Is Mark Teixeira REALLY worse than last year?

So yesterday Blaine made some claims that I led him down the wrong path by encouraging him to draft Mark Teixeira. He cited many valid stats showing how Teixeira's production has dropped thus far this year. I will now examine those stats to see if differences between this year and last year are statistically significant.

Batting Average
Batting Average can be treated as a Bernoulli Distribution. This means that every at bat will be given a value of 0 (out) or 1 (hit). The true probability of him getting a 1, or a hit, is his true batting average. If you then take the sum of each trial and add them up (total number of hits) and divide by number of trials (total number of at bats), you are left with the hitter's batting average. We will do a t-test (with level 0.05) to determine if the difference between batting average this year and batting average last year is statistically significant. Our null hypothesis will be that his true batting average this year is .308, which it was last year. Our alternative hypothesis is that his true mean batting average is less than .308.

Data values:
Sample Proportion = X/n = 14/77 = 0.182
Sample Variance = p(1-p) = 14/77(1-14/77) = 0.14876
n = 77
t-critical value with 76 df at 0.05 level = -1.9917
t = (0.182 - 0.308)/sqrt(0.14876/77) = -2.87

Because our t of -2.87 is less than our t-critical value of -1.9917, we reject the null hypothesis in favor of the alternative, concluding that Mark Teixeira's true batting average for 2009 is less than last year's average of .308.

On Base Percentage
On-Base percentage can be treated the same as Batting Average in that they are both proportions, but now 0 means not getting on base, while 1 means getting on base. We will perform the same t-test as before with significance level 0.05. Our null hypothesis is that Teixeira's true on base percentage is .410 as it was last year, and our alternative hypothesis is that Teixeira's true on base percentage is less than .410.

Data Values:
Sample Proportion = X/n = 35/99 = 0.354
Sample Variance = p(1-p) = 35/99(1 - 35/99) = 0.228548
n = 99
t-critical value with 98 df at 0.05 level = -1.9845
t = (0.354 - 0.410)/sqrt(0.228548/99) = -1.166

Because our t value is greater than our t-critical value of -1.9845, we fail to reject the null hypothesis and conclude that Teixeira's on base percentage is not statistically significantly different than last year.

Slugging Percentage
Slugging Percentage must be treated slightly differently than OBP or AVG, as it is not a proportion. Rather, Slugging Percentage is essentially an average of total bases obtained in each at bat, 0 for not getting on base, 1 for a single, 2 for a double, 3 for a triple, 4 for a home run. So, we will do the same t-test (level 0.05) but our sample variance will be calculated differently. Note that I am violating the assumption of normally distributed data (which come to think of it, I think I also violated in the previous two problems), but the test is still somewhat valid and useful. Our null hypothesis is that Teixeira's true slugging percentage is .552, as it was last year, and our alternative hypothesis is that his true slugging percentage is less than .552.

Data Values:
Sample Mean = 26/77 = 0.338
Sample Variance = 0.88273
n = 77
t-critical value with 76 df at 0.05 level = -1.9917
t = (0.338-0.552)/(0.88273/sqrt(77)) = -2.13

Our t of -2.13 is less than our t-critical value of -1.9917, so we (barely) reject our null hypothesis in favor of the alternative that Mark Teixeira's true slugging percentage is less than last year's.

Conclusion
So, it appears that Teixeira's AVG and SLG are significantly worse than last year. However, OBP does not appear to have suffered as much, and as that is the statistic that matters in our fantasy league, all is not lost for Blaine. Still, Teixeira's numbers really are noticeably worse than last year, and statistically significant or not, Blaine should be slightly worried. Even I can't always be right I suppose (but I am in first place nonetheless).

Interesting Baseball Fact of the Day: Today is former Major Leaguer Ben Grieve's birthday. Lifetime on his birthday, Grieve was 5-17 (.294) with 2 2B, 1 HR, 4 RBI, 3 R, 5 K, and 2 BB.