Question

A first year statistics class took part in a simple experiment. Each person took their pulse rate. They then flipped a coin and, if it came up tails, they ran in place for one minute. Everyone then took their pulse a second time. Of the 92 students in the class, 32 ran in place. In repetitions of this experiment, the number that run in place should have a binomial distribution with n = 92 trials and probability of success p = 1/2. a. Simulate 100 repetitions of this experiment using the commands given below. In how many of the 100 experiments did 32 or fewer people have to run? b. Make a histogram of the 100 observations and describe its important characteristics (shape, location, spread and outliers). c. Using software, calculate the probability of getting 32 or fewer tails in 92 tosses of a fair coin. d. Does it seem likely that only 32 of the 92 students got tails? Give a reasonable explanation for what happened.

Answer 1

The number of students that run in place has a binomial distribution with trials and probability of success

a) The experiment is simulated 100 times. R code below.
set.seed((57778))
n <- 100
n32 <- 0
n32_less <- 0
num_Tails <- array(dim = n)
for (i in 1:n)
{
X <- sample(c("T","H"),size=92,replace = TRUE, prob=c(0.5,0.5))
Tails <- length(X[which(X=="T")])
if (Tails==32)
{
n32 <- n32 + 1
}
if (Tails <=32)
{
n32_less <- n32_less + 1
}
num_Tails[i] <- Tails
}
n32/n
n32_less/n

In 1 of the 100 experiments did 32 or fewer people have to run.

b) The histogram of the observations is plotted below.

Histogram 30 35 40 45 50 Trial

R code below.

plot(1:1)
dev.new()
hist(num_Tails, col="sky blue", lwd=2, xlab="Trial", ylab = "Number of tails", main="Histogram")

Observations below 32 and above 55 are outliers. They are very unlikely. The histogram looks like Binomial distribution.

The distribution peaks around 46.

c)  The probability of getting 32 or fewer tails in 92 tosses of a fair coin is ${\color{Blue} P\left ( X\leqslant 32 \right )=0.01}$

d) The probability of getting 32 of the 92 students got tails is ${\color{Blue} P\left ( X= 32 \right )=0.01}$

Since the coin is fair in 92 tosses the mean of the distribution is 46. So most probably 40 to 50 tails are most likely.