Question

using a 5% significance level.
a. what type of test will be used in this problem? select an answer
b. enter the null hypothesis for this test. $h_0$ ? ?
c. enter the alternative hypothesis for this test. $h_1$: ? ?
d. is the original claim located in the null or alternative hypothesis? select an answer
e. what is the test - statistic for the given statistics?
f. what is the p - value for this test?
g. what is the decision based on the given statistics? select an answer
h. what is the correct interpretation of this decision? using a 5% level of significance, there select an answer sufficient evidence to select an answer the claim that more than 50% of homes were heated by natural gas.

Explanation:

Step1: Identify test type

Since we are testing a claim about a proportion (more than 50% of homes heated by natural gas), it is a one - proportion z - test.

Step2: Formulate null hypothesis

The null hypothesis for a claim about a proportion $p$ is usually of the form $H_0:p = p_0$. Here, $p_0 = 0.5$, so $H_0:p=0.5$.

Step3: Formulate alternative hypothesis

The claim is that more than 50% of homes were heated by natural gas, so the alternative hypothesis is $H_1:p > 0.5$.

Step4: Locate original claim

The original claim "more than 50% of homes were heated by natural gas" is in the alternative hypothesis.

Step5: Calculate test - statistic (assuming sample proportion $\hat{p}$, sample size $n$)

The test - statistic for a one - proportion z - test is $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$. But since sample statistics are not given, we cannot calculate the exact value.

Step6: Calculate p - value

For a right - tailed test with test - statistic $z$, the p - value is $P(Z>z)$ where $Z$ is a standard normal random variable. Without the value of $z$, we cannot calculate the exact p - value.

Step7: Make a decision

If the p - value is less than the significance level $\alpha = 0.05$, we reject the null hypothesis. If $p\geq0.05$, we fail to reject the null hypothesis.

Step8: Interpret the decision

If we reject the null hypothesis, there is sufficient evidence to support the claim that more than 50% of homes were heated by natural gas. If we fail to reject the null hypothesis, there is not sufficient evidence to support the claim that more than 50% of homes were heated by natural gas.

Answer:

a. One - proportion z - test
b. $H_0:p = 0.5$
c. $H_1:p>0.5$
d. Alternative hypothesis
e. Cannot be calculated without sample statistics
f. Cannot be calculated without test - statistic
g. Cannot be determined without p - value
h. If reject $H_0$: There is sufficient evidence to support the claim that more than 50% of homes were heated by natural gas. If fail to reject $H_0$: There is not sufficient evidence to support the claim that more than 50% of homes were heated by natural gas.