Chapter 26, Problem 39E

a)

To determine

To find the z-value and shows that square of z value is equal to calculated chi-square.

Answer to Problem 39E

Z-statistic is 3.75.

Explanation of Solution

Given:

  

Formula:

Sample proportion:

  $\widehat{p}=\frac{x}{n}$

Z-statistic:

  $z=\frac{{\widehat{p}}_1-{\widehat{p}}_2}{\sqrt{\overline{p}(1-\overline{p})\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$

First, need to calculate sample proportions.

  ${\widehat{p}}_1=\frac{87}{95}=0.9158$

  ${\widehat{p}}_2=\frac{83}{117}=0.7094$

Pooled proportion is,

  $\overline{p}=\frac{87+83}{95+117}=0.8019$

Therefore, z-statistic is,

  $z=\frac{0.9158-0.7094}{\sqrt{0.8019(1-0.8019)\left(\frac{1}{95}+\frac{1}{117}\right)}}=3.75$

Therefore, z=2.75

Now,

  $z^2={3.75}^2=14.06\approx {\chi }^2$

b)

To determine

To show that p-value from z statistic and chi-square test are same.

Answer to Problem 39E

Both p-values are almost same.

Explanation of Solution

Given:

  

  ${\chi }^2$ = 14.06 and P-value = 0.0002

Z-statistic is 3.75.

Using, excel formula,

=2*(1-NORMSDIST(3.75))

P-value = 0.0002

Both p-values are almost same.