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 0.01 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?
\[
\begin{array}{lllllll}
752 & 667 & 1264 & 673 & 639 & 565
\end{array}
\]
The P-value is
(Round to four decimal places as needed.)
State the final conclusion that addresses the original claim.
$\mathrm{H}_{0}$. There is evidence to support the claim that the sample is from a population with a mean less than 1000 hic.
What do the results suggest about the child booster seats meeting the specified requirement?
A. There is strong evidence that the mean is less than 1000 hic, but one of the booster seats has a measurement that is greater than 1000 hic.
B. The results are inconclusive regarding whether one of the booster seats could have a measurement that is greater than 1000 hic.
C. The requirement is met since most sample measurements are less than 1000 hic.
Next
Answer
The results suggest that not all of the child booster seats meet the specified requirement. Therefore, the answer is \(\boxed{\text{A. There is strong evidence that the mean is less than 1000 hic, but one of the booster seats has a measurement that is greater than 1000 hic.}}\)
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis (H0) is that the mean hic measurement is equal to 1000, while the alternative hypothesis (H1) is that the mean hic measurement is less than 1000.
Step 2 ：Calculate the sample mean and standard deviation. The sample mean is 760.0 and the sample standard deviation is 254.1511361375353.
Step 3 ：Calculate the test statistic using the formula for a one-sample t-test, which is (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). The test statistic is -2.31.
Step 4 ：Calculate the P-value using the t-distribution. The P-value is 0.0343.
Step 5 ：Compare the P-value with the significance level (0.01). Since the P-value is greater than the significance level, we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the mean hic measurement is less than 1000.
Step 6 ：Interpret the results in the context of the original claim. One of the booster seats has a hic measurement that is greater than 1000. This suggests that not all of the child booster seats meet the specified requirement.
Step 7 ：The null and alternative hypotheses are: \(H_0\): \(\mu = 1000\) \(H_1\): \(\mu < 1000\)
Step 8 ：The test statistic is \(\boxed{-2.31}\) and the P-value is \(\boxed{0.0343}\).
Step 9 ：Since the P-value is greater than the significance level of 0.01, we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the mean hic measurement is less than 1000.
Step 10 ：The results suggest that not all of the child booster seats meet the specified requirement. Therefore, the answer is \(\boxed{\text{A. There is strong evidence that the mean is less than 1000 hic, but one of the booster seats has a measurement that is greater than 1000 hic.}}\)
