Download The Central Limit Theorem: Homework

The Central Limit Theorem: Homework EXERCISE 2 Determine which of the following are true and which are false. Then, in complete sentences, justify your answers. a. When the sample size is large, the mean of is approximately equal to the mean of X. As random variables, the means are the same. When the sample is large, the probability
distribution for the sample mean becomes increasingly concentrated around the shared
mean. b. When the sample size is large, is approximately normally distributed. This is true by the Central Limit Theorem. c. When the sample size is large, the standard deviation of is approximately the same as the standard deviation of X. This is not true. The standard deviation of the sample mean becomes increasingly small
as the sample size grows. EXERCISE 6 Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from 2 to 6 pounds. We randomly survey 64 homes with children. a.
b.
c.
d.
e.
f.
g.
In words, X = the weight of open boxes of cereal in a home with children. X ∼ U(2,6) µx = 4 σx = 1.15 In words, ΣX = the total weight of open boxes in homes from the sample. ΣX ∼ N(64(4),8(1.15)) = N(256,9.2) Find the probability that the total weight of open boxes is less than 250 pounds. normalcdf(-10,250,256,9.2) = .257 Find the 35th percentile for the total weight of open boxes of cereal. invnormal(.35,256,9.2) = 252.5
EXERCISE 8 According to the Internal Revenue Service, the average length of time for an individual to complete (record keep, learn, prepare, copy, assemble and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is 2 hours. Suppose we randomly sample 36 taxpayers. a. In words, X = the time it takes an individual to complete form 1040. b. In words, = the average time it takes the people in the sample to complete form 1040. c.
∼ N(10.53,2/6)=N(10.53,1/3) d. Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences. I would be very surprised because 12 is more than 3 standard deviations from the mean
of the sample average. To quantify this: normalcdf(12,100,10.53,1/3) = .000005. e. Would you be surprised if one taxpayer finished his Form 1040 in more than 12 hours? In a complete sentence, explain why. Not so much, this is less than a single standard deviation from the mean. The
distribution is unkown, but an estimate might be normalcdf(12,100,10.53,2) = .231 EXERCISE 10 The attention span of a two year‐old is exponentially distributed with a mean of about 8 minutes. Suppose we randomly survey 60 two year‐olds. a. In words, X = the attention span of a two year old. b. X ∼ Exp(1/8) c. In words, = the average attention span of the children in the sample. d.
∼ N(8,8/sqrt(60))=N(8,1.03) e. Before doing any calculations, which do you think will be higher: •
•
the probability that an individual attention span is less than 10 minutes or the probability that the average attention span for the 60 children is less than 10 minutes? Why? Since the region less than 10 includes the mean 8, and the sample average is approaching the mean with higher sample sizes, the probability for the average should be higher. f.
Calculate the probabilities in part (e). P(X<10) = .713 and P(
< 10) = .974 g. Explain why the distribution for is not exponential. The central limit theorem says that no matter what the original distribution, the
distribution of sample averages will become increasingly normal as the sample size
increases.