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?
\[
\begin{array}{lllllll}
712 & 627 & 1096 & 582 & 570 & 541
\end{array}
\]
What are the hypotheses?
A. $\mathrm{H}_{0}, \mu> 1000$ hic $\mathrm{H}_{1}: \mu< 1000$ hic
C. $\mathrm{H}_{0}: \mu=1000$ hic $\mathrm{H}_{1}, \mu< 1000$ hic
Identify the test statistic. $t=\square$ (Round to three decimal places as needed.)
Identify the P-value.
B. $\mathrm{H}_{0}, \mu=1000$ hic $\mathrm{H}_{1} \cdot \mu \geq 1000 \mathrm{hic}$
D. $\mathrm{H}_{0}: \mu< 1000$ hic $H_{1}: \mu \geq 1000$ hic
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$11 / 20 / 2023$
Answer
The final conclusion that addresses the original claim is that the results suggest that all of the child booster seats meet the specified requirement. Therefore, \(\boxed{\text{The null and alternative hypotheses are } H_{0}: \mu=1000 \text{ hic and } H_{1}: \mu<1000 \text{ hic. The test statistic is } t=-3.664. \text{ The P-value is approximately 0.007. Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This suggests that the mean hic measurement is less than 1000, which would suggest that all of the child booster seats meet the specified requirement.}}\)
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis is that the mean hic measurement is equal to 1000, denoted as \(H_{0}: \mu=1000\) hic. The alternative hypothesis is that the mean hic measurement is less than 1000, denoted as \(H_{1}: \mu<1000\) hic.
Step 2 ：Calculate the test statistic. This is done by first calculating the sample mean and standard deviation. Then, subtract the population mean (1000 in this case) from the sample mean and divide by the standard deviation divided by the square root of the sample size. The test statistic is \(t=-3.664\) (rounded to three decimal places).
Step 3 ：Calculate the P-value using the test statistic and the t-distribution. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The P-value is approximately 0.007.
Step 4 ：Make a conclusion based on the P-value. If the P-value is less than the significance level (0.05 in this case), reject the null hypothesis. This suggests that the mean hic measurement is less than 1000, which would suggest that all of the child booster seats meet the specified requirement.
Step 5 ：The final conclusion that addresses the original claim is that the results suggest that all of the child booster seats meet the specified requirement. Therefore, \(\boxed{\text{The null and alternative hypotheses are } H_{0}: \mu=1000 \text{ hic and } H_{1}: \mu<1000 \text{ hic. The test statistic is } t=-3.664. \text{ The P-value is approximately 0.007. Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This suggests that the mean hic measurement is less than 1000, which would suggest that all of the child booster seats meet the specified requirement.}}\)
