Question

test the hypothesis using the p - value approach

$h_{0}: p = 0.45$ versus $h_{1}: p < 0.45$
$n = 150$, $x = 62$, $\alpha = 0.05$

perform the test using the p - value approach.

$p$-value $=\square$ (round to four decimal places as needed.)

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$. Given $x = 62$ and $n=150$, so $\hat{p}=\frac{62}{150}\approx0.4133$.

Step2: Calculate the test - statistic

The formula for the test - statistic $z$ in a one - sample proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Here, $p = 0.45$, $\hat{p}=0.4133$, and $n = 150$.

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Step3: Calculate the P - value

Since the alternative hypothesis is $H_1:p\lt0.45$ (left - tailed test), the P - value is $P(Z\lt z)$. Using the standard normal distribution table or a calculator, for $z=-0.90$, $P(Z\lt - 0.90)=0.1841$

Answer:

$0.1841$