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Chapter 5: Q14E (page 307)

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 mean and is shown below.

μX=xpx=10.2+20.3+30.2+40.2+50.1=2.7Em=Empm=1×0.04+1.5×0.12+2×0.17+2.5×0.20+3×0.20+0.04×0.18+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 of μandEmis 2.7 each, somis an unbiased estimator of.

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Most popular questions from this chapter

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.

Plastic fill process. University of Louisville operators examined the process of filling plastic pouches of dry blended biscuit mix (Quality Engineering, Vol. 91, 1996). The current fill mean of the process is set at μ= 406 grams, and the process fills standard deviation is σ= 10.1 grams. (According to the operators, “The high level of variation is since the product has poor flow properties and is, therefore, difficult to fill consistently from pouch to pouch.”) Operators monitor the process by randomly sampling 36 pouches each day and measuring the amount of biscuit mix in each. Considerx the mean fill amount of the sample of 36 products. Suppose that on one particular day, the operators observe x= 400.8. One of the operators believes that this indicates that the true process fill mean for that day is less than 406 grams. Another operator argues thatμ = 406, and the small observed value is due to random variation in the fill process. Which operator do you agree with? Why?

Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 2.5 (p. 72). The researchers obtained a random sample of 106 social robots through a Web search and determined the number that was designed with legs but no wheels. Let p^represent the sample proportion of social robots designed with legs but no wheels. Assume that in the population of all social robots, 40% are designed with legs but no wheels.

a. Give the mean and standard deviation of the sampling distribution of p^.

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

c. Find P(p^>.59).

d. Recall that the researchers found that 63 of the 106 robots were built with legs only. Does this result cast doubt on the assumption that 40% of all social robots are designed with legs but no wheels? Explain.

Question: The standard deviation (or, as it is usually called, the standard error) of the sampling distribution for the sample mean, x¯ , is equal to the standard deviation of the population from which the sample was selected, divided by the square root of the sample size. That is

σX¯=σn

  1. As the sample size is increased, what happens to the standard error of? Why is this property considered important?
  2. Suppose a sample statistic has a standard error that is not a function of the sample size. In other words, the standard error remains constant as n changes. What would this imply about the statistic as an estimator of a population parameter?
  3. Suppose another unbiased estimator (call it A) of the population mean is a sample statistic with a standard error equal to

σA=σn3

Which of the sample statistics,x¯or A, is preferable as an estimator of the population mean? Why?

  1. Suppose that the population standard deviation σis equal to 10 and that the sample size is 64. Calculate the standard errors of x¯and A. Assuming that the sampling distribution of A is approximately normal, interpret the standard errors. Why is the assumption of (approximate) normality unnecessary for the sampling distribution ofx¯?

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviours of CPA corporations, Exercise 1.28 (p. 51). Corporate sustainability, recall, refers to business practices designed around social and environmental considerations. The level of support senior managers has for corporate sustainability was measured quantitatively on a scale ranging from 0 to 160 points. The study provided the following information on the distribution of levels of support for sustainability:μ=68 , σ=27. Now consider a random sample of 45 senior managers and let x represent the sample mean level of support.

a. Give the value of μx¯, the mean of the sampling distribution ofx¯ , and interpret the result.

b. Give the value ofσx¯ , the standard deviation of the sampling distribution of x¯, and interpret the result.

c. What does the Central Limit Theorem say about the shape of the sampling distribution ofx¯ ?

d. Find Px¯>65.
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