Answered: The International Business Society…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

The International Business Society conducts a series of managerial surveys to detect challenges that international managers encounter and what strategies help them overcome those challenges. A previous study showed that employee retention is the most critical organizational challenge currently confronted by international managers. One strategy that may impact employee retention is a successful employee recognition program. The International Business Society randomly selected 423 small firms and 192 large firms. The study showed that 326 of the 423 small firms have employee retention programs compared to 167 of the 192 large organizations.

At the 0.01 level of significance, is there evidence of a significant difference between small firms and large firms concerning the proportion that has employee recognition program?
Find the p-value and interpret its meaning?
Construct and interpret a 99% confidence interval estimate for the difference between small firms and large firms concerning the proportion of employee recognition program.

Please solve part 2 and 3

Expert Solution

Step 1

International Business Society wishes to test the difference between small firms and large firms concerning the proportion that has employee recognition program.

Let X₁and X₂ be the number of small and large firms that conducts employee recognition program out of a sample of n₁ and n₂ respectively.

Then the sample proportion p₁ for the small firm is $\frac{X_{1}}{n_{1}}$ .

And the sample proportion p₂for the large firm is $\frac{X_{2}}{n_{2}}$ .

Step 2

Let P₁and P₂be the population proportions for the proportions of small and large firms conducting employee recognition program respectively

Then the Null hypothesis and alternate hypothesis be:

Ho : P₁= P₂

Ha : P₁≠ P₂

Then the test statistics for the difference of proportions is :

_{$Z = \frac{p_{1} - p_{2}}{\sqrt{P Q (\frac{1}{n_{1}} + \frac{1}{n_{2}})}} ~ N (0, 1)$}

where P is the unbiased estimate of population proportion ( based on both the samples ) given by :

_{$\overset{⏞}{P} = \frac{X_{1} + X_{2}}{n_{1} + n_{2}} \hat{Q} = 1 - \hat{P}$}

Step 3

Thus,the unbiased estimate of population proportion is calculated as,

$\begin{array}{rcl} \overset{⏞}{P} & = & \frac{326 + 162}{423 + 192} \\ = & \frac{488}{615} \\ = & 0.7935 \end{array}$

$\begin{array}{rcl} \overset{⏞}{Q} & = & 1 - \overset{⏞}{P} \\ = & 1 - 0.7935 \\ = & 0.2065 \end{array}$

And the sample proportions p₁and p₂are:

$\begin{array}{rcl} p_{1} & = & \frac{X_{1}}{n_{1}} \\ = & \frac{326}{423} \\ = & 0.7707 \end{array}$ $\begin{array}{rcl} p_{2} & = & \frac{X_{2}}{n_{2}} \\ = & \frac{162}{192} \\ = & 0.8438 \end{array}$

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