Question 1202874: Mary Jo Fitzpatrick is the vice president for Nursing Services at St. Luke’s Memorial Hospital. Recently she noticed in the job postings for nurses that those who are unionized seem to offer higher wages. She decided to investigate and gathered the following information.
groups: union, nonunion.
sample size: 40, 45.
sample mean wage: $20.75, $19.80.
population standard deviation: $2.25, $1.90.
the decision rule is to reject H0 if z is greater than what number?
what is the test statistic and p-value?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this appears to be a two sample z-test.
the test is for the difference between the mean each group.
group 1 mean is 20.75 = x1
group 2 mean is 19.80 = x2
group 1 standard deviation = 2.25 = d1
group 2 standard deviation = 1.90 = d2
group 1 stmple size is 40 = n1
group 2 sample size is 45 = n2
the difference in the means is x1 - x2 = 20.75 - 19.80 = .95
the standard error of the test is equal to sqrt(d1^2/n1 + d2^2/n2) = sqrt(2.25^2/40 + 1.90^2/45) = .4547358818.
z-score of the test is (.95 / .4547358818 = 2.089
the 95% one tailed confidence interval has critical z-score of 1.645.
2.089 is outside of this range, so the results would be considered significant and the null hypothesis would be rejected in factor of the alternate hypothesis.
the null hypothesis assumes the union wages are not greater than non-union wages.
the alternate hypothesis assumes they are greater.
the conclusion is that the sample differences are greater enough so that it would be reasonable to conclude that they are, in fact, greater, and that rthe difference is not just due to random varitions of the mean in different samnples.
i confirmed through the use of the online statistics calculator at https://www.statology.org/two-sample-z-test-calculator/
the calculator states that the one tailed p-value of the test, which is the area to the right of the z-score of 2.089 is equal to .01839.
this is less than the critical p-value of .05 which is another indication that the results are significant.
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