Answers to Example Problems

  1. Do various occupational groups differ in their diets? In a British study of this question, two of the groups compared were 98 drivers and 83 conductors of London double-decker buses. The article reporting the study gives the data as "Mean daily consumption (± s. e.)." Some of the study results appear below. [From Marr and Heady, Human Nutrition: Applied Nutrition 40A (1986), pp. 347-364.]
  2. Drivers
    Total calories
    2821 ± 44
    2844 ± 48
    Alcohol (g)
    0.24 ± 0.06
    0.39 ± 0.11

    (a) What does "s. e." stand for? Give the mean and standard deviation for each of the four sets of measurements.

    "s. e." is the standard error. Since they have given you the standard error, you have to do some calculation if you want the standard deviation. You have to multiply the standard error by the square root of the sample size to get the standard deviation. In other words, they could have provided the following table:

    Total calories (mean ± sd)
    2821 ± 436
    2844 ± 437
    Alcohol (mean ± sd) (g)
    0.24 ± 0.59
    0.39 ± 1.00


    (b) Is there significant evidence at the 5% level that conductors consume more calories per day than do drivers?

    The easiest way to do this is to look at the original table with standard errors and note that the difference in between Drivers and Conductors is only 23 calories and that this difference is much less than even one standard error in either sample. There is no way this will result in a significant difference from a significance test.

    If you want to actually do a t-test, you just divide the difference by the pooled standard error (see p. 169 in The Cartoon Guide). In this case, the pooled standard error is 65, so the t-statistic would be 23/65 = 0.35. This result needs to be greater than 2 for us to conclude that there is a significant difference between the two populations. (To get the pooled standard error, square each individual standard error, add them, and then take the square root.)

    (c) How significant is the difference in alcohol consumption between the two groups? Give either a P value or some equivalent statistic.

    Using the same method as described in part (b), we can find a t-statistic. The pooled standard error is 0.125; the difference between the means is 0.15. The t-statistic is 0.15/0.125 = 1.20. Again, this does not reach our 95% confidence level. If you use a t-table or Excel to find the P value, it should come out near P = 0.12. This means that we only have evidence of significant difference to the 88% confidence level.

  3. There are four major blood types in humans: O, A, B, and AB. In a study conducted using blood samples from the Blood Bank of Hawaii in 1950, individuals were classified according to blood type and ethnic group. Summarize the data. Is there evidence to conclude that blood type and ethnic group are related? Explain how you arrive at your conclusion.
  4. Blood Type Hawaiian Hawaiian-white Hawaiian-Chinese White


    1903 4469 2206 53,759
    A 2490 4671 2368 50,008
    B 178 606 568 16,252
    AB 99 236 243 5001

    The first step is to convert the above table to a frequency table showing the frequency of each blood type within each population.

    Blood Type

    ME ~ 0.014

    ME ~ 0.010
    ME ~ 0.013
    ME ~ 0.003


    0.407 0.448 0.410 0.430
    A 0.533 0.468 0.440 0.400
    B 0.038 0.061 0.105 0.130
    AB 0.021 0.024 0.045 0.040

    where n is the total sample size for that population and ME is the margin of error. The margin of error is just twice the standard error assuming a frequency of 0.5 ( sqrt[ 0.5 (1-0.5) / n ] ), which should be fine for looking at the O and A blood types. For B and AB types, we should make adjustments for the small frequencies, this would make the margins of error about half as big.

    Clearly there are areas of significant difference. Hawiians have significantly higher rates of blood type A compared to all other populations surveyed. Hawaiians and Hawaiian-Chinese have significantly lower rates of blood type O compared to the other groups. What is interesting is that the rate of blood type O is significantly higher in Hawaiian-whites compared to either Hawaiians or Whites. Hawaiian-Chinese appear to have lower rates of blood types O and A and higher rates of B and AB. Whites have the lowest rate of blood type A, significantly lower than all other groups; and they have higher rates of O and B blood types.

    There are more specific tools for analyzing the differences in these data, but general comments like these are sufficient for this program.

  5. According to the National Center for Education Statistics, the average base salary for the 1999-2000 school year for male full-time teachers in public elementary and secondary school was $41,104 while the average base salary for female full-time teachers was $39,475. The standard error on each figure was reported as $198 and $126, respectively. Are the base salaries of male teachers significantly different from female teachers? Given that most public school salaries are set by a pay scale which assigns salaries by years of education and years of experience and which makes no reference to gender, can you come up with any reason for the gender difference in average salaries?

    Using the same methods as used above in #1, you can see that the difference between the salaries is $1,629 which is much more than two or three or even five times either of the standard errors given. So the difference in base salaries is clearly significant. In fact, a t-test would yield a t-statistic of about 1629 / 235 = 7. This is way past 2 standard errors of difference. But public schools set salaries based only on years of experience and years of education. Why is there a significant difference based on gender? It must be the case that, on average, males have more experience and/or more education. There is one other factor. Usually middle and high school teachers get paid more than elementary school teachers. So another hypothesis would be that a greater percentage of male teachers are middle and high school teachers as compared to the percentages for female teachers.