Step 1 :State the null hypothesis and the alternative hypothesis. The null hypothesis is that the proportion of homes in Oregon that are heated by natural gas is the same as the US reported proportion, i.e., \(0.51\). The alternative hypothesis is that the proportion of homes in Oregon that are heated by natural gas is different from the US reported proportion.
Step 2 :Calculate the test statistic using the formula for the z-score, which is (sample proportion - population proportion) / standard error. The standard error can be calculated using the formula \(\sqrt{(p*(1-p))/n}\), where p is the population proportion and n is the sample size.
Step 3 :Calculate the p-value using the z-score. If the p-value is less than the significance level (in this case, \(0.01\)), we reject the null hypothesis.
Step 4 :The test statistic is approximately \(-2.620\) and the p-value is approximately \(0.009\). Since the p-value is less than the significance level of \(0.01\), we reject the null hypothesis.
Step 5 :There is enough evidence to support the claim that the proportion of homes in Oregon that are heated by natural gas is different from the US reported proportion of \(51\%\).
Step 6 :The correct hypotheses are \(H_0: p = 0.51\) and \(H_a: p \neq 0.51\). The test statistic is approximately \(\boxed{-2.620}\) and the p-value is approximately \(\boxed{0.009}\). Based on these results, we reject the null hypothesis. There is enough evidence to support the claim that the proportion of homes in Oregon that are heated by natural gas is different from the US reported proportion of \(51\%\).