Question

Researchers were interested in the short-term effect that caffeine has on heart rate. They enlisted a group of volunteers and measured each person's resting heart rate. Then they had each subject drink 6 ounces of coffee. Nine of the subjects were given coffee containing caffeine, and 11 were given decaffeinated coffee. After 10 minutes each person's heart rate was measured again. The data in the table show the change in heart rate; a positive number means that heart rate went up, and a negative number means that heart rate went down. \(^{47}\) (a) Use these data to construct a \(90 \%\) confidence interval for the difference in mean effect that caffeinated coffee has on heart rate, in comparison to decaffeinated coffee. [Note: Formula (6.7.1) yields 17.3 degrees of freedom for these data. (b) Using the interval computed in part (a) to justify your answer, is it reasonable to believe that caffeine may not affect heart rates? (c) Using the interval computed in part (a) to justify your answer, is it reasonable to believe that caffeine may affect heart rates? If so, by how much? (d) Are your answers to (b) and (c) contradictory? Explain. $$ \begin{array}{|ccc|} \hline \text { Caffeine } & \text { Decaf } \\ \hline & 28 & 26 \\ & 11 & 1 \\ & -3 & 0 \\ & 14 & -4 \\ & -2 & -4 \\ & -4 & 14 \\ & 18 & 16 \\ & 2 & 8 \\ & 2 & 0 \\ & & 18 \\ & & -10 \\ \hline n & 9 & 11 \\ \bar{y} & 7.3 & 5.9 \\ s & 11.1 & 11.2 \\ \text { SE } & 3.7 & 3.4 \\ \hline \end{array} $$

Short answer

After calculating the 90% confidence interval and interpreting it, if the interval includes 0, it is reasonable to believe that caffeine may not have an effect. Otherwise, it suggests that caffeine does affect heart rates, and the interval indicates the magnitude of that effect. The answers to (b) and (c) depend on the interval and are not necessarily contradictory.

Step-by-step solution

Calculate the standard error of the difference between means

The standard error (SE) of the difference between two means can be estimated by using the formula SE = \( \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \), where \( s_1 \) and \( s_2 \) are the sample standard deviations and \( n_1 \) and \( n_2 \) are the sample sizes for the two groups.

Calculate the degrees of freedom

The degrees of freedom for the two-sample t-test can be estimated using the formula (6.7.1) or Welch's formula, which is more complicated but it has been provided in the exercise as 17.3 degrees of freedom.

Determine the critical t-value

Using a t-distribution table or a calculator, find the t-value that corresponds to a 90% confidence level and 17.3 degrees of freedom for a two-tailed test.

Construct the 90% confidence interval for the difference in means

The confidence interval is calculated using the formula: \(\text{CI} = (\bar{y}_1 - \bar{y}_2) \pm t \times SE\), where \(\bar{y}_1\) and \(\bar{y}_2\) are the sample means, \(t\) is the critical t-value, and \(SE\) is the standard error from Step 1.

Interpret the confidence interval

If the 90% confidence interval includes the value 0, it suggests that there is a chance that caffeine may not have an effect on heart rates. If the interval does not contain 0, then it implies that there is a significant difference in the effect of caffeine versus decaf on heart rates.

Answer the questions based on the interval

(b) If 0 is within the interval, then it is reasonable to believe that caffeine may not affect heart rates. (c) If 0 is not within the interval, it is reasonable to believe that caffeine does affect heart rates, and the interval provides an estimate of how much. (d) The answers to (b) and (c) are not contradictory; they are based on whether the confidence interval contains 0.

Key concepts

Two-Sample T-Test

Understanding the two-sample t-test is crucial when researchers want to compare the means of two groups. This statistical test is particularly useful for small sample sizes or when the population standard deviations are unknown and assumed to be unequal.
In the case of the caffeine study, researchers were comparing the change in heart rate for two different groups: one drinking caffeinated coffee, and the other decaffeinated coffee. The two-sample t-test would help determine if there's a statistically significant difference in the mean change of heart rate between these two independent groups.
To conduct a two-sample t-test, essential steps need to be taken. We start by calculating the standard error of the difference, find the degrees of freedom, determine the critical t-value from the t-distribution, and finally construct the confidence interval for the difference in means. By comparing this interval to a specific value (often 0), researchers can draw conclusions about the impact of caffeine on heart rate.

Standard Error Calculation

The standard error (SE) of the difference between two means is a measure of the variability of the sample means. It reflects how much the sample means would differ from the true population means. In our caffeine study, calculating the SE was the first step in exploring the data.
The formula for the SE of the difference between two independent means is given by: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \], where \( s_1 \) and \( s_2 \) represent the sample standard deviations, and \( n_1 \) and \( n_2 \) are the sample sizes. It's important to note that the larger the sample sizes and the smaller the standard deviations, the smaller the SE, which indicates more precise estimates of the population means. In this instance, we used the given standard deviations for both caffeinated and decaffeinated groups to calculate the SE, resulting in a more accurate confidence interval.

Degrees of Freedom

Degrees of freedom (df) are a vital concept in statistical tests, including the two-sample t-test. They are used to determine the exact distribution to refer to when deciding if a test statistic is significant. The df determine the shape of the t-distribution we're using to find critical t-values for our confidence interval.
For independent samples, when standard deviations are not assumed equal, we often calculate df using Welch's formula, which accounts for the variance and sample size of each group. In our exercise, the textbook provided 17.3 degrees of freedom. It's a decimal value because the formula takes into account the variation within each group and the sample sizes, providing a more precise measure for our t-distribution. This is critical for accurate and reliable inference about the means from our two groups in the caffeine study.

Statistical Significance

Statistical significance is the likelihood that the difference in sample findings reflects a true difference rather than random chance. Significance is generally determined by a p-value, which is inferred from the critical t-value in the context of a confidence interval.
In the context of our caffeine study, we're looking at whether the difference in heart rate changes between both groups is significant. After computing the confidence interval, we can assess statistical significance. If the confidence interval does not include 0, it implies that there is a statistically significant difference in the effect of caffeinated versus decaffeinated coffee on heart rates.
When we interpret the confidence interval from the original exercise, the inclusion or exclusion of 0 lets us answer whether caffeine may or may not affect heart rates. This evaluation helps researchers make data-driven conclusions while accounting for the natural variability present in any sample.