Final answer:
The correct hypotheses for the test are H0: p = 0.45 and HA: p ≠ 0.45. The test statistic is 2.3708 and the p-value is approximately 0.0177.
Step-by-step explanation:
To test the claim that the proportion of homes in Oregon that were heated by natural gas is different than what was reported, we need to set up the hypotheses:
H0: p = 0.45 (The proportion of homes in Oregon heated by natural gas is the same as reported)
HA: p ≠ 0.45 (The proportion of homes in Oregon heated by natural gas is different than reported)
Next, we need to find the test statistic and p-value:
The test statistic is calculated using the formula: z = (p - P) / sqrt(P(1 - P) / n),
where p is the sample proportion, P is the proportion reported by the US Department of Energy, and n is the sample size.
Using the given values, we have: p = 169/315 = 0.537, P = 0.45, and n = 315.
Substituting these values into the formula, we get: z = (0.537 - 0.45) / sqrt(0.45(1 - 0.45) / 315) = 2.3708
The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. To find the p-value, we can use a standard normal distribution table or a calculator. In this case, the p-value is approximately 0.0177.