Question

In: Statistics and Probability

A contractor desires to build new homes with fireplaces. He reads in a survey that more...

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

Solutions

Expert Solution

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.


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