To determine if there is strong evidence against the claim that the percentage of new construction with serious code violations is 8%, we will conduct a hypothesis test for a proportion.

Step 1: State the Hypotheses

• Null hypothesis ($H_0$): $p = 0.08$ (The true proportion of homes with serious violations is 8%)
• Alternative hypothesis ($H_a$): $p > 0.08$ (The true proportion of homes with serious violations is greater than 8%)

Step 2: Collect the Sample### Step 2: Collect the Sample Data

• Sample size ($n$) = 200
• Number of homes with serious violations ($X$) = 21
• Sample proportion ($\hat{p}$) = $\frac{X}{n} = \frac{21}{200} = 0.105$

Step 3: Calculate the Test Statistic

We'll use a one-sample z-test for the proportion.

The test statistic for a proportion is given by:

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

Where:

• $\hat{p}$ is the sample proportion
• $p_0$ is the population proportion under the null hypothesis
• $n$ is the sample size

Substitute the values:

$z = \frac{0.105 - 0.08}{\sqrt{\frac{0.08 \times 0.92}{200}}}$

Calculate the denominator:

$\sqrt{\frac{0.08 \times 0.92}{200}} = \sqrt{\frac{0.0736}{200}} = \sqrt{0.000368} \approx 0.01917$

Now, calculate the test statistic:

$z = \frac{0.105 - 0.08}{0.01917} \approx \frac{0.025}{0.01917} \approx 1.303$

Step 4: Determine the p-value

The p-value for a one-tailed z-test can be found using standard normal distribution tables or statistical software. For $z \approx 1.303$, the p-value is approximately 0.0963.

Step 5: Make a Decision

At a significance level ($\alpha$) of 0.05, we compare the p-value to $\alpha$:

• If the p-value $\leq$ $\alpha$, we reject the null hypothesis.
• If the p-value > $\alpha$, we fail to reject the null hypothesis.

In this case, the p-value (0.0963) is greater than the significance level (0.05), so we fail to reject the null hypothesis.

Conclusion

There is not strong evidence against the claim that the percentage of new construction with serious code violations is 8%.

Therefore, the correct answer is:

b. No, because the p-value is 0.0963

Would you like more details or have any questions? Here are some additional questions you might be interested in:

1. How do you interpret p-values in hypothesis testing?
2. What is the difference between one-tailed and two-tailed tests?
3. How do you calculate the sample size needed for a specific confidence level?
4. What are the assumptions of the z-test for proportions?
5. How can you use statistical software to find p-values?

Tip: Always check if the assumptions for the statistical test you are using are met before performing the test.