8 MATH1127: Introduction to Statistics (60022) Lesson 9.4 Rare Events, the Sample, Decision and Conclusion Unit 3 Chapter 9: Lesson 9.4 Assi You are conducting a study to see if the proportion of women over 40 who regularly have mammograms is significantly less than 0.26 . You use a significance level of $\alpha=0.01$. \[ \begin{array}{l} H_{0}: p=0.26 \\ H_{1}: p<0.26 \end{array} \] You obtain a sample of size $n=159$ in which there are 24 successes. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic $=$ What is the $p$-value for this sample? (Report answer accurate to four decimal places.) $p$-value $=$ The $p$-value is... less than (or equal to) $\alpha$ greater than $\alpha$ This test statistic leads to a decision to... reject the null accept the null fail to reject the null
Solution
Step 1 ：We are given that the sample size, \(n = 159\), the number of successes is 24, and the hypothesized population proportion, \(p_0 = 0.26\).
Step 2 ：We first calculate the sample proportion, \(\hat{p}\), which is the number of successes divided by the sample size. So, \(\hat{p} = \frac{24}{159} = 0.1509433962264151\).
Step 3 ：We then calculate the test statistic, \(z\), using the formula: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\).
Step 4 ：Substituting the given values into the formula, we get: \(z = \frac{0.1509433962264151 - 0.26}{\sqrt{\frac{0.26(1 - 0.26)}{159}}}\).
Step 5 ：Solving the above expression, we get \(z = -3.1350754494877946\).
Step 6 ：Rounding to three decimal places, the test statistic for this sample is \(\boxed{-3.135}\).
