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 these 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? 773 756 1290 659 650 625. 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. d. there is not strong evidence that the mean is less than 1000 hic, and one of the booster seats has a

Explanation:

Step1: Calculate sample mean and standard deviation

Given data \(x = [773,756,1200,669,650,625]\)
Sample size \(n = 6\)
Sample mean \(\bar{x}=\frac{773 + 756+1200+669+650+625}{6}=\frac{4673}{6}\approx778.83\)
Sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
\(\sum_{i=1}^{6}(x_{i}-\bar{x})^{2}=(773 - 778.83)^{2}+(756 - 778.83)^{2}+(1200 - 778.83)^{2}+(669 - 778.83)^{2}+(650 - 778.83)^{2}+(625 - 778.83)^{2}\)
\(=(- 5.83)^{2}+(-22.83)^{2}+(421.17)^{2}+(-109.83)^{2}+(-128.83)^{2}+(-153.83)^{2}\)
\(=33.99+521.21+177384.17+12062.63+16597.69+23663.46\)
\(=230263.15\)
\(s=\sqrt{\frac{230263.15}{5}}\approx214.61\)

Step2: State hypotheses and calculate test - statistic

Null hypothesis \(H_{0}:\mu\geq1000\)
Alternative hypothesis \(H_{1}:\mu < 1000\)
Test - statistic \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\mu = 1000\), \(\bar{x}\approx778.83\), \(s\approx214.61\), \(n = 6\)
\(t=\frac{778.83 - 1000}{214.61/\sqrt{6}}=\frac{-221.17}{87.67}\approx - 2.52\)

Step3: Calculate P - value

Degrees of freedom \(df=n - 1=5\)
For a one - tailed \(t\) - test with \(t=-2.52\) and \(df = 5\), using a \(t\) - table or calculator, the P - value is between \(0.025\) and \(0.05\)

Step4: Make a conclusion

Since the significance level \(\alpha=0.01\) and \(P - value>0.01\), we fail to reject the null hypothesis \(H_{0}\)

Answer:

D. 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.