Question

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?
644 612 1138 590 557 683
what are the hypotheses?
a. ( h_0:mu>1000 ) hic
( h_1:mu<1000 ) hic
b. ( h_0:mu = 1000 ) hic
( h_1:mu<1000 ) hic
c. ( h_0:mu<1000 ) hic
( h_1:mugeq1000 ) hic
d. ( h_0:mu = 1000 ) hic
( h_1:mugeq1000 ) hic
identify the test statistic.
( t=-3.34 ) (round to three decimal places as needed.)
identify the p - value.
the p - value is 0.0103
(round to four decimal places as needed.)
state the final conclusion that addresses the original claim.
reject ( h_0 ). there is sufficient 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. the requirement is met since most sample measurements are less than 1000 hic.
b. 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
c. there is not strong evidence that the mean is less than 1000 hic, and one of the booster seats has a measurement that is greater than 1000 hic
d. the results are inconclusive regarding whether one of the booster seats could have a measurement that is greater than 1000 hic

Explanation:

• Option A: Just because most sample measurements are less than 1000 hic does not ensure all meet the requirement. The hypothesis test is about the population mean, not individual values.
• Option B: The hypothesis test shows there is sufficient evidence (\(P - value=0.0103<0.05\)) that the population mean is less than 1000 hic. But looking at the data, 1138 hic is greater than 1000 hic.
• Option C: Since we rejected \(H_0\) (there is sufficient evidence for \(\mu < 1000\) hic), this option is incorrect.
• Option D: The hypothesis test gives a conclusion about the population mean, and we can also observe individual values from the data.

Answer:

B. 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.