Question

1. customers use the starbucks app for a variety of reasons, such as viewing the menu, ordering ahead, and paying for their order. of the customers at starbucks, 61.4% of them use the app. suppose that a random sample of 20 customers at starbucks is selected.

a. consider the probability that exactly 12 people in the sample use the starbucks app. define the random variable of interest and state how the random variable is distributed.
b. determine the probability that exactly 12 people in the sample use the starbucks app.

Explanation:

Step1: Define random variable

Let $X$ = number of customers in the sample who use the Starbucks app.

Step2: State distribution

$X$ follows a binomial distribution: $X \sim Binomial(n=20, p=0.614)$, since we have independent trials with two outcomes (uses app or not), fixed sample size $n=20$, and constant success probability $p=0.614$.

Step3: Recall binomial probability formula

The formula for exact binomial probability is:
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Step4: Substitute values for $k=12$

Substitute $n=20$, $k=12$, $p=0.614$, $1-p=0.386$:
$$P(X=12) = \binom{20}{12} (0.614)^{12} (0.386)^{8}$$
First calculate $\binom{20}{12} = \frac{20!}{12!8!} = 125970$
Then calculate $(0.614)^{12} \approx 0.00317$, $(0.386)^{8} \approx 0.000437$
Multiply the terms: $125970 \times 0.00317 \times 0.000437 \approx 0.175$

Answer:

a. Random variable: $X$ = number of sample customers using the Starbucks app. $X$ follows a binomial distribution: $X \sim Binomial(n=20, p=0.614)$
b. $\approx 0.175$