Question

In the year 2029, Anjail Lateef is a leading nurse anesthetist.    Anjail is presently doing research on the effectiveness of a certain type of anesthesia on reducing the mean length of time that it takes patients to recover from surgery (presently the mean length of time is 2 weeks; σ = 0.4 weeks).    Anjail uses this anesthesia on random selection of 49 patients and finds the sample mean to be 1.85 weeks.    Using α = 0.05, answer the following questions.

a) What is the setup for your null and alternative hypothesis?

b) What is the value of the test statistic, Z*?

c) Using the appropriate level of confidence, what is the confidence interval?

d) What is the interpretation (not your conclusions) of the confidence interval (in the context of the problem) you obtained in part c?

e) What is the p value?

f) What is the interpretation (not your conclusions) of the p value in the context of the problem you found in part e?

g) What are your conclusions in the context of the problem? Relate your conclusions to the test statistic and critical value, confidence interval and p value.  

Answer 1

a)

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ=2

Ha: μ<2

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

b)

Test Statistics

The z-statistic is computed as follows:

c)

Confidence Interval

The 95% confidence interval is 1.738<μ<1.962.

d)

We would reject the null hypothesis since 2 does not lie in the above confidence interval.

e)

The p-value is p = 0.0043

f)

since p=0.0043<0.05, it is concluded that the null hypothesis is rejected.

g)

Since it is observed that z=−2.625<zc​=−1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0043, and since p=0.0043<0.05, it is concluded that the null hypothesis is rejected.It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 2, at the 0.05 significance level.

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