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Chapter 11: Q. 11.114 (page 450)

Racial Crossover. Refer to Exercise $11.108$ and find and interpret a $95 %$ confidence interval for the difference between the stroke incidences of white and African-American elderly.

Short Answer

Expert verified

There is $95 %$ the interval for the difference between the proportion is $(- 0.029, 0.015)$

Step by step solution

01

Given Information

The given values are,

$x_{1} = 529, n_{1} = 4989, x_{2} = 103, n_{2} = 906 .$

02

Explanation

The formula for ${\tilde{p}}_{1}$ is given by,

${\tilde{p}}_{1} = \frac{x_{1}}{n_{1}}$

The formula for ${\tilde{p}}_{2}$ is given by,

${\tilde{p}}_{2} = \frac{x_{2}}{n_{2}}$

The formula for $z,$ is given by,

$z = \frac{{\tilde{p}}_{1} - {\tilde{p}}_{2}}{\sqrt{{\tilde{p}}_{p} (1 - {\tilde{p}}_{p})} \sqrt{\frac{1}{n_{1}} + \frac{1}{m_{2}}}}$

The value of ${\tilde{p}}_{1}$ is calculated as,

${\tilde{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{529}{4989} = 0.106$

The value of ${\tilde{p}}_{2}$ is calculated as,

${\tilde{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{103}{906} = 0.113$

The critical value of $Z$ for two-tail test and $5 %$ level of significance is $1.96$ .

The $95 %$ confidence interval is.

$({\tilde{p}}_{1} - {\tilde{p}}_{2}) 卤 z_{\frac{a}{2}} \sqrt{\frac{{\tilde{p}}_{1} (1 - {\tilde{p}}_{1})}{n_{1}} + \frac{{\tilde{p}}_{1} (1 - {\tilde{p}}_{1})}{n_{2}}} = (0.106 - 0.113)$

$卤 1.96 \sqrt{\frac{0.106 (1 - 0.106)}{4989} + \frac{0.113 (1 - 0.113)}{906}}$

$= - 0.007 卤 0.022$

Therefore, there is $95 %$ interval for the difference between the proportion is $(- 0.029, 0.015)$

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Most popular questions from this chapter

a. Determine the sample proportion.

b. Decide whether using the one-proportion $z -$ test is appropriate.

c. If appropriate, use the one-proportion $z -$ test to perform the specified hypothesis test.

$x = 3$

$n = 100$

$H_{0} : p = 0.04$

$H_{a} : p 鈮� 0.04$

$伪 = 0.10$

a. Determine the sample proportion.

b. Decide whether using the one-proportion $z -$ test is appropriate.

c. If appropriate, use the one-proportion $z -$ test to perform the specified hypothesis test.

$x = 16$

$n = 20$

$H_{0} : p = 0.7$

$H_{a :} p 鈮� 0.7$

$a = 0.05$

Eating Out Vegetarian. Refer to the study on ordering vegetarian considered in Examples 11.8-11.10.

a. Obtain the margin of error for the estimate of the difference between the proportions of men and women who sometimes order veg by taking half the length of the confidence interval found in Example $11.10$ on page 473. Interpret your answer in words.

b. Obtain the margin of error for the estimate of the difference between the proportions of men and women who sometimes order veg by applying Exercise 11.132.

11.94 Drowning Deaths. In the article "Drowning Deaths of Zero to Five Year Old Children in Victorian Dams, 1989-2001" (Australian Journal of Rural Health, Vol. 13, Issue 5, pp. 300-308), L. Bugeja and R. Franklin examined drowning deaths of young children in Victorian dams to identify common contributing factors and develop strategies for future prevention. Of $11$ young children who drowned in Victorian dams located on farms, $5$ were girls, At the $5 %$ significance level, do the data provide sufficient evidence to conclude that, of all young children drowning in Victorian dams located on farms, less than half are girls?

Buckling Up. Refer to Exercise $11.109$ and find and interpret a $99 %$ confidence interval for the difference between the proportions of seat-belt users for drivers in the age groups $16 - 24$ years and 25-69 years.

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