STAT 516 hw 1

Author

Karl Gregory

Students in a statistics class were asked to feel their pulses and count the number of heartbeats they felt during two thirty-second time periods separated by several minutes. For each student the two counts were added together in order to obtain a beats per minute (bpm) measurement. These bpm measurements are stored in the vector hr in the R code below:

hr <- c(75,77,86,108,61,46,82,78,81,
        79,55,67,63,94,76,56,69,79,
        66,78,77,91,82,80,73,62,76,80,83)

1)

Check whether it appears the measurements come from a normal distribution by making a normal quantile-quantile plot.

2)

Report the sample mean ${\bar{X}}_{n}$ , sample variance $S_{n}^{2}$ , and sample standard deviation $S_{n}$ of the measurements. Give also the sample size $n$ .

3)

Assume that the resting heart rate measurements are drawn from a normal distribution with unknown mean, but with a known standard deviation of $16$ beats per minute (bpm). Give a $99 %$ confidence interval for the mean resting heart rate based on the measurements from the students.

4)

Now give a $99 %$ confidence interval for the mean resting heart rate based on the measurements of the students, using the estimated standard deviation $S_{n}$ in place of $σ$ . That is, assume now that $σ$ is unknown and must be estimated with $S_{n}$ .

5)

Suppose a researcher wished to estimate the mean resting heart rate of this population of students with a margin of error no greater than $2$ beats per minute with $95 %$ confidence. Using $S_{n}$ as a guess for $σ$ , suggest a sample size which should be just large enough for the researcher.

6)

A researcher wishes to know whether the mean resting heart rate for this population of students exceeds $73$ bpm. Give

The null and alternate hypotheses of interest.
The value of the test statistic.
The critical value for testing the hypothesis at significance level $α = 0.05$ .
The p-value for testing these hypotheses based on the data.
The decision whether or not to reject $H_{0}$ .

7)

A researcher wishes to know whether the mean resting heart rate for this population of students equal to $70$ bpm. Give:

The null and alternate hypotheses of interest.
The value of the test statistic.
The critical value for testing the hypothesis at significance level $α = 0.10$ .
The p-value for testing these hypotheses based on the data.
The decision whether or not to reject $H_{0}$ .

8)

Suppose a researcher wants to test whether the mean resting heart rate in this population of students is greater than $75$ bpm. If the true mean is $77$ bpm or greater, the researcher would like to detect this with probability at least $90 %$ while using a significance level of $α = 0.05$ . Using the sample standard deviation as an estimate of the unknown population standard deviation $σ$ , give a recommended sample size for the researcher.

9)

Suppose a researcher wants to test whether the mean resting heart rate in this population of students is equal to $78$ bpm. If the true mean lies $2$ bpm or more away from this value, the researcher would like to detect this with probability at least $80 %$ while using a significance level of $α = 0.01$ . Using the sample standard deviation as an estimate of the unknown population standard deviation $σ$ , give a recommended sample size for the researcher.