ou wish to test the following claim π1<π2 at a significance level of 0.05. Ho:π1=π2 Ha:π1<π2 You obtain a sample from the first population with 115 successes and 99 failures. You obtain a sample from the second population with 138 successes and 71 failures. Use the Theory-based inference applet to conduct this test of significance. (a) What is the z-score for this sample? z-score = (Report answer accurate to two decimal places.) (b) What is the p-value for this sample? p-value = (Report answer accurate to four decimal places.) (c) The p-value is... less than (or equal to) α greater than α (d) This p-value leads to the following conclusion: we have statistically significant evidence to support the alternative hypothesis the null hypothesis is true the null hypothesis is plausible the alternative hypothesis is true (e) As such, the final conclusion is that... We conclude that the first population proportion is equal to the second population proportion. We conclude that the first population proportion is less than the second population proportion. We have statistically significant evidence that the first population proportion is less than the second population proportion. It is plausible that the first population proportion is equal to the second population proportion.

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

You wish to test the following claim π1<π2 at a significance level of 0.05.

Ho:π1=π2
Ha:π1<π2

You obtain a sample from the first population with 115 successes and 99 failures. You obtain a sample from the second population with 138 successes and 71 failures. Use the Theory-based inference applet to conduct this test of significance.

(a) What is the z-score for this sample?
z-score = (Report answer accurate to two decimal places.)

(b) What is the p-value for this sample?
p-value = (Report answer accurate to four decimal places.)

(c) The p-value is...

less than (or equal to) α

greater than α

(d) This p-value leads to the following conclusion:

we have statistically significant evidence to support the alternative hypothesis
the null hypothesis is true
the null hypothesis is plausible
the alternative hypothesis is true

(e) As such, the final conclusion is that...

We conclude that the first population proportion is equal to the second population proportion.
We conclude that the first population proportion is less than the second population proportion.
We have statistically significant evidence that the first population proportion is less than the second population proportion.
It is plausible that the first population proportion is equal to the second population proportion.

Expert Solution

Step 1

Number of favorable cases, $x_{1} = 115$

Sample Size, $n_{1} = 214$

Number of favorable cases, $x_{2} = 138$

Sample Size, $n_{2} = 209$

Significance Level, $α = 0.0500$

$H_{o} : p_{1} = p_{2} H_{1} : p_{1} < p_{2}$

$\begin{aligned} so, {\hat{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{115}{214} = 0.5374 \\ so, {\hat{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{138}{209} = 0.6603 \\ so, \bar{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} = \frac{253}{423} = 0.5981 \end{aligned}$