Question

The US Department of Energy reported that 49% of homes were heated by natural gas. A random sample of 338 homes in Oregon found that 148 were heated by natural gas. Test the claim that proportion of homes in Oregon that were heated by natural gas is different than what was reported. Use a 5% significance level. What are the correct hypotheses

Answer 1

`z=\frac{0.438 -0.49}{\sqrt{(0.49(1-0.49))/(338)}}=-1.912`

The p value for this case would be given by:

`p_v =2*P(z<-1.912)=0.056`

For this case the p value is higher than the significance level so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true proportion is not significantly different from 0.49 or 49%

Explanation:

Information given

n=338 represent the random sample taken

X=148 represent the homes in Oregon were heated by natural gas

`\hat p=(148)/(338)=0.438` estimated proportion of homes in Oregon were heated by natural gas

`p_o=0.49` is the value that we want to test

represent the significance level

z would represent the statistic

`p_v` represent the p value

Hypothesis to test

We want to check if the true proportion of interest is 0.49, then the system of hypothesis are.:

Null hypothesis:
`p=0.49`

Alternative hypothesis:
`p \\eq 0.49`

The statitic is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

Replacing we got:

`z=\frac{0.438 -0.49}{\sqrt{(0.49(1-0.49))/(338)}}=-1.912`

The p value for this case would be given by:

`p_v =2*P(z<-1.912)=0.056`

For this case the p value is higher than the significance level so then we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true proportion is not significantly different from 0.49 or 49%