Download will be between 67 and 69

Central Limit Theorem Worksheet
1. Suppose 𝑥 has a distribution with a mean of 20 and a standard deviation of 3.
Random samples of 𝑛 = 36 are drawn.
a) Find the mean and standard deviation for an 𝑥̅ distribution
b) Find the z – value corresponding to 𝑥̅ = 19
c) Find 𝑃(𝑥̅ < 19)
2. Suppose 𝑥 has a distribution with a mean of 30.4 and a standard deviation of 4.1.
Random samples of 𝑛 = 50 are drawn.
a) Find the mean and standard deviation for an 𝑥̅ distribution
b) Find 𝑃(29.5 ≤ 𝑥̅ ≤ 31.5)
3. Coal is carried from a mine in West Virginia to a power plant in New York in hopper
cars on a long train. The automatic hopper car loader is set to put 75 tons of coal
into each car. The actual weights of coal loaded into each car is normally distributed,
with mean 𝜇 = 75 tons and standard deviation 𝜎 = 0.8 ton.
What is the probability that 20 cars chosen at random will have a mean load weight
𝑥̅ less than 74.5 tons of coal ?
4. The heights of 18 – year – old men are approximately normally distributed, with
mean 𝜇 = 68 inches and a standard deviation 𝜎 = 3 inches.
If nine 18 – year – old males are selected, what is the probability that the mean
height 𝑥̅ will be between 67 and 69 inches ?
5. Let 𝑥 represent the dollar amount spent on supermarket impulse buying in a 10 –
minute (unplanned) shopping interval. The mean of the 𝑥 distribution is about $20
and the standard deviation is about $7.
a) Find the mean and standard deviation for a random sample of 𝑛 = 100 shoppers
of the 𝑥̅ distribution.
b) What is the probability that 𝑥̅ is between $18 and $22 ?