Exercise 27 Page 632

Below we have the results of a survey, in which people were asked whether they prefer sports drink X or sports drink Y. We want to know how big the sample needs to be in order for us to be sure that sports drink X is truly preferred by more than half of the population.

Sports Drink X	Sports Drink Y
52 %	48 %

First, recall that when a sample of size n is taken from a large population, the margin of error can be approximated with the following formula. Margin of error = ± 1/sqrt(n) Now, if the percent of the people choosing sports drink X over sports drink Y is 52 %, then the percent of the population who would respond the same way is likely to be less than the margin of error away from 52 %. Therefore, it is likely to be between the two following values. 52 %- 1/sqrt(n) and 52 %+ 1/sqrt(n) In the best scenario, 52 %+ 1sqrt(n) of the population prefers sports drink X. In the worst scenario, only 52 %- 1sqrt(n) prefers the drink. Therefore, to be confident that sports drink X is truly preferred, the lower margin would need to be higher than 50 %. 52 %- 1/sqrt(n) > 50 % Let's solve the inequality.

52 %- 1/sqrt(n) > 50 %

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Solve for n

AddIneq

LHS+ 1/sqrt(n)>RHS+ 1/sqrt(n)

52 % > 1/sqrt(n) + 50 %

SubIneq

LHS-50 %>RHS-50 %

2 % > 1/sqrt(n)

WriteDec

Write as a decimal

0.02 > 1/sqrt(n)

MultIneq

LHS* sqrt(n) > RHS * sqrt(n)

0.02sqrt(n) > 1

DivIneq

.LHS /0.02.>.RHS /0.02.

sqrt(n) > 1/0.02

RaiseIneq

LHS^2>RHS^2

n > (1/0.02)^2

UseCalc

Use a calculator

n > 2500

We found that at least 2500 people need to participate in the survey in order for us to be confident that sports drink X is truly preferred by more than half of the population.