Question 1181350: In 2001, the mean household expenditure for energy was $1493, according to a survey. An economist wanted to know whether this amount has changed significantly from its 2001 level. In a random sample of 35 households, he found the mean expenditure (in 2001 dollars) for energy during the most recent year to be $1618, with a standard deviation $321. Is there evidence to conclude that the mean household expenditure for energy has increased since 2001? Use the α = 0.01 level of significance.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to conduct a hypothesis test to answer this question:
**1. State the Hypotheses:**
* **Null Hypothesis (H₀):** The mean household expenditure for energy has not increased since 2001. (μ ≤ $1493)
* **Alternative Hypothesis (H₁):** The mean household expenditure for energy has increased since 2001. (μ > $1493) This is a right-tailed test.
**2. Determine the Level of Significance:**
α = 0.01
**3. Calculate the Test Statistic:**
Since the sample size is greater than 30, we can use a z-test. The formula for the z-statistic is:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
z = (1618 - 1493) / (321 / sqrt(35))
z = 125 / (321 / 5.916)
z = 125 / 54.26
z ≈ 2.30
**4. Determine the Critical Value:**
For a right-tailed test with α = 0.01, we look up the z-score that corresponds to 0.99 in the standard normal distribution table (or use a calculator). The critical value is approximately 2.33.
**5. Make a Decision:**
Our calculated z-statistic (2.30) is *less than* the critical value (2.33). Therefore, we *fail to reject* the null hypothesis.
**6. State the Conclusion:**
There is *not* sufficient evidence at the 0.01 level of significance to conclude that the mean household expenditure for energy has increased since 2001. Although the sample mean is higher, the difference is not statistically significant.
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