Minor League Baseball Attendance. The International League...

Minor League Baseball Attendance. The International League of Triple-A minor league baseball consists of 14 teams organized into three divisions: North, South, and West. The following data show the average attendance for the 14 teams in the International League. Also shown are the teams' records; W denotes the number of games won, L denotes the number of games lost, and PCT is the proportion of games played that were won.$$ \begin{array}{lllllc} \text { Team Name } & \text { Division } & \text { W } & \text { L } & \text { PCT } & \text { Attendance } \\ \text { Buffalo Bisons } & \text { North } & 66 & 77 & .462 & 8812 \\ \text { Lehigh Valley IronPigs } & \text { North } & 55 & 89 & .382 & 8479 \\ \text { Pawtucket Red Sox } & \text { North } & 85 & 58 & .594 & 9097 \\ \text { Rochester Red Wings } & \text { North } & 74 & 70 & .514 & 6913 \\ \text { Scranton-Wilkes Barre Yankees } & \text { North } & 88 & 56 & .611 & 7147 \\ \text { Syracuse Chiefs } & \text { North } & 69 & 73 & .486 & 5765 \\ \text { Charlotte Knights } & \text { South } & 63 & 78 & .447 & 4526 \\ \text { Durham Bulls } & \text { South } & 74 & 70 & .514 & 6995 \\ \text { Norfolk Tides } & \text { South } & 64 & 78 & .451 & 6286 \\ \text { Richmond Braves } & \text { South } & 63 & 78 & .447 & 4455 \\ \text { Columbus Clippers } & \text { West } & 69 & 73 & .486 & 7795 \\ \text { Indianapolis Indians } & \text { West } & 68 & 76 & .472 & 8538 \\ \text { Louisville Bats } & \text { West } & 88 & 56 & .611 & 9152 \\ \text { Toledo Mud Hens } & \text { West } & 75 & 69 & .521 & 8234 \end{array} $$ a. Use $\alpha=.05$ to test for any difference in the mean attendance for the three divisions. b. Use Fisher's LSD procedure to determine where the differences occur. Use $\alpha=.05$.

Minor League Baseball Attendance. The International League of Triple-A minor league baseball consists of 14 teams organized into three divisions: North, South, and West. The following data show the average attendance for the 14 teams in the International League. Also shown are the teams' records; W denotes the number of games won, L denotes the number of games lost, and PCT is the proportion of games played that were won.$$
\begin{array}{lllllc}
\text { Team Name } & \text { Division } & \text { W } & \text { L } & \text { PCT } & \text { Attendance } \\
\text { Buffalo Bisons } & \text { North } & 66 & 77 & .462 & 8812 \\
\text { Lehigh Valley IronPigs } & \text { North } & 55 & 89 & .382 & 8479 \\
\text { Pawtucket Red Sox } & \text { North } & 85 & 58 & .594 & 9097 \\
\text { Rochester Red Wings } & \text { North } & 74 & 70 & .514 & 6913 \\
\text { Scranton-Wilkes Barre Yankees } & \text { North } & 88 & 56 & .611 & 7147 \\
\text { Syracuse Chiefs } & \text { North } & 69 & 73 & .486 & 5765 \\
\text { Charlotte Knights } & \text { South } & 63 & 78 & .447 & 4526 \\
\text { Durham Bulls } & \text { South } & 74 & 70 & .514 & 6995 \\
\text { Norfolk Tides } & \text { South } & 64 & 78 & .451 & 6286 \\
\text { Richmond Braves } & \text { South } & 63 & 78 & .447 & 4455 \\
\text { Columbus Clippers } & \text { West } & 69 & 73 & .486 & 7795 \\
\text { Indianapolis Indians } & \text { West } & 68 & 76 & .472 & 8538 \\
\text { Louisville Bats } & \text { West } & 88 & 56 & .611 & 9152 \\
\text { Toledo Mud Hens } & \text { West } & 75 & 69 & .521 & 8234
\end{array}
$$
a. Use $\alpha=.05$ to test for any difference in the mean attendance for the three divisions.
b. Use Fisher's LSD procedure to determine where the differences occur. Use $\alpha=.05$.

Key Concepts

Assumptions in ANOVA

The validity of the ANOVA results depends on several key assumptions: the observations should be independent, the data within each group should be normally distributed, and the variances across groups should be homogeneous. Violating these assumptions can lead to unreliable results and may require alternative statistical methods or transformations.

Hypothesis Testing

Hypothesis testing is a core statistical procedure used to decide whether to reject a null hypothesis based on sample data. In scenarios like this, it typically begins with a null hypothesis stating that there is no difference in means across the groups. A significance level (alpha) is set, and test statistics are computed to determine if the observed differences could have occurred by random chance.

Fisher's Least Significant Difference (LSD) Procedure

Fisher's LSD is a post hoc analysis method employed after finding a significant result in ANOVA. It facilitates the identification of which specific pairs of group means differ significantly by performing pairwise comparisons. This method calculates a minimum significant difference based on the pooled error variance from the ANOVA and then checks if the absolute difference between any two means exceeds this threshold.

One-Way Analysis of Variance (ANOVA)

One-Way ANOVA is a statistical test used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. In the context of this problem, it is used to assess if the average attendance differs among the three divisions. It involves comparing the variance between groups with the variance within groups to decide if the group means are significantly different.

Transcript

00:01 So we have our anova test for the first, and we're looking at comparing the mean attendance for some minor league teams for the north, for the south, and for the west.

00:14 And we'll assume they're all equal, and then our alternative is that not all mean attendances are equal.

00:23 And so i'm doing my anova test on my calculator.

00:26 I put my data into list one, list two, and list three.

00:30 And the f statistic that i got for this data is 6 .9578 and the p value for this, and we have two degrees of freedom and 11 in the denominator.

00:48 And our p value for this is 0 .111, which is smaller than our 0 .05 significance level.

00:58 Therefore we have evidence to reject the null.

01:02 So we know that we have sufficient evidence to say that not all these means are equal.

01:07 So now we'll move on to part b and find where those differences are.

01:11 And so we need to know the means of each of these values.

01:15 And we're going to assume, and i have the three means.

01:18 The first mean for the north, the x bar of the north, is 7 ,702 .1 .6 with the six repeating the x bar for the south is oh and we have six pieces of data here by the way and for the south in the north the west we have four pieces of data and we have 5 ,565 .5 for the south of a sample size of four and the x bar for the west is 8 ,429 .75 and again we have four here so we can see that it would appears though this smallest and this largest would end up being significant and then we'll have to go through and we'll quick do these tests so we want to know do we have evidence that the north is different from the south do we have evidence that the north is different from the west and do we have evidence that the south is different from the west and alternately each of these will be and i'm just going to put a not equal to, not equal to, not equal to, just to save myself some time.

02:32 And so we have three test statistics.

02:34 We'll quick find these, and we know that the t values are all going to have 11 degrees of freedom since we had a total of 14 pieces of data minus three.

02:45 And that first test statistic is going to be to take the difference between these means, 7 ,702 .16 repeating, minus 5 ,5 ,000 .65 .5.

02:59 And then we'll have the mean square error.

03:03 And the mean square error for this is, wow, it's big.

03:06 We have a 1 -301 -392 .6.

03:12 And then we multiply that by one over.

03:15 And the sample size of the first one was six, and the second one was a four.

03:19 That will be our first test statistic.

03:21 And let me just write down the other ones while i'm at it, and then we'll make our decision.

03:24 So this is for this, i'll call this test one.

03:29 This is for test one.

03:30 This will be for test two.

03:32 This will be for test two.