Research Question: MLB Advanced Media, L.P. indicated that Major League Baseball had approximately a mean of 81 wins per team in the past 10 years. A sample of 20 teams from 2008 revealed the mean amount of wins was 78 with a standard deviation of 10.66. Can we conclude that the mean number of wins in 2008 is the same as the mean number of wins in Major League Baseball in the past 10 years? Use the hypothesis testing procedure and the 0.05 significance level.
Data: 95, 74, 93, 80, 67, 71, 56, 77, 73, 67, 82, 75, 83, 81, 67, 79, 88, 100, 81, 71
Step 1: State the null and alternate hypothesis: The null hypothesis is that the average number of wins in Major League Baseball is 81. The alternate hypothesis is that the average number of wins in Major League Baseball is not 81. Symbolically, these statements are written as follows:
Ho: ? = 81
H?: ? â? 81
Step 2: Select the level of significance: We decide on the 0.05 significance level.
Step 3: Select the test statistic: The test statistic in this situation is the t distribution.
t = Sample Mean â?" ? â?? 78 - 81 â?? -3 / 2.39 â?? -1.26
s / sqrt (n) 10.66/4.47
Step 4: Develop the decision rule: Remember that the significance level stated in the problem is 0.05. The critical values of t are given in Appendix F. The number of degrees of freedom is (n-1) = (20-1) = 19. We have a two-tailed test, so we find the portion of the table labeled â??two-tailed.â? Locate the column for the 0.05 significance level. Read down the column until it intersects the row with 19 degrees of freedom. The value is 2.093. The decision rule is: Reject the null hypothesis if the computed value of t is to the left of -2.093, or to the right of 2.093.
Step 5: Make a decision regarding the null hypothesis, and interpret the results:
The value of t lies between the two critical values: -2.093 and 2.093. The null hypothesis is not rejected at the 0.05 significance level. We conclude the population mean in Major League Baseball could be 81 wins. The evidence fails to show the average wins to be different.
Step by step method for computing test statistic for One sample t test is given in the answer.