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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

Consider the following probability distribution:

  1. Findand蟽2.
  2. Find the sampling distribution of the sample mean x for a random sample of n = 2 measurements from this distribution
  3. Show that xis an unbiased estimator of 渭. [Hint: Show that.]鈭�x=鈭�xpx=渭.]
  4. Find the sampling distribution of the sample variances2for a random sample of n = 2 measurements from this distribution.

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.

Question: The standard deviation (or, as it is usually called, the standard error) of the sampling distribution for the sample mean, x炉 , 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

蟽X炉=蟽n

  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

蟽A=蟽n3

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

  1. Suppose that the population standard deviation 蟽is equal to 10 and that the sample size is 64. Calculate the standard errors of x炉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炉?

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.

Improving SAT scores. Refer to the Chance(Winter2001) examination of Scholastic Assessment Test (SAT)scores of students who pay a private tutor to help them improve their results, Exercise 2.88 (p. 113). On the SAT鈥擬athematics test, these students had a mean score change of +19 points, with a standard deviation of 65 points. In a random sample of 100 students who pay a private tutor to help them improve their results, what is the likelihood that the sample mean score change is less than 10 points?
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