Question

An appliance manufacturer claims to have developed a compact microwave oven that consumes a mean of no more than 260 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. A consumer group has decided to try to discover if the claim appears true. They take a sample of 20 microwave ovens and find that they consume a mean of 265 W. For a test with a level of significance of 0.05, the critical value would be ________.

Answer 1

To find the critical value for the hypothesis test, we first need to identify the hypotheses and the sampling distribution.

1. **State the Hypotheses:**
- Null Hypothesis (\(H_0\)): \(\mu \leq 260\) (the mean power consumption is no more than 260 W)
- Alternative Hypothesis (\(H_a\)): \(\mu > 260\) (the mean power consumption is greater than 260 W)

Since we are testing whether the mean is greater than 260 W, this is a right-tailed test.

2. **Level of Significance:**
- \(\alpha = 0.05\)

3. **Determine the Critical Value:**
Since the population standard deviation is known and the sample size is 20, we will use the z-test for this analysis.

We need to find the z-value that corresponds to the right-tailed area of 0.05 in the standard normal distribution. This is found using a z-table or a calculator:

The critical z-value for \( \alpha = 0.05 \) in a right-tailed test is approximately:
\[
z = 1.645
\]

Thus, the critical value for this hypothesis test at a significance level of 0.05 is:
\[
\boxed{1.645}
\]