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Chapter 5: 5-50E (page 321)

Question:Who prepares your tax return? As part of a study on income tax compliance (Behavioral Research and Accounting, January 2015), researchers found that 37% of adult workers prepare their own tax return. Assume that this percentage applies to all U.S. adult workers. Now consider a random sample of 270 adult workers.

a. Find the probability that more than 112 of the workers prepare their own tax return.

b. Find the probability that between 100 and 150 of the workers prepare their own tax return

Short Answer

Expert verified
  1. The probability that more than 112 of the workers prepare their own tax return is zero.
  2. The probability that between 100 and 150 of the workers prepare their own tax return is zero.

Step by step solution

01

Given Information

The proportion of adult workers is given by

The number of sample size is

The standard deviation of the sampling distribution is computed as

蟽p^=p1-pn=0.37脳0.63270=0.0017

So, p^ is normally distributed with mean of 0.37 and a standard deviation of 0.0017.

02

(a) Compute the probability

The probability that more than 112 of the workers is computed as

03

(b) Identifying the probability

The probability that between 100 and 150 of the workers is computed as

100<p^<150=p100-0.370.0017<p^-p蟽p^<150-0.370.0017=p58605.88<z<88017.64=pz猢�88017.64-pz猢�58605.88=1-1=0

Hence, the probability is 0.

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Most popular questions from this chapter

Switching banks after a merger. Banks that merge with others to form 鈥渕ega-banks鈥� sometimes leave customers dissatisfied with the impersonal service. A poll by the Gallup Organization found 20% of retail customers switched banks after their banks merged with another. One year after the acquisition of First Fidelity by First Union, a random sample of 250 retail customers who had banked with First Fidelity were questioned. Letp^ be the proportion of those customers who switched their business from First Union to a different bank.

  1. Find the mean and the standard deviation of role="math" localid="1658320788143" p^.
  2. Calculate the interval Ep^卤2蟽p^.
  3. If samples of size 250 were drawn repeatedly a large number of times and determined for each sample, what proportion of the values would fall within the interval you calculated in part c?

Video game players and divided attention tasks. Human Factors (May 2014) published the results of a study designed to determine whether video game players are better than non鈥搗ideo game players at crossing the street when presented with distractions. Participants (college students) entered a street-crossing simulator. The simulator was designed to have cars traveling at various high rates of speed in both directions. During the crossing, the students also performed a memory task as a distraction. The researchers found that students who are video game players took an average of 5.1 seconds to cross the street, with a standard deviation of .8 second. Assume that the time, x, to cross the street for the population of video game players has , Now consider a sample of 30 students and let x represent the sample mean time (in seconds) to cross the street in the simulator.

a. Find Px炉>5.5

b. The 30 students in the sample are all non鈥搗ideo game players. What inference can you make about and/or for the population of non鈥搗ideo game players? 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?

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.

Variable life insurance return rates. Refer to the International Journal of Statistical Distributions (Vol. 1, 2015) study of a variable life insurance policy, Exercise 4.97 (p. 262). Recall that a ratio (x) of the rates of return on the investment for two consecutive years was shown to have a normal distribution, with 渭=1.5, 蟽=0.2. Consider a random sample of 100 variable life insurance policies and letx炉represent the mean ratio for the sample.

a. Find E(x) and interpret its value.

b. Find Var(x).

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d. Find the z-score for the value x炉=1.52.

e. Find Px炉>1.52

f. Would your answers to parts a鈥揺 change if the rates (x) of return on the investment for two consecutive years was not normally distributed? Explain.

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