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?
\[
\begin{array}{llllll}
688 & 767 & 1248 \quad 635 \quad 634 \quad 570 \text { 만 }
\end{array}
\]
What are the hypotheses?
A.
\[
\begin{array}{l}
H_{0}: \mu=1000 \text { hic } \\
H_{1}: \mu< 1000 \text { hic }
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: \mu=1000 \text { hic } \\
H_{1}: \mu \geq 1000 \text { hic }
\end{array}
\]
B. $\mathrm{H}_{0}: \mu> 1000$ hic $\mathrm{H}_{1}: \mu< 1000$ hic
D. $\mathrm{H}_{0}: \mu< 1000$ hic $\mathrm{H}_{1}: \mu \geq 1000$ hic
Identify the test statistic.
$t=-2.387$ (Round to three decimal places as needed.)
Identify the P-value.
The P-value is
(Round to four decimal places as needed.)
Answer
The final answer is: The hypotheses are: \[\begin{array}{l} H_{0}: \mu=1000 \text { hic } \\ H_{1}: \mu<1000 \text { hic } \end{array}\] The test statistic is \(\boxed{-2.387}\). The P-value is \(\boxed{0.0313}\). Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This suggests that the sample data provides strong evidence to support the claim that the mean hic measurement is less 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 hic. The alternative hypothesis (H1) is that the mean hic measurement is less than 1000 hic. So, the hypotheses are: \[\begin{array}{l} H_{0}: \mu=1000 \text { hic } \\ H_{1}: \mu<1000 \text { hic } \end{array}\]
Step 2 ：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 \(-2.387\).
Step 3 ：Calculate the P-value, which is the probability of obtaining a result as extreme as the observed data, assuming that the null hypothesis is true. The P-value is approximately \(0.0313\).
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), we reject the null hypothesis and conclude that the data provides strong evidence to support the alternative hypothesis. If the P-value is greater than the significance level, we fail to reject the null hypothesis and conclude that the data does not provide strong evidence to support the alternative hypothesis. Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This suggests that the sample data provides strong evidence to support the claim that the mean hic measurement is less than 1000 hic.
Step 5 ：The final answer is: The hypotheses are: \[\begin{array}{l} H_{0}: \mu=1000 \text { hic } \\ H_{1}: \mu<1000 \text { hic } \end{array}\] The test statistic is \(\boxed{-2.387}\). The P-value is \(\boxed{0.0313}\). Since the P-value is less than the significance level of 0.05, we reject the null hypothesis. This suggests that the sample data provides strong evidence to support the claim that the mean hic measurement is less than 1000 hic.
