11: Variances and means (Key Odd)

Review Questions

  1. Mean square error, mean square, residual error, variance within groups, "squared standard deviation"; population variance (parameter) = 2; sample variance (statistic) = s2
  2. "Root mean square error"; population standard deviation = ; sample standard deviation = s
  3. When the variable has a Normal distribution. 
  4. Chebychev's inequality. 
  5. Additional measures of spread: (1) inter-quartile range  (2) range. 
  6. Boxplot visual clues for quantifying variability: hinge spread (i.e., IQR); "whiskers-spread"; range (which is often equal to the whisker's spread)
  7. The sum of squares is the sum of the squared of deviation around the distribution's mean.
  8. Recall that  s2  = SS / (n - 1). Therefore, SS = (n - 1)s2  
  9. F-ratio test (and Levene's test)
  10. Because pooling the variances suppress the non-uniformity of population variances.
  11. False. The standard error is a measure of the mean's precision. 
  12. df1 = 11 - 1 = 10, df2 = 10 - 1 = 9, and df = 10 + 9 = 19.
  13. yes, and yes. 
  14. t16,.975 = 2.12
  15. H02122
  16. Under the alternative hypothesis,  the variance in population 1 is larger than (or different from) the variance in population 2.
  17. H0: 1 = 2 versus H1: 1   2  
  18. Population mean difference = 1 - 2; sample mean difference = 1 - 2 
  19. How to compare group variability (spreads) 
    (a) Descriptively (e.g., compare sample standard deviations or IQRs
    (b) Graphically (e.g., side-by-side boxplots) 
    (c) Testing (e.g., F ratio test; Levene's test)
  20. How to compare group averages (central locations)
    (a) Descriptively (e.g., compare 1 and 2); 
    (b) Graphically (e.g., side-by-side stemplots, side-by-side boxplots, mean SE plots) 
    (c) Testing (e.g., t test)
    (d) Confidence interval for 12  
  21.  It goes by various names including "Welch's modified t" and "the unequal variance t test". The general problem of comparing means from populations with unequal variances is called the Behrens-Fisher problem.

Exercises

11.1  Comparing means depends on within group variability. ...we are confident the difference observed in Comparison B is real, while the observed difference in Comparison A might be due to chance fluctuation. Conduct t tests (for both comparisons) to confirm this suspicion. Calculations are shown below. Notice that even though Comparison A and Comparison B both compare a mean of 70 to a mean of 50, Comparison A derives P = 0.081, while Comparison B derives P = 0.002. Return to the stemplots in the exercise to see how this relates to the variability within groups.  

11.3 Linoleic acid and LDL cholesterol 

(A) Stemplot - The distribution for Group 1 is located toward the higher values (locations of medians underlined). Both distributions may sport modest positive skews, but there are  no apparent outliers. The ranges are visible.

 Group 1 | | Group 2
---------|-|---------
         |4|0

         |4|5
         |5|04  

     9888|5|6
    43100|6|04
       75|6|
        0|7|
         (1) 

(B) Descriptive statistics

Group

n

mean  (mmol/m3)

s (mmol/m3)

1 (Cases, Rassias data)

12

6.192

0.3919

2 (Controls, fictitious)

7

5.271

0.8381 

(C) F-ratio test of H0: 1222 ; Fstat = 0.83812 / 0.39192  = 4.573 w/ df1 = 6 and df2 = 11; 0.01 < P < 0.025. Significant: Yes!

(D) H0: 1 = 2  vs. H1: 1  2  by Welch modified t test; SEmean dif  = (.39192 / 12 + .83812 / 7) = 0.3364; tstat = (6.192 - 5.271) / 0.3364 = 2.74; df by conservative hand-based method = 6 (dfWelch = 7.56); 0.025 < P < 0.05; the observed difference seems to be significant (reject H0)

11.5 Body weight and pituitary adenoma. H02122 ; Fstat = (21.4)2 / (12.4)2 = 2.98 with 11 and 4 degrees of freedom. P = 0.15. The evidence against H0 is not significant.

11.7 Heart size and congestive heart failure.  

GROUP

n

Mean

Std. Deviation

Std. Error Mean

1

11

450.00

139.338

42.012

2

10

317.00

47.093

14.892

11.9 Efficacy of echinacea (severity of symptoms.) 

Comment: The published source (Taylor et al., 2003) gives P  = 0.68 for a Cox regression for censored data. Our method is a simplification and uses rounded summary statistics ( reported in Table 2 of Taylor et al., 2003 ) but comes up with an equivalent answer. (I requested the original data from the author via email on 8/31/06, but have yet to receive a response.)

11.11 The effect of calcium supplementation on blood pressure. See the boxplot shown below. Notice that the calcium-supplemented group has a higher average and (perhaps?) more variation. Also notice the outside value in the placebo group.