Question

I'm struggling with this problem. How would you calculate the P-Value and what would it tell you?

In 2008, 17% of homes in North County had been contacted by a member of the Solar Solutions sales team. Recently a sample of 220 homes across several North County neighborhoods indicated that 21% had been contacted. At alpha = 0.05 can it be supported that the true proportion of Solar Solutions contacted homes has actually increased?

Answer 1

Sol:

here when stattemnt says

In 2008, 17% of homes in North County had been contacted by a member of the Solar Solution

he is referrring to population proportion,p

p=17%=0.17

And also,

a sample of 220 homes across several North County neighborhoods indicated that 21% had been contacted.

n=220

p^=sample proportion=21%=0.21

we need to check the true proportion of Solar Solutions contacted homes has actually increased

set null hypothesis

H0:p=0.17

Alternative hypothesis

H1:p>0.17

alpha=0.05

decsion rule is

if p<0.05 reject null hypothesis and support the claim

if p>0.05 do not reject HO and do not support the claim

First calculate tst statistic as

Z=p^-p/sqrt(p(1-p)/n)

=0.21-0.17/sqrt(0.17*(1-0.17)/220)

Z=1.579

TO GET P VALUE IN R type below one line code

pnorm(1.579,mean=0,sd=1,lower.tail = FALSE)

0.05716803

p=0.05716803 that is only in 0.057*100=6% cases results is true and

that is the probability is gretaer than 0.05 that the observed sample prportion is would have occured by chance if the null hypothesis were true

p>0.05

Do not reject Null hypothesis

conclusion:

there is no sufficient statistical evidence at 5% level of significance to conclude that the true proportion of Solar Solutions contacted homes has actually increased