### Step-by-Step Solution
Given Data:
- Sample data: [tex]\( \{810, 744, 1203, 604, 624, 527\} \)[/tex]
- Desired population mean (μ) for safety: 1000 hic
- Significance level (α): 0.05
Step 1: State the Hypotheses
The hypotheses to test the claim that the sample is from a population with a mean less than 1000 hic are:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( \mu = 1000 \)[/tex] hic
- Alternative hypothesis ([tex]\(H_1\)[/tex]): [tex]\( \mu < 1000 \)[/tex] hic
So, the correct answer is C:
[tex]\[ \text{C. } H_0: \mu = 1000 \text{ hic}, \; H_1: \mu < 1000 \text{ hic} \][/tex]
Step 2: Calculate the Sample Statistics
From the given data:
- Sample mean ([tex]\(\bar{x}\)[/tex]): 752.0 hic
- Sample standard deviation (s): 243.204 hic
- Sample size (n): 6
Step 3: Calculate the Test Statistic
The test statistic for a t-test is calculated using the formula:
[tex]\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \][/tex]
Here:
[tex]\[
\bar{x} = 752.0, \; \mu = 1000, \; s = 243.204, \; n = 6
\][/tex]
Plugging the values into the formula, we get:
[tex]\[
t = \frac{752.0 - 1000}{243.204 / \sqrt{6}} \approx -2.498
\][/tex]
So, the test statistic is:
[tex]\[ t = -2.498 \][/tex]
Step 4: Calculate the P-value
Given the t-statistic of -2.498 and degrees of freedom (df) equal to [tex]\( n - 1 = 5 \)[/tex]:
- The P-value is approximately 0.027 from the t-distribution table or using statistical software.
Step 5: Compare the P-value with the Significance Level
- Significance level (α): 0.05
- P-value: 0.027
Since [tex]\( \text{P-value} < \alpha \)[/tex], we reject the null hypothesis.
Conclusion:
Given that the P-value (0.027) is less than the significance level of 0.05, we reject [tex]\( H_0 \)[/tex]. This means there is sufficient evidence to support the claim that the mean hic measurement of the population is less than 1000. Therefore, based on the sample data, the results suggest that all of the child booster seats tested meet the specified safety requirement.
