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, value, and state the final conclusion that addresses the original claim. A safety administration conducted crash fests 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 salety 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 populatio with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified requirement? \[ \begin{array}{lllllll} 738 & 644 & 1260 & 642 \quad 555 \quad 505 & 0 \end{array} \] Identify the test statistic. $t=\square$ (Round to three decimal places as needed) Identify the P-value. The P value is $\square$ (Round to four decimal places as needed.) State the final conclusion that addresses the original claim. $H_{0}$. 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?
Solution
Step 1 ：Identify the null hypothesis (H0) as \(\mu = 1000\) hic and the alternative hypothesis (H1) as \(\mu < 1000\) hic.
Step 2 ：Calculate the sample mean (\(\bar{x}\)) as \(\frac{738 + 644 + 1260 + 642 + 555 + 505 + 0}{7} = 620.571\).
Step 3 ：Calculate the sample standard deviation (s) as \(\sqrt{\frac{(738-620.571)^2 + (644-620.571)^2 + (1260-620.571)^2 + (642-620.571)^2 + (555-620.571)^2 + (505-620.571)^2 + (0-620.571)^2}{6}} = 394.261\).
Step 4 ：Calculate the test statistic (t) as \(\frac{620.571 - 1000}{394.261 / \sqrt{7}} = -2.428\).
Step 5 ：Find the P-value as 0.0254.
Step 6 ：Since the P-value (0.0254) is less than the significance level (0.05), reject the null hypothesis.
Step 7 ：\(\boxed{\text{There is evidence to support the claim that the sample is from a population with a mean less than 1000 hic.}}\)
Step 8 ：\(\boxed{\text{The results suggest that not all child booster seats meet the safety requirement of less than 1000 hic.}}\)
