Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim.
A safety administration conducted crash tests of child booster seats for cars. Listed below are results from those​ tests, with the measurements given in hic​ (standard head injury condition​ units). The safety requirement is that the hic measurement should be less than 1000 hic. Use a 0.05 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified​ requirement?
735     693     1196     692     590     768

To test the given claim, we will perform a hypothesis test for the population mean. Here are the steps involved:

1. **Identify the null and alternative hypotheses:**
   - Null hypothesis (\(H_0\)): \(\mu \geq 1000\) (The mean HIC is at least 1000)
   - Alternative hypothesis (\(H_1\)): \(\mu < 1000\) (The mean HIC is less than 1000)

2. **Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)):**
   - Sample dataTo test the given claim, we will perform a hypothesis test for the population mean. Here are the steps involved:

1. **Identify the null and alternative hypotheses:**
   - Null hypothesis (\(H_0\)): \(\mu \geq 1000\) (The mean HIC is at least 1000)
   - Alternative hypothesis (\(H_1\)): \(\mu < 1000\) (The mean HIC is less than 1000)

2. **Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)):**
   - Sample  735, 693, 1196, 692, 590, 768
   - Sample size (\(n\)): 6

   \[
   \bar{x} = \frac{735 + 693 + 1196 + 692 + 590 + 768}{6}
   \]
   \[
   \bar{x} = \frac{4674}{6} = 779
   \]

   Now calculate the sample standard deviation (\(s\)):

   \[
   s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
   \]

   \[
   s = \sqrt{\frac{(735-779)^2 + (693-779)^2 + (1196-779)^2 + (692-779)^2 + (590-779)^2 + (768-779)^2}{5}}
   \]

   \[
   s = \sqrt{\frac{(1936) + (7396) + (174636) + (7569) + (35721) + (121)}{5}}
   \]

   \[
   s = \sqrt{\frac{225379}{5}} = \sqrt{45075.8} \approx 212.31
   \]

3. **Calculate the test statistic (t):**
   \[
   t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
   \]
   where \(\mu_0 = 1000\)

   \[
   t = \frac{779 - 1000}{212.31 / \sqrt{6}}
   \]
   \[
   t = \frac{779 - 1000}{86.687} \approx -2.55
   \]

4. **Determine the P-value:**
   Using the t-distribution table or a calculator, find the P-value for \(t = -2.55\) with \(df = n - 1 = 5\).

   For \(t = -2.55\) and \(df = 5\), the P-value is approximately 0.026.

5. **State the final conclusion:**
   - Significance level (\(\alpha\)): 0.05
   - Since the P-value (0.026) is less than the significance level (0.05), we reject the null hypothesis.

   The results suggest that the mean HIC measurement is less than 1000. Thus, it indicates that the child booster seats meet the specified safety requirement.

### Summary:
- **Null Hypothesis (\(H_0\))**: \(\mu \geq 1000\)
- **Alternative Hypothesis (\(H_1\))**: \(\mu < 1000\)
- **Test Statistic**: \(t = -2.55\)
- **P-value**: 0.026
- **Conclusion**: Reject the null hypothesis. The sample data suggests that the mean HIC measurement is less than 1000.

Would you like more details or have any questions?

Here are five questions you might want to ask:
1. How do you calculate the sample standard deviation step-by-step?
2. What is the significance level in hypothesis testing?
3. Can you explain how to use the t-distribution table to find the P-value?
4. What are Type I and Type II errors in hypothesis testing?
5. How does the central limit theorem apply to hypothesis testing?

**Tip:** Always check if the sample data follows a normal distribution when performing a t-test, as this is one of the key assumptions for the validity of the test.

Explore a hypothesis test example using crash test data of child booster seats for car safety. Analyze the sample mean, sample standard deviation, and t-test statistic to determine if the mean HIC measurement meets safety requirements.

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