Question

1. (S7.2) According to the Mars company, 20% of Peanut Butter M&Ms are orange. Calculate and interpret what you’d expect if you took a sample of 100 Peanut Butter M&Ms:

    ??_??=                       ????=         

       We expect               % of Peanut Butter M&Ms to be orange, give or take                                                                                                               %.

2. (S7.3) ) At a particular movie theater, one out of four patrons request extra butter on their popcorn.

1. Suppose you randomly sample 117 patrons buying popcorn at the movie theater. What are the approximated parameters of the sampling distribution of ??̂ for samples of n = 117 where “success” is defined as requesting extra butter?

                                                                   ??_?? =                              ??????     ???? =

2. What conditions need to be satisfied for CLT to apply?

3. What is the probability that 30% or more of the patrons from your sample of 117 request extra butter?

Answer 1

Q7.2) The mean and standard deviation of the sampling distribution of proportion here is computed as:

$\mu_p = p = 0.2$

$SS_p = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.2*0.8}{100}} = 0.04$

The interpretation of this is that we expect 20% of Peanut butter M&Ms to be orange, give or take 4% here.

Q7.3) a) Using the same methodology, the distribution here is obtained as:

$\mu_p = 1/4 = 0.25$

$SS_p = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.25*0.75}{117}} = 0.0400$

b) We have here:
np = 0.25*117 = 29.25 > 5
n(1-p) = 117*(1 - 0.25) = 87.75 > 5

Therefore the conditions are satisfied here.

c) The probability that 30% or more of the patrons from your sample of 117 request extra butter is computed here as:

P( p >= 0.3)

Converting it to a standard normal variable, we have here:

$P(Z > \frac{0.3 - 0.25}{0.04})$

Getting it from the standard normal tables, we have here:

Therefore 0.1056 is the required probability here.