What is the alternative and null hypothesis in this problem?

Question

in the midst of a labor management negotiations, the president of the union claims that blue collar workers who annual income is $100,000) of the company are underpaid, since the average greater manila area blue collar income exceeds 100,000. management claims the companys workers are well paid, since the average GMA blue collar income is less than 100,000. to help resolve the conflict, a labor arbitrator decides to conduct a survey among 400 GMA blue collar workers to determine if their mean income is different from 100,000 assuming the standard dev is 4,000 can the arbitrator conclude at 5% significance level that the mean is different from 100,000 if the sample mean is 102,500

Answer 1

because of the phrase "determine if their mean income is different from" we have a two tail hypothesis test.

Hypothesis Test for mean:

Assuming you have a large enough sample such that the central limit theorem holds, or you have a sample of any size from a normal population with known population standard deviation, then to test the null hypothesis

H0: μ ≤ Δ or

H0: μ ≥ Δ or

H0: μ = Δ

Find the test statistic z = (xbar - Δ ) / (sx / √ (n))

where xbar is the sample average

sx is the sample standard deviation, if you know the population standard deviation, σ , then replace sx with σ in the equation for the test statistic.

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|

If the p-value is less than or equal to the significance level α, i.e., p-value ≤ α, then we reject the null hypothesis and conclude the alternate hypothesis is true.

If the p-value is greater than the significance level, i.e., p-value > α, then we fail to reject the null hypothesis and conclude that the null is plausible.  Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.

The hypothesis test in this question is:

H0: μ = 1e+05 vs. H1: μ ≠ 1e+05

The test statistic is:

z = ( 102500 - 1e+05 ) / ( 4000 / √ ( 400 ))

z = 12.5

The p-value = P( Z > |z| )

= P( Z < -12.5 ) + P( Z > 12.5 )

= 2 * P( Z < -12.5 )

= 7.465129e-36

Since the p-value is less than the significance level we reject the null hypothesis and conclude the alternate hypothesis μ ≠ 1e+05 is true.

Consider the hypothesis as a trial against the null hypothesis.  the data is evidence against the mean.  you assume the mean is true and try to prove that it is not true.  After finding the test statistic and p-value, if the p-value is less than or equal to the significance level of the test we reject the null and conclude the alternate hypothesis is true.  If the p-value is greater than the significance level then we fail to reject the null hypothesis and conclude it is plausible.  Note that we cannot conclude the null hypothesis is true, just that it is plausible.

If the question statement asks you to determine if there is a difference between the statistic and a value, then you have a two tail test, the null hypothesis, for example, would be μ = d vs the alternate hypothesis μ ≠ d

if the question ask to test for an inequality you make sure that your results will be worth while.  for example.  say you have a steel bar that will be used in a construction project.  if the bar can support a load of 100,000 psi then you'll use the bar, if it cannot then you will not use the bar.

if the null was μ ≥ 100,000 vs the alternate μ < 100,000 then will will have a meaningless test.  in this case if you reject the null hypothesis you will conclude that the alternate hypothesis is true and the mean load the bar can support is less than 100,000 psi and you will not be able to use the bar.  However, if you fail to reject the null then you will conclude it is plausible the mean is greater than or equal to 100,000.  You cannot ever conclude that the null is true.  as a result you should not use the bar because you do not have proof that the mean strength is high enough.

if the null was μ ≤ 100,000 vs. the alternate μ > 100,000 and you reject the null then you conclude the alternate is true and the bar is strong enough; if you fail to reject it is plausible the bar is not strong enough, so you don't use it.  in this case you have a meaningful result.  

Any time you are defining the hypothesis test you need to consider whether or not the results will be meaningful.

Answer 2

Null: The mean is 100,000

Alternative: The mean is different from 100,000