3 A researcher wishes to conduct a hypothesis test to invest

3. A researcher wishes to conduct a hypothesis test to investigate whether the mean exercise time of office workers in Vancouver is less than 4 hours per week. A random sample of 36 office workers in Vancouver is taken and the exercise time per week of each of the sampled worked is recorded. Their average weekly exercise time is 3.5 hours, and the standard deviation is 1 hour. Twenty of the sampled office workers exercise for less than 4 hours per week.

(a) Construct a 95% confidence interval for the mean weekly exercise time among the all office workers in Vancouver. Also provide an interpretation to the interval in the context of this question.

(b) Write down the null and alternative hypotheses that the researcher should use in his hypothesis test. Define any notation used.

(c) Compute the test statistic based on the data collected from the sampled workers.

Solution

(a) Given a=1-0.95=0.05, Z(0.025) = 1.96 (from standard normal table)

So the lower bound is

xbar - Z*s/vn =3.5 - 1.96*1/sqrt(36) =3.173333

So the upper bound is

xbar + Z*s/vn =3.5 + 1.96*1/sqrt(36) =3.826667

We have 95% confidnet that the popluation mean will be within this interval.

----------------------------------------------------------------------------------------------------------------

(b) Let mu be the population mean

Null hypothesis: mu=4

Alternative hypothesis: mu<4

----------------------------------------------------------------------------------------------------------------

(c) test statistic:

Z=(xbar-mu)/(s/vn)

=(3.5-4)/(1/sqrt(36))

=-3