Question

The significance level is 0.042% and alternative hypothesis is: p<0.7.

Answer 1

In hypothesis testing, the significance level (often denoted as \(\alpha\)) defines the threshold for rejecting the null hypothesis (H0). A significance level of 0.042% (or 0.00042 in decimal form) indicates a very stringent criterion for significance.

If your alternative hypothesis (H1) is \(p < 0.7\), then your null hypothesis would typically be \(H0: p \geq 0.7\). In this context:

- **Null Hypothesis (H0)**: The population proportion \(p\) is equal to or greater than 0.7.
- **Alternative Hypothesis (H1)**: The population proportion \(p\) is less than 0.7.

Here’s how you might proceed with the hypothesis test:

1. **Collect Data**: Gather your sample data to calculate the sample proportion (\( \hat{p} \)).

2. **Choose the Right Test**: Since you're dealing with proportions, you would typically use a one-sample z-test for proportions.

3. **Calculate the Test Statistic**: The test statistic for a proportion can be calculated using the formula:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

where:
- \(\hat{p}\) is the sample proportion,
- \(p_0\) is the hypothesized population proportion (0.7 in this case), and
- \(n\) is the sample size.

4. **Determine the Critical Value**: For a significance level of 0.042%, you can find the critical z-value for a one-tailed test. You would look up the z-score that corresponds to the area of 0.042% in the left tail of the standard normal distribution. This will be a very small negative z-value because you are looking for a very stringent criterion for rejecting the null hypothesis.

5. **Make a Decision**:
- If your calculated z-statistic is less than the critical z-value, you reject the null hypothesis in favor of the alternative hypothesis.
- If your calculated z-statistic is greater than the critical z-value, you do not reject the null hypothesis.

6. **Draw Conclusions**: Based on your decision, you will either have evidence to support that the population proportion is indeed less than 0.7, or you do not have sufficient evidence to reject the null hypothesis.

It is also important to note that a significance level of 0.042% is quite low, indicating that you are requiring strong evidence to support the claim that \(p < 0.7\).