Question

Jerry in Minnesota has an incredible team of athletes who are equally adept at both sports of baseball and hockey. For most of his athletes, there are two options: a career in baseball or a career in hockey. Accordingly, Jerry reviews information and derives the following summary measures concerning weekly salaries based on a random sample of 31 for each such sport. For the analysis, Jerry uses a historical (population) standard deviation of $21.5 for baseball players and $19.9 for hockey players.

Sample 1 represents baseball players and Sample 2 represents hockey players.)

Baseball	Hockey
x¯1x¯1 = $250.7	x¯2x¯2 = $235.2
n1 = 31	n2 = 31

Set up the hypotheses to test whether the mean salaries differs (i.e. any difference) between baseball salaries and hockey salaries.

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

• Compute the value of the test statistic and the corresponding p-value.

• At the 5% significance level, what is the conclusion to the test? What if the significance level is 1%?

• (Click to select)  Do not reject  Reject   H0.At either the 5% or 1% significance levels, we  (Click to select)  can  cannot  conclude the mean salaries differs between baseball players and hockey players.

• Now provide confidence interval information from the previous question. Specifically:

  a.    What is the value of the point estimate of the difference between the two population means?

  b.    What is the margin of error at 90% confidence?

          (± what value; please provide to 2 decimals; e.g. "1234.56")

  c.    With that margin of error, what is the low number in the confidence interval?

  d.    With that margin of error, what is the high number in the confidence interval?

Answer 1

H0: = 0

HA: 0

The test statistic z = ()/

                             = (250.7 - 235.2)/sqrt((21.5)^2/31 + (19.9)^2/31)

                             = 2.95

P-value = 2 * P(Z > 2.95)

             = 2 * (1 - P(Z < 2.95))

             = 2 * (1 - 0.9984)

             = 0.0032

At 5% significance level, since the P-value is less than the significance level(0.0032 < 0.05), so we should reject the null hypothesis.

At 1% significance level, since the P-value is less than the significance level(0.0032 < 0.01), so we should reject the null hypothesis.

Reject H0. At either the 5% or 1% significance levels, we can conclude that the mean salaries differs between baseball players and hockey players.

a) = 250.7 - 235.2 = 15.5

b) At 90% confidence interval the critical value is z_0.05 = 1.645

Margin of error = z_0.05 *

                            = 1.645 * sqrt((21.5)^2/31 + (19.9)^2/31)

                            = 8.66

c) The lower limit = () - E

                            = 15.5 - 8.66 = 6.84

d) The upper limit = () + E

                             = 15.5 + 8.66 = 24.16