statistical math question?

Question

The management of White Industries is considering a new method of assembling its golf cart. The present method requires 42.3 minutes, on the average, to assemble a cart. The mean assembly time for a random sample of 24 carts, using the new method, was 40.6 minutes, and the standard deviation of the sample was 2.7 minutes. Using the .10 level of significance, can we conclude that the assembly time using the new method is faster?

Answer 1

Yes.

Z = (xbar-mu)/(sd/sqrt(n))
= (40.6-42.3)/(2.7/sqrt(24))
= -3.084543

Compare to a t-distribution with 23 degrees of freedom:

p = 0.002617878 (one-sided) < 0.10

Answer 2

Hypothesis Test for mean:

assuming you have a large enough sample such that the central limit theorem holds and the mean is normally distributed then to test the null hypothesis H0: μ = Δ

find the test statistic z = (xBar - Δ) / (sx / sqrt(n))

where xbar is the sample average
sx is the sample standard deviation
n is the sample size

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.

H1: μ > Δ; p-value is the area to the right of z
H1: μ < Δ; p-value is the area to the left of z
H1: μ ≠ Δ; p-value is the area in the tails greater than |z|

for a small sample test for the mean every thing is the same save the test statistic is a t statistic with n - 1 degrees of freedom. However, in this case the underlying distribution must be normal for this test to be valid.


for your question we have:

H0: μ ≥ 42.3 vs. H1: μ < 42.3

the test statistic is a t statistic with 23 degrees of freedom

t = (40.6 - 42.3) / (2.7 / sqrt(24)) = -3.084543

the p-value of the test is:

P( t_23 < -3.084543) = 0.002617878

since the p-value is less than the significance level we reject the null hypothesis and conclude the alternate hypothesis is true. Yes the new method is faster.

Answer 3

Hi. Depends on whether you mean the "mean", "median", or "average". They are not the same thing. Also if the faster time results in lower quality or rework then all bets are off.