to be 70%. What inference would a statistician make?

2. A fruit-filled cereal is packaged in boxes that contain an average of (mu) = 450 grams of fruit and for which the standard deviation is (sigma) = 12 grams.
a. What is the probability that a single box contains at least 454 grams of fruit?
b. A sample of 36 boxes is randomly selected.
i. Write the notation for the distribution of xbar.
ii. Find the probability that the sample mean will be at least 454 grams of fruit. Is it unusual for a sample of 36 cereal boxes of this type to have a sample average amount of fruit more than 454 grams?
iii. Find the probability that the sample mean amount of fruit will be between 446 and 455 grams.
iv. Find the probability that the sample mean amount of fruit is not between 446 and 455 grams. Is it unusual for a sample mean amount of fruit to not be between 446 and 455 grams?
v. Find the z-score for the sample mean values xbar = 454, 446, and 455 grams

3. A standard mullet is defined to be a hair cut with the following characteristics: short hair on the top and sides, and long hair in the back (at least 4 inches). An extreme mullet is when the hair in the back exceeds 10 inches. Let X = the length of a standard 4 to 10 inch mullet. Suppose X is uniformly distributed from 3 to 10 inches with a standard deviation of (sigma) = 4 inches. a. What is the probability that a randomly selected standard mullet is more than 8 inches? Also show the appropriate curve with shading and the notation for the distribution of X.
b. What is the probability that a sample of 49 mullets has a mean length more than 8 inches. Also show the appropriate curve with shading and the notation for the distrbution of Xbar.

4i. A value has a z-score of z = 4.55. The value of x associated with this z-score is (a) much smaller than the mean (b) much larger than the mean (c) equal to the mean (d) slightly smaller than the mean (e) slightly larger than the mean (f) cannot be determined becauce we do not know the mean and standard deviation.
4ii. A value has a z-score of z = 4.55. Approximately, what is the area to the left of z = 4.55.

5. A distribution of xbar is normally distributed with mean (mu) = 100 and standard deviation = (sigma)/sqrt(n) = 50/sqrt(25). The area to the left of a particular value of xbar is 0.01.
What is this particular value of xbar?

6. Suppose X is normally distrbuted. Describe the difference betwen the mean of individual observations and the mean of sample averages. Describe the difference between the spread of the individual observations and the spread of the sample averages (be specific)