STAT 301 - Business Statistics
Lecture Review #10 - Sampling Distributions

Review questions:

1) What is a sampling distribution in statistics?

2) What is the law of large numbers?

3) What does the central limit theorem tell us about a sampling distribution?

4) According to the law of large numbers, what is the standard deviation (or standard error) …
     4a) … associated with a sample mean?
     4b) … associated with a sample proportion?

Computational exercises:

Remember that you should show your work on these problems; do not simply copy my solutions.

1) Automated machinery at the factory that makes Chocolate Frosted Sugar Bombs cereal is used to fill the cereal boxes. The amount of cereal put into a (nominally) 16 ounce box of cereal is normally distributed, with a mean of 16.2 ounces and a standard deviation of .1 ounces.

a) What percentage of the individual boxes of cereal weigh less than 16 ounces?

b) The cereal is distributed to wholesalers in cartons of 16 boxes. What is the probability that the average cereal box in a carton weighs less than 16 ounces?

c) What is the probability that the average box in a carton weighs 16.25 ounces or more?

2) A recent anthropological study of the aboriginal dwarf pygmies living on the island of Kafoonistan has shown that adult males in the tribe have an average height of 60", with a standard deviation of 4".

a) What percentage of the men of the tribe are over six feet (72") tall?

b) Seven of the men, selected at random, are measured. What is the probability that their average height is 58 inches or less?

c) A random sample of 42 (of course) men is taken. What is the probability that their average height is between 59.8 and 60.2 inches?

3) Suppose a fair coin is tossed 10,000 times. What is the probability of obtaining …

a) … between 4900 and 5100 'heads'?

b) … more than 5500 'heads'?

4) In a typical Major League Baseball season, a top player will have approximately 600 times at bat. However, the 2020 season was anything but typical — because of COVID threats the season was considerably shortened, and so a top player might have only approximately 200 times at bat. At the beginning of the season this prompted some speculation that we might see a "400 hitter" that year.

For those not into spiritual things: a "400 hitter" is someone whose batting average is 0.400 or higher — that is, the proportion of times in which they get a hit is at least 40%. This is exceedingly rare and special — the last time it happened was in 1941 (when Ted Williams batted .406). But the thought was that the shorter season would allow for larger sampling error in batter performance. Let’s work out some probabilities associated with hitting 400.

a) Suppose a player is a true .300 hitter — that is, he has a 30% chance of getting a hit, every time at bat. If he has 600 at bats over the course of a season, what is the probability of hitting .400 or higher?

b) Now suppose that there are only 200 times at bat in a season. Now what is the probability that this same player hits .400 or higher?

SOLUTIONS:
1a) z = -2, prob = .0228
1b) z = -8, prob = approx. 0
1c) z = 2, prob = .0228
2a) prob = .0013
2b) prob = .093
2c) prob = .254 (not huge – but a pretty hefty probability for such a narrow range of possibilities)
3a) .9544
3b) approx. 0
4a) z = 5.345, prob = 4.515 E-08 (that is 0.00000004515). That’s really rare.
4b) z = 3.086, prob = 0.0010. That’s still pretty rare – but it at least something that’s plausible. (And remember: there are many players, each of whom would have this probability.)