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? 597     655     1081    573    517    555

Answer 1

Solution:-

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

• State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

  Null hypothesis: μ >= 1000

  Alternative hypothesis: μ < 1000

  Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.
• Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.
• Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

  SE = s / sqrt(n) = 209.86853 / sqrt(6) = 85.678

  DF = n - 1 = 6 - 1 = 5

  t = (x - μ) / SE = (663 - 1000)/85.678 = -3.9

  where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

  Here is the logic of the analysis: Given the alternative hypothesis (μ < 1000), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

  The observed sample mean produced a t statistic test statistic of -3.9. We use the t Distribution Calculator to find P(t < -3.9) = 0.005705.
• Interpret results. Since the P-value (0.005705) is less than the significance level (0.05), we cannot accept the null hypothesis.
• Conclusion. Reject null hypothesis. We have significant proof to prove the claim that, the safety requirement is less than 1000 hic