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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.

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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.

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Most popular questions from this chapter

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