Question

Show your calculations in detail please..

1. Suppose that a researcher is conducting a study on fast food habits and assumes that 55% of adults prefer hamburger to pizza. He randomly selects 10 adults and ask whether they prefer hamburger to pizza or not. Based on this information, answer the following questions.

a) (20 pts) Define the random variable (in words) and check whether it is a binomial random variable or not and write your conclusion. Explain your answer.

b) (10 pts) What is the probability that at least 8 adults prefer hamburger?

c) (10 pts) What is the probability that as most 3 adults prefer hamburger?

d) (15 pts) What is the expected number of people who prefer hamburger.

e) (15 pts) Calculate the standard deviation of this random variable and interpret it.

Answer 1

Question (a)

The random variable here is number of people who prefer hamburger to pizza

The conditions for a random vairable to be a binomail random variable is

1. The number of trails are fixed. Here number of trails is 10. So the condition is satiisfied

2. Every trail has only two outcomes. Here the outcomes are Yes or No when asked about whether they prefer hamburger to pizza or not. So the condition is satiisfied

3. The probability of success p is equal for all the trails, Here 55% of adults prefer hamburger to pizza. So it is same for all the trails. So the condition is satiisfied

4. The trials are independent of each other. Here the researcher is selecting 10 different people. Hence their choices wull be independent of one another. So the condition is satiisfied

So the random variable is a binomail random variable

Question (b)

Given number of people he selected n = 10

Probability of success p = 0.55

q = 1 - p

= 0.45

Probability of x successes in n trials in a binomial distribution = * px * qn-x where = n! / x! (n-x)!

Here we need P(x8) which is P(x=8) + P(x=9) + P(x=10)

So P(x8) = * 0.558 * 0.4510-8 + * 0.559 * 0.4510-9 + * 0.5510 * 0.4510-10

= 0.076303 +0.020724 + 0.002533

= 0.099560

probability that at least 8 adults prefer hamburger = 0.099560

Question (c)

Given number of people he selected n = 10

Probability of success p = 0.55

q = 1 - p

= 0.45

Probability of x successes in n trials in a binomial distribution = * px * qn-x where = n! / x! (n-x)!

Here we need P(x3) which is P(x=0) + P(x=1) + P(x=2) + P(x=3)

So P(x3) = * 0.550 * 0.4510-0 + * 0.551 * 0.4510-1 + * 0.552 * 0.4510-2 +   * 0.553 * 0.4510-3

= 0.000341 + 0.004162 + 0.022890 + 0.074603

= 0.101995

probability that atmost 3 adults prefer hamburger = 0.101995

Question (d)

Expected number of people who prefer hamburger. = n*p

= 10 *0.55

= 5.5

Question (e)

Standard deviation of this random variable = n * p * q

=10*0.55*0.45

=2.475

= 1.573213

Standard deviation gives a measure of how the far or near the values are away from the mean or expected value of the data

Here the expected number of people who prefer hamburger is 5.5

So the standard deviation tells us that the values of the random variables here are within 1.573123 of the mean 5.5