In: Statistics and Probability
A contractor desires to build new homes with fireplaces. He reads in a survey that more than 80% of all homes buyers want a fireplace. To test this claim, he selected a sample of 30 home buyers and found that 20 wanted a fire place. At a= 0.05, should the claim be rejected?
a. sate the null and alternatives hypothesis and identify the claim
b. calculate the value of test statistics
c. find the critical value
make a decision about the claim at a= 0.05
Solution:
a. sate the null and alternatives hypothesis and identify the claim
Here, we have to use z test for population proportion.
Null hypothesis: H0: About 80% of all homes buyers want a fireplace.
Claim: Alternative hypothesis: Ha: More than 80% of all homes buyers want a fireplace.
H0: p = 0.80 versus Ha: p > 0.80
This is an upper tailed or right tailed (one tailed) test.
b. calculate the value of test statistics
The test statistic formula is given as below:
Z = (p̂ - p)/sqrt(pq/n)
Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size
p̂ = x/n
x = number of items of interest
We are given
Level of significance = α = 0.05
Sample size = n = 30
Number of interested items = x = 20
p = 0.80, q = 1 – p = 1 – 0.80 = 0.20
p̂ = x/n = 20/30 = 0.666666667
Z = (0.666666667 – 0.80)/sqrt(0.80*0.20/30)
Z = -1.8257
P-value = 0.9661
(by using z-table)
c. find the critical value
Critical value = 1.6449
(by using z-table)
d. make a decision about the claim at a= 0.05
P-value > α = 0.05
Test statistic = Z = -1.8257 < Critical value = 1.6449
So, we do not reject the null hypothesis
There is insufficient evidence to conclude that More than 80% of all homes buyers want a fireplace.