Question

Recycling Do most teens recycle? To find out, an AP Statistics class asked an SRS of 100 students at their school whether they regularly recycle. How many students in the sample would need to say Yes to provide convincing evidence that more than half of the students at the school recycle? The Fathom dotplot below shows the results of taking 200 SRSs of 100 students from a population in which the true proportion who recycle is 0.50. (a) Suppose 55 students in the class’s sample say Yes. Explain why this result does not give convincing evidence that more than half of the school’s students recycle. (b) Suppose 63 students in the class’s sample say Yes. Explain why this result gives strong evidence that a majority of the school’s students recycle.

Answer 1

A.)

According to given data Suppose 55 students in the class’s sample.

n = 100 and x = 55

⇒ p = $\frac{55}{100}$ = 0.55

hence p' = 0.50

so, p > 0.50

Z = $\frac{p-p'}{\sqrt{} \frac{p'(1-p')}{n} }$

Z = $\frac{0.55-0.50}{\sqrt{} \frac{0.50(1-0.50)}{100} }$

so, Z = 1

let level of significance α = 0.05

critical Z value at α = 0.05 for one tail test is 1.645

since Z = 1 < z' = 1.645

do not reject at α = 0.05

there is not sufficient evidence that more than half of the school student recycle.

B.)

According to given data Suppose 63 students in the class’s

given data,

X = 63 and n = 100

now P = 63/100 = 0.63

Z = $\frac{0.63-0.50}{\sqrt{} \frac{0.50(1-0.50)}{100} }$ = 2.6

since, Z = 2.6 > Z' = 1.645

⇒ reject at  α = 0.05

there is sufficient evidence that more than half or majority school student recycle.

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