Question

A bus company official claims that the mean waiting time for a bus during peak hours is less than 10 minutes. Karen took a bus during peak hours on 18 different occasions. Her mean waiting time was 9.1 minutes with a standard deviation of 2.1 minutes. At the 0.05 significance level, test the claim that the mean waiting time is less than 10 minutes by providing each of the following:
a. the null and alternative hypothesis
b. the test statistic
c. the pvalue
d. the conclusion in non technical terms
e. a 99% confidence interval
f. does your confidence interval support the conclusion from part D? explain

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 10
Alternative Hypothesis, Ha: μ < 10

b)

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (9.1 - 10)/(2.1/sqrt(18))
t = -1.818

c)

P-value Approach
P-value = 0.0434

d)

As P-value < 0.05, reject the null hypothesis.

e)
sample mean, xbar = 9.1
sample standard deviation, s = 2.1
sample size, n = 18
degrees of freedom, df = n - 1 = 17

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.898

ME = tc * s/sqrt(n)
ME = 2.898 * 2.1/sqrt(18)
ME = 1.4

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (9.1 - 2.898 * 2.1/sqrt(18) , 9.1 + 2.898 * 2.1/sqrt(18))
CI = (7.67 , 10.53)

f)

No, because confidence iterval contain 10