I need help with the graph and everything else tbh...

Question

To introduce his class to binomial distributions, Mr. Miller does the "Pop

quiz" activity at the beginning of this section (page 431). Each student in the

class guesses an answer from A through E on each of the 10 multiple-choice

questions. Mr. Miller determines the "correct" answer for each of the 10

questions randomly so that A, B, C, D, or E was equally likely to be chosen.

Hannah is one of the students in this class. Let Y = the number of questions that

Hannah answers incorrectly.

1. Use technology to make a histogram of the probability distribution of Y. Describe

its shape.

2. Calculate and interpret the mean of Y.

3. Calculate and interpret the standard deviation of Y.

4. On page 440, we defined X = the number of correct answers that Hannah got on

the quiz. How do the shape, center, and variability of the probability distribution

of X compare to your answers for Questions 1 to 3?

Answer 1

Hi Sylvian,

I don't know what technology you use, but I used R to get your initial histogram for 1. I cannot post a graph due to character limits, but code is below So:

#1.

success<-0:10

plot(success, dbinom(success, size=10, prob=0.8), type='h')

size=number of questions, 10 in this case

"success"=probability of success, which. ironically, means "incorrect answer" in our case.

A note on the binomial distribution: we can define success and failure any way we want. Anyway:

size=10

success=0.8--probability that Hannah will answer any question wrong is 4/5, or 0.8.

Distribution appears slightly left-skewed.

#2. For any binomial distribution, if we are looking for number of successes:

mu=np

n=number of trials

p=probability of success

Again, we are ironically defining success as missing multiple-choice questions, so:

n=10

p=0.8

mu=10*0.8= 8

#3. Formula for standard deviation of binomial distribution is:

sigma=sqrt(n*p*(1-p)) where:

sigma=standard deviation

n=number of trials

p=probability of "success"

Again, we define success ironically here as incorrect answers, so:

n=10

p=0.8

sigma=sqrt(10*0.8*0.2)

sigma=1.265

#4. Center--essentially mean--should be much lower for x here than for y in the previous scenario. Recall:

mu=np, so, if we define success as correct answer for x:

mu_x=10*0.2=2

mu_y=10*0.8=8

Variability (spread) is generally measured by standard deviation, which is identical. Recall:

sigma=sqrt(n*p*(1-p))

It will not matter which probability we call p because of the inclusion of 1-p.

sigma_x=sqrt(10*0.2*0.8)=1.265

sigma_y=sqrt(10*0.8*0.2)=1.265

So, spread or standard deviation should be essentially identical for X and Y.

As for shape, I can't post a graph here due to character limits, but just use the same R code as in part #1 and change the probability to 0.2. You will get a right-skewed histogram. Shape of X is slightly right-skewed while shape of y is slightly left-skewed.

I hope this helps.