Question

A direct mail company wishes to estimate the proportion of people on a large mailing list that will purchase a product. Suppose the true proportion is 0.07. If 343 are sampled, what is the probability that the sample proportion will be less than 0.11

Answer 1

The probability that the sample proportion will be less than 0.11 = .9982

Explanation:

Given -

Suppose the true proportion is 0.07 .

true proportion
`(\\u _{\widehat{p}})` = p = 0.07

q = 1 - p = 1 - 0.07 = 0.93

n = 343

Standard deviation
`(\sigma _{\widehat{p}})` =
`\sqrt{(p* q)/(n)}` =
`\sqrt{(0.07* 0.93)/(343)}` = .0137

the probability that the sample proportion will be less than 0.11 =

`P(\widehat{p}< 0.11)` =
`P(\frac{\widehat{p} - \\u _{\widehat{p}}}{\sigma _{\widehat{p}}}<( 0.11 - 0.07)/(.0137))` using[
`z = \frac{\widehat{p} - \\u _{\widehat{p}}}{\sigma _{\widehat{p}}}`]

=
`(z< 2.91)`

= .9982