/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 The professor swims Here are dat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 16

The professor swims Here are data on the time (in minutes) Professor Moore takes to swim 2000 yards and his pulse rate (beats per minute) after swimming on a random sample of 23 days: $$ \begin{array}{lrrrrrr} \hline \text { Time: } & 34.12 & 35.72 & 34.72 & 34.05 & 34.13 & 35.72 \\ \text { Pulse: } & 152 & 124 & 140 & 152 & 146 & 128 \\ \text { Time: } & 36.17 & 35.57 & 35.37 & 35.57 & 35.43 & 36.05 \\ \text { Pulse: } & 136 & 144 & 148 & 144 & 136 & 124 \\ \text { Time: } & 34.85 & 34.70 & 34.75 & 33.93 & 34.60 & 34.00 \\ \text { Pulse: } & 148 & 144 & 140 & 156 & 136 & 148 \\ \text { Time: } & 34.35 & 35.62 & 35.68 & 35.28 & 35.97 & \\ \text { Pulse: } & 148 & 132 & 124 & 132 & 139 & \\ \hline \end{array} $$ Is there statistically significant evidence of a negative linear relationship between Professor Moore's swim time and his pulse rate in the population of days on which he swims 2000 yards? Carry out an appropriate significance test at the \(\alpha=0.05\) level.

Short Answer

Expert verified
Conduct a t-test on the correlation coefficient; if the calculated t-value is less than the critical value at \( \alpha=0.05 \), we have evidence of a negative relationship.

Step by step solution

01

Understand the Hypotheses

We need to test if there is a significant negative linear relationship between swimming time and pulse rate. Our null hypothesis \( H_0 \) states that there is no linear relationship, which means the correlation coefficient \( \rho = 0 \). Our alternative hypothesis \( H_a \) states that there is a negative relationship, meaning \( \rho < 0 \).
02

Calculate the Correlation Coefficient

Using the given data, calculate the Pearson correlation coefficient \( r \). This will measure the strength and direction of the linear relationship between swim time and pulse rate. For precision, use software or statistical tools to compute \( r \).
03

Conduct the Significance Test

Use the correlation coefficient \( r \) to perform a t-test for the significance of \( r \). The test statistic is \( t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \), where \( n \) is the number of paired samples (23 in this case). Calculate \( t \) using the value of \( r \) obtained.
04

Determine the Critical Value and Decision

For a one-tailed test at \( \alpha = 0.05 \) with \( n-2 = 21 \) degrees of freedom, obtain the critical value from the t-distribution table. If the calculated \( t \) is beyond the negative critical value, reject the null hypothesis. Otherwise, we fail to reject \( H_0 \).
05

Conclusion

State whether there is statistically significant evidence of a negative linear relationship based on whether \( H_0 \) was rejected. If rejected, there is evidence of such a relationship between swim time and pulse rate.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Linear Relationship
A linear relationship describes how two variables relate to each other by using a straight line. In this exercise, we are examining the relationship between Professor Moore's swim time and his pulse rate. A linear relationship suggests that as one variable changes, the other changes at a constant rate.

To visualize a linear relationship, imagine plotting swim time on the x-axis and pulse rate on the y-axis of a graph. A line that goes in a downward slope, from left to right, indicates a negative linear relationship. This would mean that as swim time increases, the pulse rate decreases (or vice versa).

Establishing whether a linear relationship exists involves statistical tests, particularly when we hypothesize potential links as part of our analysis.
Correlation Coefficient
The correlation coefficient is a numerical measure detailing the strength and direction of a linear relationship between two variables. This statistic, often represented by the letter 'r', can range between -1 and 1. An 'r' close to -1 indicates a strong negative linear relationship, while an 'r' close to 1 suggests a strong positive relationship. If 'r' is around 0, it implies little to no linear relationship.

For Professor Moore's data, calculating the correlation coefficient helps us understand whether swim time and pulse rate are linked. To find 'r', the Pearson correlation formula is used, which considers each pair of swim times and pulse rates. Statistical software often aids in this computation to increase accuracy and efficiency, especially with larger data sets.
Significance Test
A significance test assesses whether the observed relationship in data is due to chance or can be considered statistically significant. For this exercise, we use a significance test to determine if the correlation between swim time and pulse rate is meaningful.

We employ a t-test alongside the correlation coefficient to establish this significance. The test compares the correlation ('r') obtained from our data set against the null hypothesis (which claims 'no relationship'). The resulting t-value, derived from a specific formula, helps in making this comparison.

