Question

consider the following hypotheses. ( h_{0}: p = 0.44 ) ( h_{1}: p
eq 0.44 ) given that ( hat{p} = 0.39 ), ( n = 500 ), and ( alpha = 0.10 ), answer the following questions. a. determine the critical value(s) and the test statistic. what conclusion should be drawn? b. determine the p - value for this test. (note ( x=hat{p} n )) calculate the test statistic. ( z_{0}=-2.25 ) (round to two decimal places as needed.) choose the correct answer below. a. do not reject ( h_{0} ). there is insufficient evidence that ( p
eq 0.44 ). b. do not reject ( h_{0} ). there is sufficient evidence that ( p
eq 0.44 ). c. reject ( h_{0} ). there is sufficient evidence that ( p
eq 0.44 ). d. reject ( h_{0} ). there is insufficient evidence that ( p
eq 0.44 ). b. p - value ( = ) (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for the p - value in a two - tailed z - test

For a two - tailed test with test statistic \(z_0\), the p - value is \(p = 2\times(1 - P(Z<|z_0|))\), where \(Z\) is a standard normal random variable.

Step2: Find the probability \(P(Z < |z_0|)\)

Given \(z_0=- 2.25\), then \(|z_0| = 2.25\). Using the standard normal table or a calculator with a normal distribution function (\(P(Z < 2.25)\)), we know that \(P(Z < 2.25)=0.9878\) (from standard normal table: \(P(Z < 2.25)\) corresponds to the value in the row \(2.2\) and column \(0.05\)).

Step3: Calculate the p - value

Substitute into the p - value formula: \(p = 2\times(1 - 0.9878)\).

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Answer:

\(0.024\)