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 stat 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 inju 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 pop mean less than 1000 hic. do the results suggest that all of the child booster seats meet the specified requirement? 755 626 1290 582 612 485 what are the hypotheses? a. ( h_{0}: mu=1000 ) hic ( h_{1}: mu geq 1000 ) hic b. ( h_{0}: mu<1000 ) hic ( h_{1}: mu geq 1000 ) hic c. ( h_{0}: mu=1000 ) hic ( h_{1}: mu<1000 ) hic d. ( h_{0}: mu>1000 ) hic ( h_{1}: mu<1000 ) hic identify the test statistic. ( t=) (round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample mean

The sample data is \(x = [755,626,1290,582,612,485]\).
The formula for the sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
\(\sum_{i=1}^{6}x_{i}=755 + 626+1290+582+612+485=4350\), \(n = 6\).
\(\bar{x}=\frac{4350}{6}=725\).

Step2: Calculate the sample standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\).
\((755 - 725)^{2}=900\), \((626-725)^{2}=9801\), \((1290 - 725)^{2}=319225\), \((582-725)^{2}=20449\), \((612 - 725)^{2}=12769\), \((485-725)^{2}=57600\).
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=900+9801+319225+20449+12769+57600 = 420744\).
\(s=\sqrt{\frac{420744}{6 - 1}}=\sqrt{84148.8}\approx290.1\).

Step3: Calculate the test - statistic

The formula for the \(t\) - test statistic in a one - sample \(t\) - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\).
Here, \(\mu = 1000\), \(\bar{x}=725\), \(s\approx290.1\), \(n = 6\).
\(t=\frac{725-1000}{290.1/\sqrt{6}}\).
First, calculate \(290.1/\sqrt{6}\approx290.1/2.45\approx118.4\).
Then \(t=\frac{725 - 1000}{118.4}=\frac{- 275}{118.4}\approx - 2.323\).

Answer:

The test statistic \(t\approx - 2.323\)