Problem
You are conducting a study to see if the accuracy rate for fingerprint identification is significantly less than 0.47 . You use a significance level of $\alpha=0.002$. \[ \begin{array}{l} H_{0}: p=0.47 \\ H_{1}: p<0.47 \end{array} \] You obtain a sample of size $n=252$ in which there are 98 successes. What is the test statistic for this sample? (Report answer accurate to three decimal places.) \[ \text { test statistic }= \] What is the p-value for this sample? (Report answer accurate to four decimal places.) \[ \text { p-value }= \] 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 As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the accuracy rate for fingerprint identification is less than 0.47 . There is not sufficient evidence to warrant rejection of the claim that the accuracy rate for fingerprint identification is less than 0.47 . The sample data support the claim that the accuracy rate for fingerprint identification is less than 0.47 . There is not sufficient sample evidence to support the claim that the accuracy rate for fingerprint identification is less than 0.47 . Question Help: Video
Solution
Step 1 ：Given values are \(p_{hat} = \frac{98}{252}\), \(p_0 = 0.47\), and \(n = 252\).
Step 2 ：Calculate the test statistic using the formula \(Z = \frac{p_{hat} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\).
Step 3 ：Substitute the given values into the formula to get \(Z = \frac{0.3888888888888889 - 0.47}{\sqrt{\frac{0.47(1-0.47)}{252}}}\).
Step 4 ：The calculated test statistic is -2.579845852193562.
Step 5 ：However, the question asks for the answer accurate to three decimal places.
Step 6 ：Therefore, the test statistic is -2.580.
Step 7 ：The final answer is \(\boxed{-2.580}\).
