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Chapter 5: Q36E (page 320)

Suppose a random sample of n measurements is selected from a binomial population with the probability of success p = .2. For each of the following values of n, give the mean and standard deviation of the sampling distribution of the sample proportion,

  1. n = 50
  2. n = 1,000
  3. n = 400

Short Answer

Expert verified

A sampling distribution is a statistic that calculates the chance of an occurrence depending on information from a tiny subset of a significant population.

Step by step solution

01

(a) The n = 50 calculations are given below

For various values of n, we must calculate the mean as well as standard deviations of the sampling range of the probability value. If we consider p as a proportion, the sample mean may be regarded as a normal distribution.

The calculation is given below:

Mean=pStandardDeviation=PQnP=Numberofsuccess.Q=1-P=Numberoffailures.

localid="1662358414393" n=50Mean=0.2StandardDeviation=PQn=0.2脳0.850

=0.056

02

(b) The n = 1,000 calculations are given below

The calculation is given below:

n=1000Mean=0.2StandardDeviation=PQn=0.2脳0.81000

=0.0126

03

(c) The n = 400 calculations are given below

The calculation is given below:

n=400Mean=0.2StandardDeviation=PQn=0.2脳0.8400

=0.02

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Most popular questions from this chapter

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

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.

Refer to Exercise 5.18. Find the probability that

  1. x炉is less than 16.
  2. x炉is greater than 23.
  3. x炉is greater than 25.
  4. x炉falls between 16 and 22.
  5. x炉 is less than 14.

Suppose a random sample of n = 500 measurements is selected from a binomial population with probability of success p. For each of the following values of p, give the mean and standard deviation of the sampling distribution of the sample proportion,p^.

  1. p= .1
  2. p= .5
  3. p= .7

Consider the following probability distribution:

a. Calculate for this distribution.

b. Find the sampling distribution of the sample mean for a random sample of n = 3 measurements from this distribution, and show that is an unbiased estimator of .

c. Find the sampling distribution of the sample median for a random sample of n = 3 measurements from this distribution, and show that the median is a biased estimator of .

d. If you wanted to use a sample of three measurements from this population to estimate , which estimator would you use? Why?
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