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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
a. The sample proportion (\( \hat{p} \)) of students who passed the introductory chemistry is 0.7 or \(70\% \).\n b. The proportion of students who pass the introductory chemistry under the null hypothesis (\( p_0 \)) is 0.62 or \(62\% \).\n c. The test statistic can be calculated using the given formula, which then quantifies the level of departure of our sample proportion from the proportion under the null hypothesis.

Step by step solution

01

Calculation of Sample Proportion (\( \hat{p} \))

Calculate the sample proportion (\( \hat{p} \)) by dividing the number of students who passed the course by the total number of students. In this case, \( \hat{p} = \frac{140}{200} = 0.7 \) or \(70\% \)
02

Determination of Proportion under Null Hypothesis (\( p_0 \))

The proportion of students who pass the course under the null hypothesis (\( p_0 \)) is given as the initial passing rate. Hence, \( p_0 = 0.62 \) or \( 62\% \)
03

Calculation of Test Statistic

The test statistic (z) is calculated by the formula \( z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0}(1 - p_{0})}{n}}} \). Substituting the values, we get \( z = \frac{0.7 - 0.62}{\sqrt{\frac{0.62 * 0.38}{200}}} \). After computing, we get the value of the test statistic. The test statistic allows us to quantify how much our sample proportion deviates from the proportion under the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion
The sample proportion (\(\hat{p}\)) is a way to express the percentage of a particular outcome in a sample group. In our context, it helps us assess how many students out of a total group passed the introductory chemistry courses. Calculating it is straightforward. You simply divide the number of students who passed the course by the total number of students in the sample.

For instance, in our exercise, 140 out of 200 students passed, which gives:
  • \(\hat{p} = \frac{140}{200} = 0.7\)
  • This means 70% of the students in the sample successfully passed the course.
Understanding this proportion is crucial when considering whether an intervention, like tutoring, has impacted success rates. It's essentially a snapshot of the group's performance.
Null Hypothesis
In hypothesis testing, the null hypothesis is an initial assumption that there is no effect or no difference. In our example, it implies that the use of embedded tutors doesn't change the passing rate of students in the introductory chemistry courses.

The null hypothesis is often represented by a symbol, \(p_0\).For this scenario, \(p_0\) represents the historical passing rate of students, which is given as 62% or 0.62.

Formally, the null hypothesis is:
  • The passing rate remains 62%, irrespective of using tutors.
We test this hypothesis to see if the change in teaching method has had a significant effect by comparing it to the sample proportion.
Test Statistic
The test statistic is a number calculated from the sample data, which in turn helps determine how far a sample statistic diverges from what was expected under the null hypothesis. A commonly used test statistic in proportion tests is the z-score.

It is calculated using the formula:
  • \(z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0}(1 - p_{0})}{n}}}\)
  • Here, \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and \(n\) is the sample size.
If our z-score is large (positive or negative), it suggests that the sample results are significantly different from what was expected under the null hypothesis. In our exercise, we calculated a z-score to determine the impact of embedded tutors on student performance.
Introductory Chemistry Courses
Introductory chemistry courses often serve as the foundation for further studies in the field. These entry-level courses lay down the basics of chemical principles, including topics like atomic structure, bonding, and reactions, which are crucial for understanding more advanced subjects.

Improving success rates in these courses is important as they typically have high enrollment and high impact due to their foundational nature.
  • Because of their difficulty, such courses often face high failure rates, encouraging educational interventions.
  • Strategies like employing embedded tutors can support students to better grasp complex content, aiming to boost their passing rates.
Studying the effects of these interventions can help educators enhance teaching methods, ultimately leading to more students advancing in their chemistry education.

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Most popular questions from this chapter

A Gallup poll conducted in 2017 found that 648 out of 1011 people surveyed supported same-sex marriage. An NBC News/Wall Street Journal poll conducted the same year surveyed 1200 people and found 720 supported same-sex marriage. a. Find both sample proportions and compare them. b. Test the hypothesis that the population proportions are not equal at the 0.05 significance level.

Embedded Tutors A college chemistry instructor thinks the use of embedded tutors (tutors who work with students during regular class meeting times) will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is \(62 \%\). The instructor will use embedded tutors in all sections of introductory chemistry and record the percentage of students passing the course. State the null and alternative hypotheses in words and in symbols. Use the symbol \(p\) to represent the passing rate for all introductory chemistry courses that use embedded tutors.

Freedom of Religion A Gallup poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of religion was secure or threatened in the country today. In 2016,2089 out of 3072 students surveyed said that freedom of religion was secure or very secure. In 2017,1929 out of 3014 students felt this way. a. Determine whether the proportion of college students who believe that freedom of religion is secure or very secure in this country has changed from 2016 . Use a significance level of \(0.05\). b. Use the sample data to construct a \(95 \%\) confidence interval for the difference in the proportions of college students in 2016 and 2017 who felt freedom of religion was secure or very secure. How does your confidence interval support your hypothesis test conclusion?

Choosing a Test and Naming the Population(s) In each case, choose whether the appropriate test is a one-proportion z-test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.

Refer to Exercise \(8.97 .\) Suppose 14 out of 20 voters in Pennsylvania report having voted for an independent candidate. The null hypothesis is that the population proportion is \(0.50 .\) What value of the test statistic should you report?
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