Question

A new bus route has been established between downtown Denver and Englewood (a suburb of Denver). Dan has taken the bus to work for many years. For the old bus route, he knows from long experience that the mean waiting time between buses at this stop was u=20 minutes. However, a random sample of five waiting times between buses using the mean route had mean x = 15.1 minutes with sample standard deviation s = 6.2 minutes. Does this indicate that the population mean waiting time for the new route is shorter than what it used to be? Use a = 0.05. Assume that x is normally distributed.
a. What is the LEVEL of SIGNIFICANCE? State the NULL and ALTERNATE HYPOTHESES.   Will you use the a LEFT-TAILED, RIGHT-TAILED, or TWO-TAILED test?
b. Identify the sampling distribution you will use: The STANDARD NORMAL or The STUDENT'S t DISTRIBUTION? what is the value of the SAMPLE TEST STATISTIC?
c. Find ( or estimate) the P VALUE or P VALUE RANGE. Sketch the sampling distribution and SHOW THE AREA corresponding to the P value. (Alternate method: find the critical value (s) and sketch the critical regions and the sample test statistic on the normal curve.
d. Based on your answer, will you REJECT or FAIL TO REJECT THE NULL HYPOTHESIS?
e. INTERPRET your decision in the context of the application.

Answer 1

Solution-A:

LEVEL of SIGNIFICANCE=alpha=0.05

State the NULL and ALTERNATE HYPOTHESES.

NULL HYPOTHESES

Ho:

Ha:

Its a left tail test

b. Identify the sampling distribution you will use:

since sample standard deviation is known use t statistic

The STUDENT'S t DISTRIBUTION

s the value of the SAMPLE TEST STATISTIC

t=xbar-mu/s/sqrt(n)

=(15.1-20)/(6.2/sqrt(5)

t=-1.7672

SAMPLE TEST STATISTIC

t=-1.7672

c. Find ( or estimate) the P VALUE or P VALUE RANGE. S

df=n-1=5-1=4

1-right tail

=1-=T.DIST.RT(-1.7672,4)

=1-0.924033

=0.0760

p=0.0760

Solution-d:

p=0.076

p>0.05

FAIL TO REJECT THE NULL HYPOTHESIS

Solution-e:

there is no suffcient statistical evidence at 5% level of significance to conclude that

that the population mean waiting time for the new route is shorter than what it used to be.