Q 42. Significance tests A test of H0:... [FREE SOLUTION]

91影视

The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 9: Q 42. (page 582)

Significance tests A test of $H_{0} : p = 0.65$ against $H_{a} : p < 0.65$
based on a sample of size $400$ yields the standardized test statistic $z = 鈭� 1.78$ .
a. Find and interpret the $P$ -value.
b. What conclusion would you make at the $伪 = 0.10$ significance level? Would
your conclusion change if you used $伪 = 0.05$ instead? Explain your reasoning.
c. Determine the value of $\overset{铃�}{p}$ = the sample proportion of successes.

Short Answer

Expert verified

a. Null hypothesis is rejected.

b. $伪 = 0.1 : fail to reject the null hypothesis.$ and $伪 = 0.05 : reject the null hypothesis$

c. $\hat{p} = 0.6510$

Step by step solution

Part (a) Step 1: Given Information

It is given that $z = 1.78$

$n = 400, 伪 = 0.65$

$H_{0} : p = 0.65$

$H_{a} : p < 0.65$

Part (a) Step 2: Explanation

Value of $z$ score, $z = 1.78$ is:

$P (x < z) = 0.96246$

$P (x > z) = 1 - 0.96246 = 0.037538$

If $P value > 伪$ , null hypothesis is rejected.

$P = 0.037538 < 0.65$ , null hypothesis is rejected here.

$H_{0} : p = 0.65$

Part (b) Step 1: Given Information

It is given that $伪 = 0.1, 伪 = 0.05$

Part (b) Step 2: Explanation

As studied above, for $伪 = 0.65$ , hypothesis is rejected.

For $伪 = 0.1, P = 0.037538 < 伪$ , null hypothesis is rejected. $H_{0} : p = 0.65$

For $伪 = 0.05, P = 0.037538 < 伪$ , null hypothesis is rejected. $H_{0} : p = 0.5$

$P$ value changes due to significance level changes.

Part (c) Step 1: Given Information

It is given that $n = 400$

$p = 0.65$

$z = 1.78$

Part (c) Step 2: Explanation

Test statistic is calculated as:

$Z = \frac{\hat{p} - p}{\sqrt{\frac{p (1 - p)}{n}}}$

$1.78 = \frac{\hat{p} - 0.65}{\sqrt{\frac{0.65 (1 - 0.65)}{400}}}$

$1.78 = \frac{\hat{p} - 0.65}{\sqrt{\frac{0.65 (0.35)}{400}}}$

$1.78 = \frac{\hat{p} - 0.65}{5.6875 脳 10^{- 4}}$

Hence, $\hat{p} = 0.6510$

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Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

Part (c) Step 1: Given Information

Part (c) Step 2: Explanation

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