The code chunk below reads in resting heart rate data that was collected from a sample of Gustavus students via a Google form..The data is stored in a data set called hr_data. The data set contains a single variable named heart_rate. heart_rate is measured in beats per minute (bpm). The summary statistics for heart rate are displayed below.

min Q1 median Q3 max mean sd n missing
38 61.5 68 80.25 100 69.77 14.17 26 0
  1. Create a plot to display the distribution of heart rate in the code chunk below. All the packages you need todo this are already loaded in the document. Be sure your plot is fully labeled.

  1. What shape is the distribution of heart rate?

ANSWER: Normally Distributed

  1. What summary statistics would you use to describe the distribution?

ANSWER: I would use the mean and standard deviation as the data set is normally distributed

  1. The true value of the average resting heart rate for all Gustavus students (\(\mu\)) is unknown. What is a reasonable estimate of \(\mu\)?

ANSWER: 69.77

  1. Is it appropriate to use the formula, \(\bar x \pm t^* \space s / \sqrt{n}\) to calculate a 95% confidence interval? Explain your answer?

ANSWER: Yes as although the data set is less than 30 there is no extreme outliers and it appears to be normally distributed.

  1. What value of \(t^*\) would you use to calculate a 95% confidence interval for the average resting heart rate of Gustavus students?

ANSWER: 2.060

  1. Calculate a 95% confidence interval for the true average heart rate using the formula we discussed in class. (\(\bar x \pm t^* \times s / \sqrt{n}\)). The t.test() function will do this calculation for you.
t.test(hr_data$heart_rate, conf.level=.95)
## 
##  One Sample t-test
## 
## data:  hr_data$heart_rate
## t = 25.114, df = 25, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  64.04762 75.49085
## sample estimates:
## mean of x 
##  69.76923

ANSWER: (64.048, 75.49)

  1. Interpret your confidence interval from #3 in the context of the problem.

ANSWER: We are 95% confident that the true average resting heart rate of all gusties is contained within the interval 64.05 and 75.49.

According to netfit (“Your definitive guide to health and fitness.”), average men between 18 and 25 years old should have a resting heart rate between 70 and 73 bpm. Average women from 18-25 years old have resting heart rates between 74 and 78 bpm. Suppose we wanted to test the following hypotheses.

\(H_0 \space \mu = 72\)

\(H_a \space \mu \ne 72\)

  1. Translate the alternative hypothesis into words.

ANSWER: The average person does not have a resting heart rate equal to 72 bpm

  1. Calculate the test statistic and p-value to evaluate the hypotheses stated above.
t.test(hr_data$heart_rate, mu = 72, alternative = "two.sided")
## 
##  One Sample t-test
## 
## data:  hr_data$heart_rate
## t = -0.80298, df = 25, p-value = 0.4296
## alternative hypothesis: true mean is not equal to 72
## 95 percent confidence interval:
##  64.04762 75.49085
## sample estimates:
## mean of x 
##  69.76923
  1. State your conclusion of the hypothesis test in the context of the problem.

ANSWER: We fail to reject the null hypothesis as the p value of .4296 is larger than the alpha value of .05. Therefore there is convincing evidence that the average resting heartbeat rate for Gustavus students does lie around 72 bpm.

  1. Below is a randomization distribution for hypothesis test stated above.

  1. What shape is the distribution?

ANSWER: Normal Distribution

  1. At what value is the distribution centered?

ANSWER: Approximately 72 BPM

  1. How likely is it to get a sample average of 69.77 or smaller? Briefly explain your answer.

ANSWER: It is very likely as the standard deviation of this distribution is 14.16. As 66% of all data is within one standard deviations of the mean, this implies that anywhere from approximately 58 to 86 can be expected to appear with some regularity. Since we are asking for 69.77 or smaller and that is above the 58 BPM number that means that it is very likely to appear.

Completed: Mon Nov 14 09:32:17 2022