Questions: In 2001, the mean household expenditure for energy was 1493 according to data from the U.S. Energy Information Administration. An economist wanted to know whether this amount has changed significantly from its 2001 level. In a random sample of 35 households, the economist found the mean expenditure (in 2001 dollars) for energy during the most recent year to be 1618 with a standard deviation of 321. State the hypotheses the economist should use.

In 2001, the mean household expenditure for energy was 1493 according to data from the U.S. Energy Information Administration. An economist wanted to know whether this amount has changed significantly from its 2001 level. In a random sample of 35 households, the economist found the mean expenditure (in 2001 dollars) for energy during the most recent year to be 1618 with a standard deviation of 321. State the hypotheses the economist should use.

Transcript text: In 2001, the mean household expenditure for energy was $\$ 1493$ according to data from the U.S. Energy Information Administration. An economist wanted to know whether this amount has changed significantly from its 2001 level. In a random sample of 35 households, the economist found the mean expenditure (in 2001 dollars) for energy during the most recent year to be $\$ 1618$ with a standard deviation of \$321. State the hypotheses the economist should use.

Solution

Solution Steps

Step 1: State the Hypotheses

We will conduct a hypothesis test to determine if the mean household expenditure for energy has changed from its 2001 level of \$1493. The hypotheses are defined as follows:

Null Hypothesis ($H_0$): $ \mu = 1493 $ (The mean expenditure has not changed)
Alternative Hypothesis ($H_a$): $ \mu \neq 1493 $ (The mean expenditure has changed)

Step 2: Calculate the Standard Error

The standard error ($SE$) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{321}{\sqrt{35}} \approx 54.2589 \]

Step 3: Calculate the Test Statistic

The test statistic ($Z_{test}$) is calculated using the formula:

\[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{1618 - 1493}{54.2589} \approx 2.3038 \]

Step 4: Calculate the P-value

For a two-tailed test, the p-value ($P$) is calculated as follows:

\[ P = 2 \times (1 - T(|z|)) \approx 0.0212 \]

Step 5: Conclusion

To determine if we reject the null hypothesis, we compare the p-value to the significance level ($\alpha = 0.05$). Since $P \approx 0.0212 < 0.05$, we reject the null hypothesis.