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Chapter 5: Q14E (page 307)

Question: Refer to Exercise 5.5, in which we found the sampling distribution of the sample median. Is the median an unbiased estimator of the population mean m?

Short Answer

Expert verified

Yes, the median is an unbiased estimator of the population mean “m”.

Step by step solution

01

List of probabilities

The list of the probabilities found in Exercise 5.5 corresponding to the respective mean is shown below.

Mean

Probability

1

0.04

1.5

0.12

2

0.17

2.5

0.20

3

0.20

3.5

0.14

4

0.08

4.5

0.04

5

0.01

02

Determination of the biasedness of the median

The calculation of the meanandis shown below.

μx=xp(x)=1(0.2)+2(0.3)+3(0.2)+4(0.2)+5(0.1)=2.7

E(m)=Emp(m)=1×0.04+1.5×0.12+2×0.17+2.5×0.20+3×0.20+3.5×0.14+4×0.08+4.5+0.04+5×0.01=0.04+0.18+0.34+0.5+0.6+0.49+0.32+0.18+0.05=2.7

As the value oflocalid="1661429803872" μandlocalid="1661429812869" E(m)is 2.7 each, solocalid="1661429822357" mis an unbiased estimator oflocalid="1661429831113" μ.

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Most popular questions from this chapter

Refer to Exercise 5.3.

  1. Show thatx⇀is an unbiased estimator of.
  2. Findσx2.
  3. Find the probability that x will fall within2σxofμ.

Dentists’ use of laughing gas. According to the American Dental Association, 60% of all dentists use nitrous oxide (laughing gas) in their practice. In a random sample of 75 dentists, let p^represent the proportion who use laughing gas in practice.

a. Find Ep^.

b. Find σp^.

c. Describe the shape of the sampling distribution of p^.

d. Find Pp^>0.70.

A random sample of n=900 observations is selected from a population with μ=100andσ=10

a. What are the largest and smallest values ofx¯ that you would expect to see?

b. How far, at the most, would you expect xto deviate from μ?

c. Did you have to know μto answer part b? Explain.

Surface roughness of pipe. Refer to the Anti-CorrosionMethods and Materials(Vol. 50, 2003) study of the surface roughness of oil field pipes, Exercise 2.46 (p. 96). Recall that a scanning probe instrument was used to measure thesurface roughness x(in micrometers) of 20 sampled sectionsof coated interior pipe. Consider the sample mean,X¯.

  1. Assume that the surface roughness distribution has a mean of = 1.8 micrometers and a standard deviation of = .5 micrometer. Use this information to find theprobability thatexceeds 1.85 micrometers.
  2. The sample data are reproduced in the following table.Compute.
  3. Based on the result, part b, comment on the validity ofthe assumptions made in part a.

1.72

2.50

2.16

2.13

1.06

2.24

2.31

2.03

1.09

1.40

2.57

2.64

1.26

2.05

1.19

2.13

1.27

1.51

2.41

1.95

Soft-drink bottles. A soft-drink bottler purchases glass bottles from a vendor. The bottles are required to have an internal pressure of at least 150 pounds per square inch (psi). A prospective bottle vendor claims that its production process yields bottles with a mean internal pressure of 157 psi and a standard deviation of 3 psi. The bottler strikes an agreement with the vendor that permits the bottler to sample from the vendor’s production process to verify the vendor’s claim. The bottler randomly selects 40 bottles from the last 10,000 produced, measures the internal pressure of each, and finds the mean pressure for the sample to be 1.3 psi below the process mean cited by the vendor.

a. Assuming the vendor’s claim to be true, what is the probability of obtaining a sample mean this far or farther below the process mean? What does your answer suggest about the validity of the vendor’s claim?

b. If the process standard deviation were 3 psi as claimed by the vendor, but the mean were 156 psi, would the observed sample result be more or less likely than in part a? What if the mean were 158 psi?

c. If the process mean were 157 psi as claimed, but the process standard deviation were 2 psi, would the sample result be more or less likely than in part a? What if instead the standard deviation were 6 psi?
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