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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炉, the sample mean of the 40 observations.

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

Short Answer

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

\蟽n=1.2040=0.19The mean is computed as

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

The standard deviation is computed as

d=鈭�x-Ex2px=1-2.52脳0.3+2-2.52脳0.2+3-2.52脳0.2+4-2.52脳0.3=1.45=1.20

The standard deviation of x炉is

蟽n=1.2040=0.19

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

03

(b) Shape of the distribution of x¯

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

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

Suppose a random sample of n = 25 measurements are selected from a population with mean 渭and standard deviation s. For each of the following values of 渭and role="math" localid="1651468116840" 蟽, give the values of 渭蠂炉 and 蟽蠂炉.

  1. 渭=100,蟽=3
  2. 渭=100,蟽=25
  3. 渭=20,蟽=40
  4. 渭=10,蟽=100

Refer to Exercise 5.3 and find E=(x)=渭. Then use the sampling distribution ofxfound in Exercise 5.3 to find the expected value ofx. Note thatE=(x)=渭.

Question: Refer to Exercise 5.3.

  1. Show thatxis an unbiased estimator of渭.
  2. Find蟽x2.
  3. Find the x probability that x will fall within2蟽xof渭.

Will the sampling distribution of x炉always be approximately normally distributed? Explain

Suppose a random sample of n measurements is selected from a population with u=100mean and variance role="math" localid="1657967387987" 蟽2=100. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample mean.

  1. role="math" localid="1657967260825" n=4
  2. n=25
  3. n=100
  4. n=50
  5. n=500
  6. n=1000
See all solutions

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