https://file.aiquickdraw.com/mathbot/file/1726796794878kztos6ly.jpg

The image shows a hypothesis testing problem involving a claim about child booster seat safety. Here's a summary and breakdown of the task:

### Problem Summary:
- **Claim:** The safety requirement for the child booster seats is that the head injury condition (hic) measurement should be **less than 1000**.
- **Data Given:** 
  - Sample data (hic measurements): 623, 613, 1138, 553, 513, 586
- **Significance Level (α):** 0.05
- **Objective:** Test the claim that the sample comes from a population with a mean hic measurement less than 1000.

### Steps to Solve:
1. **State the Hypotheses:**
   - **Null Hypothesis (H₀):** μ ≥ 1000 (The mean hic measurement is greater than or equal to 1000).
   - **Alternative Hypothesis (H₁):** μ < 1000 (The mean hic measurement is less than 1000).

2. **Significance Level:** 
   - α = 0.05

3. **Test Statistic:**
   - Use a t-test since the population standard deviation is unknown and the sample size is small. 
   - Formula for the t-statistic:  
     \[
     t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
     \]
     Where:
     - \(\bar{x}\) = sample mean
     - \(s\) = sample standard deviation
     - \(\mu_0\) = hypothesized population mean (1000)
     - \(n\) = sample size

4. **P-value:**
   - The P-value will tell us the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

5. **Conclusion:**
   - Compare the P-value to the significance level α = 0.05. If P-value < 0.05, reject H₀; otherwise, fail to reject H₀.

### Let's calculate the necessary statistics next. I'll start by computing the sample mean, standard deviation, and then proceed to calculate the t-statistic and P-value.### Calculation Results:
- **Sample Mean (𝑥̄):** 671
- **Sample Standard Deviation (s):** 232.32
- **t-Statistic:** -3.47
- **P-value:** 0.0089

### Conclusion:
Since the P-value (0.0089) is less than the significance level (α = 0.05), we **reject the null hypothesis (H₀)**. This suggests that there is sufficient evidence to support the claim that the mean hic measurement is **less than 1000**, meaning the child booster seats meet the specified safety requirement.

Would you like more details on any of the steps or calculations?

### Follow-up Questions:
1. What would happen if the significance level were set to 0.01 instead of 0.05?
2. How would the conclusion change if one of the hic measurements were much higher?
3. Can we use a z-test instead of a t-test in this scenario? Why or why not?
4. How does the sample size affect the reliability of the hypothesis test?
5. What assumptions must be met for this t-test to be valid?

### Tip:
Always check if the data follows a normal distribution when performing a t-test with small sample sizes.

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. Use a 0.05 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. The measurements are: 623, 613, 1138, 553, 513, 586.

Test the claim that the mean head injury condition (hic) for child booster seats is less than 1000 using hypothesis testing. This problem involves a t-test with a 0.05 significance level, sample mean, standard deviation, and calculation of the test statistic. Suitable for students learning statistical methods in hypothesis testing.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation