I am very confused how to approach this question, I would appreciate a detailed step by step response if possible thank you : )

Question

Andy is a restaurant owner. He believes 75% of his customers are satisfied with the food quality of his restaurant. From a random sample of 85 customers, what are the following probabilities? (Round your answers to four decimal places, if needed.)

(a) What is the probability that less than 64 customers are satisfied with the food quality?

(b) What is the probability that at least 64 customers are satisfied with the food quality?

(c) What is the probability that the sample proportion of customers who are satisfied with the food quality is between 72% and 79%?

Answer 1

You can get exact answers for the probability of x successes in n trials with probability p using the binomial distribution. There is a formula for this - P(x) = C(n,x)p^x(1-p)^(n-x) C(n,x), read n choose x, can be written as n!/(x!(n-x)!).

To get the probability of a range of values, we need to use the cumulative binomial distribution (the sum from - infinity to x, which in this case is 0 to x).

Alternatively, we may use a normal approximation of the binomial distribution, with or without a continuity correction.

The best way to get the cumulative distribution result is to use a calculator or program like Excel that produces these results.

In Excel, it is binom.dist(x,n,p,1) - the 1 means the cumulative distribution; 1 would mean the probability of exactly x.

a) < 64 is binom.dist(63, 85, .75, 1) = 0.466738180743341; note the presence of 63 rather than 64, as it is less than 64

b) 1 - the answer to a is 1 - 0.466738180743341 = 0.533261819256658

c) The probability that there is between 72% and 79%

.72 * 85 = 61.2

.79 * 85 = 67.15

Now, it becomes a matter of how we interpret "between"

Note that 61 would be 61/85 = 71.76%, which would round to 72%, and 67 would be 78.82%, which would round to 79%.

The probability of x being between 61 and 67, inclusive, is binom.dist(67, 85, .75, 1) - binom.dist(60, 85, .75, 1) = .6199. Note how it is 60 rather than 61 in the second formula.

If we interpret the percentages as being exact rather than rounded values, the 61 value of x would not count, and the result would be binom.dist(67, 85, .75, 1) - binom.dist(61, 85, .75, 1) = .5439

The other way to do problems like this is with a normal approximation. If you think this is what is needed, let me know.