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Chapter 5: Q54SE (page 322)

Question:A random sample of 40 observations is to be drawn from a large population of measurements. It is known that 30% of the measurements in the population are 1s, 20% are 2s, 20% are 3s, and 30% are 4s.

a. Give the mean and standard deviation of the (repeated) sampling distribution ofx¯, the sample mean of the 40 observations.

b. Describe the shape of the sampling distribution ofx¯. Does youranswer depend on the sample size?

Short Answer

Expert verified
  1. The mean and standard deviation ofx¯ are 2.5 and 0.19.
  2. The shape of the sampling distribution ofx¯ is normal.

Step by step solution

01

Given Information

The sample size is 40.

According to the question, the probability distribution of x is given by

02

(a) Compute the mean and standard deviation

\σn=1.2040=0.19The mean is computed as

EX=1×0.3+2×0.2+3×0.2+4×0.3=0.3+0.4+0.6+1.2=2.5

The standard deviation is computed as

d=∑x-Ex2px=1-2.52×0.3+2-2.52×0.2+3-2.52×0.2+4-2.52×0.3=1.45=1.20

The standard deviation of x¯is

σn=1.2040=0.19

Therefore, the mean and standard deviation of x¯are 2.5 and 0.19.

03

(b) Shape of the distribution of x¯

If n is large, then the distribution of x¯follows the normal distribution with a mean E(X) and standard deviationσn

Therefore, the shape of the sampling distribution isnormal.

Here, the number of samples is 40, which is greater than 30.

Therefore, the answer depends on the sample size.

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Most popular questions from this chapter

A random sample of n = 80 measurements is drawn from a binomial population with a probability of success .3.

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