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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
Answer
Rounding to three decimal places, the test statistic for this sample is \(\boxed{-3.135}\).
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 \(\boxed{-3.135}\).
