The US Department of Energy reported that $51 \%$ of homes in the USA were heated by natural gas. A random sample of 350 homes in Oregon found that 154 were heated by natural gas. Test the claim that proportion of homes in Oregon that were heated by natural gas is different from the US reported proportion. Use a 1\% significance level. Give answers accurate to at least 3 decimal places.
What are the correct hypotheses? (Select the correct symbols and use decimal values not percentages.)
Based on the hypotheses, find the following:
Test Statistic $=$
p-value $=$
Based on the above we choose to Reject the null hypothesis
The correct summary would be: There is enough evidence to support the claim 0 that the percent of homes in Oregon heated by natural gas is different than $51 \%$.
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Answer
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 \(51\%\).
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 \(51\%\).