After computing the t-value, compare it against a critical value obtained from a t-distribution table (considering the number of samples in your data set). This comparison indicates whether to reject or fail to reject the null hypothesis.
Null Hypothesis
In hypothesis testing, the null hypothesis (H_0) is a statement that we seek to test. It suggests there is no effect or relationship between variables. In this context, Professor Moore's null hypothesis is that there is no linear relationship between his swim times and pulse rates, symbolically represented as \( \rho = 0 \).

The task is to provide evidence against this hypothesis to conclude if a negative linear relationship is present. We achieve this by calculating the correlation coefficient and performing a significance test.

If the significance test results in rejecting the null hypothesis, it implies there's likely a negative relationship present in the population of days Moore swims. Conversely, failing to reject H_0 implies insufficient evidence to claim such a relationship.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Light through the water Some college students collected data on the intensity of light at various depths in a lake. Here are their data: \begin{tabular}{cc} \hline Depth (m) & Light intensity (lumens) \\ 5 & 168.00 \\ 6 & 120.42 \\ 7 & 86.31 \\ 8 & 61.87 \\ 9 & 44.34 \\ 10 & 31.78 \\ 11 & 22.78 \\ \hline \end{tabular} (a) Make a reasonably accurate scatterplot of the data by hand, using depth as the explanatory variable. Describe what you see. (b) A scatterplot of the natural logarithm of light intensity versus depth is shown below. Based on this graph, explain why it would be reasonable to use an exponential model to describe the relationship between light intensity and depth. (c) Minitab output from a linear regression analysis on the transformed data is shown below. $$ \begin{array}{lcccc} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 6.78910 & 0.00009 & 78575.46 & 0.000 \\ \text { Depth }(\mathrm{m}) & -0.333021 & 0.000010 & -31783.44 & 0.000 \\ \mathrm{~S}=0.000055 & \mathrm{R}-\mathrm{Sq}=100.0 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=100.0 \% \end{array} $$ Give the equation of the least-squares regression line. Be sure to define any variables you use. (d) Use your model to predict the light intensity at a depth of 12 meters. Show your work.

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Which of the following is the best interpretation for the value 0.1126 in the computer output? (a) For each increase of \(\$ 1000\) in appraised value, the average selling price increases by about 0.1126 . (b) When using this model to predict selling price, the predictions will typically be off by about 0.1126 . (c) \(11.26 \%\) of the variation in selling price is accounted for by the linear relationship between selling price and appraised value. (d) There is a weak, positive linear relationship between selling price and appraised value. (e) In repeated samples of size 16 , the sample slope will typically vary from the population slope by about \(0.1126 .\)

Tattoos (8.2) What percent of U.S. adults have one or more tattoos? The Harris Poll conducted an online survey of 2302 adults during January 2008 . According to the published report, "Respondents for this survey were selected from among those who have agreed to participate in Harris Interactive surveys." 25 The pie chart at top right summarizes the responses from those who were surveyed. Explain why it would not be appropriate to use these data to construct a \(95 \%\) confidence interval for the proportion of all U.S. adults who have tattoos.

Multiple Choice: Select the best answer for Exercises Suppose that the relationship between a response variable \(y\) and an explanatory variable \(x\) is modeled by \(y=2.7(0.316)^{x}\). Which of the following scatterplots would approximately follow a straight line? (a) A plot of \(y\) against \(x\) (b) A plot of \(y\) against \(\log x\) (c) A plot of log \(y\) against \(x\) (d) A plot of log \(y\) against \(\log x\) (e) A plot of \(\sqrt{y}\) against \(x\).

Refer to the following setting. About 1100 high school teachers attended a weeklong summer institute for teaching \(\mathrm{AP}^{(\mathrm{(R)}}\) classes. After hearing about the survey in Exercise \(52,\) the teachers in the \(\mathrm{AP}^{(R)}\) Statistics class wondered whether the results of the tattoo survey would be similar for teachers. They designed a survey to find out. The class opted to take a random sample of 100 teachers at the institute. One of the questions on the survey was Do you have any tattoos on your body? (Circle one) YES \(\quad\) NO Tattoos (8.2,9.2) Of the 98 teachers who responded, \(23.5 \%\) said that they had one or more tattoos. (a) Construct and interpret a \(95 \%\) confidence interval for the actual proportion of teachers at the \(\mathrm{AP}^{\otimes}\) institute who would say they had tattoos. (b) Does the interval in part (a) provide convincing evidence that the proportion of teachers at the institute with tattoos is not 0.14 (the value cited in the Harris Poll report)? Justify your answer. (c) Two of the selected teachers refused to respond to the survey. If both of these teachers had responded, could your answer to part (b) have changed? Justify your answer.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.