Sally works at a frozen yogurt shop that has about 2,500 customers per week and wants to know how many people like vanilla frozen yogurt. She asks the question to 40 randomly selected customers. She repeats this process two times. Sally finds that 14 people said they like vanilla frozen yogurt in the first sample, 18 in the second, and 12 in third. What is the difference between the highest estimate and the lowest estimate for the number of customers per week who like vanilla frozen yogurt?

The difference is 300 customers.

Step-by-step explanation:

For each week the proportion of customers that like vanilla frozen yogurt is equal to the number of customers that said they like it, divided by the sample (40 for each week)

week 1: p1 = 14/40 = 0.35

week 2: p2 = 18/40 = 0.45

week 3: p3 = 12/40 = 0.3

Then we can calculate the average proportion as:

P = (p1 + p2 + p3)/3 = 0.367

Now, knowing that in a week the shop has about 2500 customers, we can expect that 36.7% of them like the vanilla frozen yogurt,

This is:

N = 2500*0.367 = 916.6

But we can not have a 0.6 of a customer, so we should round it to the nearest whole number, which is 917.

Now, the standard deviation of this will be:

wich is the summation of the squared difference with the mean of each value, divided by the number of repetitions minus one (3 - 1 = 2)

S = 0.075

Then the largest proportion is equal to the mean plus the standard deviation (P + S), and the smallest proportion is equal to the mean minus the standard deviation (P - S):

if we want to calculate the difference between those estimations, we need to calculate:

Diff = (P + S)*2500 - (P - S)*2500 = 2*S*2500 = 2*0.075*2500 = 300

So between the lower and the higher estimations, the difference is 300 customers.

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