Business Statistics: A First Course (7th Edition)
Business Statistics: A First Course (7th Edition)
7th Edition
ISBN: 9780321979018
Author: David M. Levine, Kathryn A. Szabat, David F. Stephan
Publisher: PEARSON
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Chapter 5, Problem 5.1LB

Given the following probability distributions:

Chapter 5, Problem 5.1LB, Given the following probability distributions: Compute the <x-custom-btb-me data-me-id='2236' class='microExplainerHighlight'>expected value</x-custom-btb-me> for each distribution.

  1. Compute the expected value for each distribution.
  2. Compute the standard deviation for each distribution.
  3. Compare the results of distributions A and B.

(a)

Expert Solution
Check Mark
To determine

To find: The expected values for both the distributions.

Answer to Problem 5.1LB

The expected values are 1 and 3 respectively.

Explanation of Solution

Given:

The probability distributions are:

    Distribution ADistribution B
    xiP(X=xi)xiP(X=xi)
    00.500.05
    10.210.1
    20.1520.15
    30.130.2
    40.0540.5

Formula used:

The formula to compute the expected mean is:

  E(X)=xi×P(X=xi)

Here, P(X=xi) is the probability associated with the xi observation.

The formula to compute the standard deviation is:

  σ=xi2×P(X=xi)(xi×P(X=xi))2

Calculation:

The expected values for both the distributions can be calculated as:

Distribution A:

  E(X)=xi×P(X=xi)=0(0.5)+1(0.2)+2(0.15)+3(0.1)+4(0.05)=0+0.2+0.30+0.3+0.2=1

Distribution B:

  E(X)=xi×P(X=xi)=0(0.05)+1(0.1)+2(0.15)+3(0.2)+4(0.5)=0+0.1+0.30+0.6+0.2=3

Thus, the required expected values are 1 and 3 respectively.

(b)

Expert Solution
Check Mark
To determine

To find: The standard deviation for each distribution.

Answer to Problem 5.1LB

The standard deviations are 1.225 and 1.225 respectively.

Explanation of Solution

The standard deviation for each distribution can be computed as:

Distribution A:

  σ=xi2×P(X=xi)(xi×P(X=xi))2=02(0.5)+12(0.20)+22(0.15)+32(0.10)+42(0.05)(1)2=1.225

Distribution B:

  σ=xi2×P(X=xi)(xi×P(X=xi))2=02(0.05)+12(0.1)+22(0.15)+32(0.2)+42(0.5)(3)2=1.225

Thus, the required standard deviations are 1.225 and 1.225 respectively.

(c)

Expert Solution
Check Mark
To determine

To compare: The provided distributions.

Explanation of Solution

From the above two parts, it is clear that the mean of distribution B is greater than the mean of distribution B but there is no difference in the standard deviation of two distributions.

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To  evaluate  the  success  of  a  1-year  experimental  program  designed  to  increase  the  mathematical achievement of underprivileged high school seniors, a random sample of participants in the program will be selected and their mathematics scores will be compared with the previous year’s  statewide  average  of  525  for  underprivileged  seniors.  The  researchers  want  to  determine  whether the experimental program has increased the mean achievement level over the previous year’s statewide average. If alpha=.05, what sample size is needed to have a probability of Type II error of at most .025 if the actual mean is increased to 550? From previous results, sigma=80.

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Business Statistics: A First Course (7th Edition)

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