Artificial Heart Part I. Medical researchers have developed a new artificial hea

Question

Artificial Heart Part I. Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient's body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation equal to 0.2 hours. Analysts wish to test the hypothesis that the mean life of battery packs is bigger than 4 hours. Using Type I error probability = 0.05,

A/ what is the numerical value of the test statistic taken to two decimal places?

B/ What is the p-value for the hypothesis test of Artificial Heart Part I? Answer to three decimal places.

C/ What will the analyst conclude? Use correct terminology. ( Accept H1 - Accept H0 - Reject H0 - Reject H1 - Fail to reject H0 - Fail to reject H1 )

Explanation / Answer

(A) The test statistic is the z-score corresponding to the
observed average life of the batteries.
If the population standard deviation is 0.2 hours,
then the standard error of the mean would be 0.2 hrs/sqrt(50)
= 0.028284 hrs.
Thus, the 0.05 hours by which the observed sample mean
exceeds the mean used in the null hypothesis is
0.05/0.028284 = 1.768 standard errors above the mean

(B) The p-value for z = 1.768 is 1-0.9615, according to the normal
distribution table, so p = 3.85% and the null hypothesis can be rejected.

(C) Type II error would occur if the true mean were 4.03 hours but the
tester failed to reject the null hypothesis. The tester would be requiring
a sample mean at least 1.645 standard errors above 4.00, so would be
requiring a sample mean of 4.04653 or more. That's 0.5844 standard errors
above the 4.03 hours, and the normal distribution table gives 72% probability
that a sample mean will be "too" low to result in the correct rejection of the
null hypothesis. The 72% is the probability of a Type II error if the true mean
is 4.03 hrs.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.