Q.21 The P-value for a chi-square goo... [FREE SOLUTION]

91影视

The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 11: Q.21 (page 695)

The $P$ -value for a chi-square goodness-of-fit test is $0.0129$ . The correct conclusion is
(a) reject $H_{0}$ at $伪 = 0.05$ ; there is strong evidence that the trees are randomly distributed.
(b) reject $H_{0}$ at $伪 = 0.05$ ; there is not strong evidence that the trees are randomly distributed.
(c) reject $H_{0}$ at $伪 = 0.05$ ; there is strong evidence that the trees are not randomly distributed.
(d) fail to reject $H_{0}$ at $伪 = 0.05$ ; there is not strong evidence that the trees are randomly distributed.
(e) fail to reject $H_{0}$ at $伪 = 0.05$ ; there is strong evidence that the trees are randomly distributed.

Short Answer

Expert verified

Correct option is

Step by step solution

Given information

Given in the question that, Researchers wondered whether the trees in a longleaf pine forest in Georgia are randomly distributed. To find out, they divided the forest into four equal quadrants. Then the researchers took a random sample of 100 trees and counted the number in each quadrant. Here are their data:

Explanation

$0.0129 .$ Table is

Quadratic	Count
$1$	$18$
$2$	$22$
$3$	$39$
$4$	$21$

$P$ -value is

Level of significance $5 %$

The null and alternative hypotheses are:

$H_{0} : p_{1} = p_{2} = p_{3} = p_{4} = 0.24$

$H_{a} :$ At least one of the $p_{i}$ is incorrect

Calculation for test statistic

The calculation for test statistic could be done as:

Observed value(O)	Expected value(E)
$18$	$0.25 (100) = 25$
$22$	$0.25 (100) = 25$
$39$	$0.25 (100) = 25$
$21$	$0.25 (100) = 25$

The test statistic is:

$蠂^{2} = 鈭� \frac{(O - E)^{2}}{E}$

$= \frac{(18 - 25)^{2}}{25} + \frac{(22 - 25)^{2}}{25} + \frac{(39 - 25)^{2}}{25} + \frac{(21 - 25)^{2}}{25}$

$= 10.8$

The $P$ -value is $0.0129$

The $P$ -value is below the significance level. Thus, the null hypothesis is rejected. Thus at $5 %$ significance level there is sufficient evidence to favor the claim about the trees.

Hence, the correct option is (c).