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Chapter 8: Problem 16

A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is \(62 \%\). During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 140 passed the course. a. What is \(\hat{p}\), the sample proportion of students who passed introductory chemistry. b. What is \(p_{0}\), the proportion of students who pass introductory chemistry if the null hypothesis is true? c. Find the value of the test statistic. Explain the test statistic in context.

Short Answer

Expert verified
The sample proportion \(\hat{p}\) is 0.70 and the null hypothesis proportion \(p_{0}\) is 0.62. After calculation, test statistic Z would be found, larger positive or negative value of Z suggests disbelief in the null hypothesis. The context of the test statistic is it measures how far the sample results are from the null hypothesis.

Step by step solution

01

Calculate Sample Proportion \(\hat{p}\)

The sample proportion is calculated by dividing the number of successful outcomes by the sample size. Here, successful outcomes are the students who passed, which is 140, and the sample size is the number of students, which is 200. So, \(\hat{p}\) would be \(140 / 200 = 0.70\).
02

Identify the Null Hypothesis Proportion \(p_{0}\)

The passing rate in the initial condition is identified as \(p_{0}\) in the null hypothesis. Here, \(p_{0} = 0.62\), which is the original pass rate without embedded tutors.
03

Calculate Test Statistic

The test statistic is calculated using the formula \( Z = ( \hat{p}- p_{0} ) / \sqrt{( p_{0}(1 - p_{0} )/n}\). Substituting the values: \(Z = (0.70 - 0.62) / \sqrt{( 0.62 * 0.38 )/200} \), where n is the sample size 200.
04

Interpret the Test Statistic

The value of the test statistic helps in deciding whether to reject the null hypothesis. A large positive or negative Z indicates that the observed sample proportion is far away from the null hypothesized proportion. Hence, if our Z score is significantly larger than what we would expect by chance (if the null hypothesis were true), we would reject the null hypothesis.

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Most popular questions from this chapter

A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get \(50 \%\) of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than \(50 \%\) in the long run. $$\begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ A student gets 30 out of 50 questions, or \(60 \%\), correct. The p-value is \(0.079\). Explain the meaning of the \(\mathrm{p}\) -value in the context of this question.

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