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.

qqnorm(scale(hr))
abline(0,1)

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 $n$ .

xbar <- mean(hr)
svar <- var(hr)
sn <- sqrt(svar)
n <- length(hr)

 Sample mean: 75.17241  
 Sample variance: 158.2192 
 Sample standard deviation: 12.57852 
 Sample size: 29

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.

sigma <- 16
alpha <- 0.01
za2 <- qnorm(1-alpha/2)
lo <- xbar - za2 * sigma / sqrt(n)
up <- xbar + za2 * sigma / sqrt(n)

The 99% confidence interval is (67.5193, 82.82553).

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 $S_{n}$ .

ta2 <- qt(1-alpha/2,n-1)
lo <- xbar - ta2 * sn / sqrt(n)
up <- xbar + ta2 * sn / sqrt(n)

The 99% confidence interval is (68.71806, 81.62677).

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.

M <- 2
alpha <- 0.05
za2 <- qnorm(1-alpha/2)
nr <- ceiling(za2^2 * sn^2 / M^2)

The required sample size is 152.

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 $H_{0}$ .

alpha <- 0.05
mu0 <- 73
Ttest <- (xbar - mu0)/(sn/sqrt(n))
ta <- qt(1-alpha,n-1)
pval <- 1 - pt(Ttest,n-1)
dec <- ifelse(pval < alpha, "Reject H0", "Fail to reject H0")

i.   Testing H0: mu <= 73 vs H1: mu > 73 
ii.  Test statistic: 0.9300621
iii. Critical value: 1.701131
iv.  p-value: 0.1801444
v.   Decision: Fail to reject H0

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 $H_{0}$ .

alpha <- 0.10
mu0 <- 70
Ttest <- (xbar - mu0)/(sn/sqrt(n))
ta2 <- qt(1-alpha/2,n-1)
pval <- 2*(1 - pt(abs(Ttest),n-1))
dec <- ifelse(pval < alpha, "Reject H0", "Fail to reject H0")

i.   Testing H0: mu = 70 vs H1: mu != 70
ii.  Test statistic: 2.214434
iii. Critical value: 1.701131
iv.  p-value: 0.03511043
v.   Decision: Reject H0

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.

alpha <- 0.05
mu_star <- 77
mu0 <- 75
gamma_star <- 0.9
beta_star <- 1 - gamma_star
za <- qnorm(1-alpha)
zb <- qnorm(1 - beta_star)
nr <- ceiling(sn^2 *(za + zb)^2/(mu_star - mu0)^2)


mu <- seq(72,80,length=500)
gm <- 1 - pnorm(za - (mu - mu0)/(sn/sqrt(nr)))
plot(gm ~mu,type ="l")
abline(h = gamma_star,lty = 3)
abline(v = mu_star,lty = 3)
abline(v = mu0,col="gray")
abline(h = alpha,col="gray")

The required sample size is 339.

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.

alpha <- 0.01
mu_star <- 80 # pick a value 2 blm away from mu0
mu0 <- 78
gamma_star <- 0.80
beta_star <- 1 - gamma_star
za2 <- qnorm(1-alpha/2)
zb <- qnorm(1 - beta_star)
nr <- ceiling(sn^2 *(za2 + zb)^2/(mu_star - mu0)^2)

mu <- seq(74,82,length=500)
gmr <- 1 - pnorm(za2 - (mu - mu0)/(sn/sqrt(nr)))
gml <- pnorm(-za2 - (mu - mu0)/(sn/sqrt(nr)))
gm <- gmr + gml
plot(gm ~mu,type ="l")
abline(h = gamma_star,lty = 3)
abline(v = mu0 + 2,lty = 3)
abline(v = mu0 - 2,lty = 3)
abline(v = mu0,col="gray")
abline(h = alpha,col="gray")

The required sample size is 462.