Question

Catherine Chao, Director of Marketing Research, needs a sample of Kansas City households to participate in the testing of a new toothpaste package. If 40% of the households in Kansas City prefer the new package, the probability that Catherine's random sample of 300 households will have a sample proportion greater than 0.45 is ___________.

Answer 1

Answer:  0.0384

Step-by-step explanation:

Given : The proportion of the households in Kansas City prefer the new package : p= 0.40

sample size : n= 300

Now, the probability that Catherine's random sample of 300 households will have a sample proportion greater than 0.45 will be :-

$P(p>0.45)=P(z>\dfrac{0.45-0.40}{\sqrt{\dfrac{0.40(1-0.40)}{300}}})\\\\=P(z>1.77)$

 [∵ $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$]

$=1-P(z\leq1.77)$    [∵ P(Z>z)=1-P(Z≤z)]

$=1-0.9616=0.0384$  [using p-value table for z]

Hence, the required probability = 0.0384