Question

A) If you would like to perform a pilot study in order to state with 90% confidence that the sample proportion of adults that purchased used items instead of new ones is within 0.044 of the actual population proportion, how large would your random sample need to be? Remember to round up the next larger whole number.

B)If a previous study found that 50% of adults purchase used items instead of new ones, how large would your random sample need to be to perform a follow up study to confirm your initial findings (note that you want to be within 0.1 of the actual population proportion with 90% confidence)?

C) A recent Harris Poll on green behavior showed that 25% of adults often purchased used items instead of new ones. If a random sample of 70 adults is used, what is the probability that fewer than 8 of the sampled adults purchase used items instead of new ones? Round to the nearest thousandth.

Answer 1

A)

Answer)

We need to use standard normal z table to estimate the sample size

Here value of p is not given so we will use the best estimate as 0.5

Error = 0.044

Critical value z from z table for 90% confidence level is 1.645

Error = z*√p*(1-p)/√n

We need to find n

0.044 = 1.645*√0.5*0.5/√n

N = 350

B)

Here

Error = 0.1

P = 0.5 (50%)

Z again = 1.645 {90% confidence level}

0.1 = 1.645*√0.5*0.5/√n

N = 68

C)

N = 70

P = 0.25

First we need to check the conditions of normality that is if n*p and n*(1-p) both are greater than 5 or not

N*p = 17.5

N*(1-p) = 52.5

Both the conditions are met so we can use standard normal z table to estimate the probability

Z = (x-mean)/s.d

Mean = n*p = 17.5

S.d = √{n*p*(1-p)} = 3.62284418654

We need to find

P(x<8)

By continuity correction

P(x<7.5)

Z = (7.5 - 17.5)/3.62284418654 = -2.76

From z table, P(z<-2.76) = 0.0029