A building inspector believes that the percentage of new construction with serious code violations may be even greater than the previously claimed 8%. She conducts a hypothesis test on 200 new homes and finds 21 with serious violations. At a 5% significance level, is this strong evidence against the 8% claim?
Group of answer choices

a. Yes, because the p-value is 0.0963

b. No, because the p-value is 0.0963

c. Yes, because the p-value is 0.034

d. No, because the p-value is over 2.0

e. No, because the p-value is 0.034

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.

Explore a hypothesis test for proportions to determine if the percentage of new construction with serious code violations differs from the claimed 8%. Includes steps for hypothesis testing, calculation of the test statistic and p-value, and decision-making process. Case study on 200 new homes with 21 found in violation. Learn more!

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