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Chapter 5: Q18E (page 314)

A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and a standard deviation equal to 16

a. Give the mean and standard deviation of the (repeated) sampling distribution of x.

b. Describe the shape of the sampling distribution of x. Does your answer depend on the sample size?

c. Calculate the standard normal z-score corresponding to a value of x = 15.5.

d. Calculate the standard normal z-score corresponding to x = 23

Short Answer

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Answer

The standard deviation is commonly utilized as a measurement of an asset's comparative volatility. The standard deviation is determined as the square root of the variation by calculating the departure of every observation point from the mean.

Step by step solution

01

Step-by-Step Solution Step 1: (a) The data is given below

The calculation is given below:

渭X=20蟽X=蟽n=1664=2

02

(b) The data is given below

Step 3: (c) The data is given below of x will have the form of a bell-shaped normally distributed. Yes, it is dependent on the scale of the sample. As the sample size improves the sample size approaches being normal.

03

(c) The data is given below

The calculation is given below:

z =X鈭�渭蟽/n=15.5鈭�202=鈭�2.25

04

(d) The data is given below

The calculation is given below:

z =X鈭�渭蟽/n=23鈭�202=1.5

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Most popular questions from this chapter

Length of job tenure. Researchers at the Terry College ofBusiness at the University of Georgia sampled 344 business students and asked them this question: 鈥淥ver the course of your lifetime, what is the maximum number of years you expect to work for any one employer?鈥� The sample resulted in x= 19.1 years. Assume that the sample of students was randomly selected from the 6,000 undergraduate students atthe Terry College and that = 6 years.

  1. Describe the sampling distribution of X炉.
  2. If the mean for the 6,000 undergraduate students is渭= 18.5 years, findPx炉>19.1.
  3. If the mean for the 6,000 undergraduate students is渭= 19.5 years, findPx炉>19.1.
  4. If,P(x炉>19.1)=0.5 what is渭?
  5. If,Px炉>19.1=0.2 is渭greater than or less than 19.1years? Explain.

Consider the population described by the probability distribution shown below.

The random variable x is observed twice. If these observations are independent, verify that the different samples of size 2 and their probabilities are as shown below.

a. Find the sampling distribution of the sample meanx.

b. Construct a probability histogram for the sampling distribution ofx.

c. What is the probability thatxis 4.5 or larger?

d. Would you expect to observe a value ofxequal to 4.5 or larger? Explain.

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鈥檚 production process to verify the vendor鈥檚 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鈥檚 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鈥檚 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?

The probability distribution shown here describes a population of measurements that can assume values of 0, 2, 4, and 6, each of which occurs with the same relative frequency:

  1. List all the different samples of n = 2 measurements that can be selected from this population. For example, (0, 6) is one possible pair of measurements; (2, 2) is another possible pair.
  2. Calculate the mean of each different sample listed in part a.
  3. If a sample of n = 2 measurements is randomly selected from the population, what is the probability that a specific sample will be selected.
  4. Assume that a random sample of n = 2 measurements is selected from the population. List the different values of x found in part b and find the probability of each. Then give the sampling distribution of the sample mean x in tabular form.
  5. Construct a probability histogram for the sampling distribution ofx.

Rental car fleet evaluation. National Car Rental Systems, Inc., commissioned the U.S. Automobile Club (USAC) to conduct a survey of the general condition of the cars rented to the public by Hertz, Avis, National, and Budget Rent-a-Car.* USAC officials evaluate each company鈥檚 cars using a demerit point system. Each car starts with a perfect score of 0 points and incurs demerit points for each discrepancy noted by the inspectors. One measure of the overall condition of a company鈥檚 cars is the mean of all scores received by the company (i.e., the company鈥檚 fleet mean score). To estimate the fleet mean score of each rental car company, 10 major airports were randomly selected, and 10 cars from each company were randomly rented for inspection from each airport by USAC officials (i.e., a sample of size n = 100 cars from each company鈥檚 fleet was drawn and inspected).

a. Describe the sampling distribution of x, the mean score of a sample of n = 100 rental cars.

b. Interpret the mean of x in the context of this problem.

c. Assume渭=30 and 蟽=60for one rental car company. For this company, findPx炉猢�45 .

d. Refer to part c. The company claims that their true fleet mean score 鈥渃ouldn鈥檛 possibly be as high as 30.鈥� The sample mean score tabulated by USAC for this company was 45. Does this result tend to support or refute the claim? Explain.
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