BIOST 510/DPHS 568 Biostatistics in Dentistry Summer 2008

Homework #2, Solutions

  1. From Day 1 lecture notes we know the sample statistics:

We don’t know σ, the true population standard deviation, so we need to use s, the sample standard deviation to estimate it. In this situation we use the confidence interval for mean with unknown variance (unknown variance also means unknown standard deviation). See section 8.2 in the coursepack.

  1. Using the formula on p83 (section 8.2) 95% confidence interval for the mean in group B is

,

where 2.03 is the 97.5th percentile of the t34 distribution.

For group C the 95% confidence interval for the mean is

,

where 2.02 is the 97.5th percentile of the t39 distribution.

  1. We want to test the hypotheses

H0: μ = 0, versus H1: μ ≠ 0,

and we’ll use significance level α = 0.05.

The t statistic is .

Since |-1.37| = 1.37 is less than 2.03 (the 97.5th percentile of the t34 distribution) we do NOT reject H0. The p-value is P(|t34| > 1.37) which is 0.18 (looked up via Excel). You can also estimate the probability using Table 4 (p179 coursepack). From Table 4 you can see that 1.37 is between the 90th and 95th percentiles of the t34 distribution, which tells you that P(t34 < 1.37) is between .90 and .95. Since P(|t34| > 1.37) = 2*P(t34 > 1.37) = 2*[1- P(t34 < 1.37)], a little algebra will conclude that P(|t34| > 1.37) is between 0.20 and 0.10.

  1. Same hypothesis, same significance level as in b.

The t statistic is .

Since 4.38 is greater than 2.02 (the 97.5th percentile of the t39 distribution) we do NOT reject H0. From Excel the p-value is P(|t39| > 4.38) = 8.65249E-05 = 0.0000865. Table 4 would tell you the p-value is less than 2*(1-.9995) = .001.

  1. The confidence intervals tell us what are possible values for the averages. In particular we see that for gum C people the average dmfs would increase (be > 0), and is probably even larger than a 1.41 dmfs increase. The average dmfs change for kids chewing gum B looks to be a lot lower, and in fact we are 95% sure that if they do increase, they probably don’t increase by more than 0.39 dmfs. Combine those two pieces of information to tell you that we’re pretty sure that kids on gum B do a lot better on average than kids on gum C (probably by more than 1 dmfs).
  1. e “Gum” data are composed of samples from 3 different populations: kids who would chew gums A, B, or C for two years. Data for groups B and C are presented in table 1 below (and also available on course webpage).
  2. Compute 95% confidence intervals for the mean two-year change in DMFS for groups B and C.
  3. Compute a hypothesis test to assess whether the average DMFS of children given gum B will change over two years. Report test hypotheses, significance level of test, T-statistic, whether or not the test rejects the null hypothesis, and the p-value (may be approximate).
  4. Repeat question 1.b. for gum C.
  5. From these data, what can you say about the overall population of children similar to these who may chew gums B or C?
  1. The “Mercury” data comes from an observational study that was investigating correlations between amount of amalgam restorations in a person’s mouth and their exposure to mercury. The “cases” are volunteers who have reported symptoms that they possibly attribute to mercury exposure. The “controls” are randomly sampled volunteers who have not reported symptoms. The data are presented in table 1 below (and also available on course webpage).
  2. Compute a 95% confidence interval for the mean daily inhaled mercury (Hg) vapor observed in the cases.
  3. Compute a 95% confidence interval for the mean daily inhaled mercury (Hg) vapor observed in the controls.
  4. Do these confidence intervals imply that the populations (cases and contols) are different? Why or why not?

Table 1: Gum Data, groups B and C
Group B / Group C
id # / DMFS / id # / DMFS
before / after / change / before / after / change
420 / 3 / 3 / 0 / 331 / 2 / 8 / 6
830 / 12 / 0 / -12 / 335 / 8 / 9 / 1
835 / 0 / 0 / 0 / 336 / 0 / 1 / 1
847 / 2 / 1 / -1 / 340 / 0 / 0 / 0
853 / 7 / 8 / 1 / 342 / 0 / 0 / 0
855 / 0 / 0 / 0 / 344 / 1 / 1 / 0
856 / 1 / 6 / 5 / 345 / 0 / 5 / 5
858 / 0 / 0 / 0 / 346 / 3 / 2 / -1
859 / 1 / 0 / -1 / 348 / 5 / 5 / 0
867 / 3 / 4 / 1 / 352 / 1 / 2 / 1
868 / 0 / 6 / 6 / 353 / 1 / 6 / 5
873 / 7 / 2 / -5 / 354 / 8 / 10 / 2
876 / 3 / 7 / 4 / 355 / 5 / 6 / 1
879 / 3 / 0 / -3 / 356 / 1 / 8 / 7
881 / 7 / 6 / -1 / 364 / 2 / 2 / 0
886 / 5 / 0 / -5 / 366 / 5 / 8 / 3
888 / 0 / 3 / 3 / 374 / 6 / 11 / 5
890 / 7 / 1 / -6 / 380 / 5 / 19 / 14
896 / 11 / 5 / -6 / 385 / 0 / 2 / 2
897 / 3 / 3 / 0 / 389 / 3 / 2 / -1
903 / 0 / 0 / 0 / 392 / 0 / 0 / 0
908 / 5 / 7 / 2 / 396 / 0 / 3 / 3
913 / 8 / 6 / -2 / 401 / 2 / 1 / -1
915 / 6 / 10 / 4 / 402 / 8 / 15 / 7
919 / 6 / 0 / -6 / 403 / 7 / 18 / 11
920 / 3 / 4 / 1 / 404 / 0 / 1 / 1
946 / 2 / 0 / -2 / 405 / 3 / 8 / 5
950 / 2 / 0 / -2 / 408 / 3 / 4 / 1
951 / 3 / 1 / -2 / 415 / 5 / 8 / 3
952 / 7 / 10 / 3 / 416 / 12 / 15 / 3
958 / 4 / 2 / -2 / 417 / 4 / 12 / 8
959 / 0 / 0 / 0 / 422 / 6 / 0 / -6
966 / 2 / 0 / -2 / 423 / 16 / 18 / 2
967 / 1 / 0 / -1 / 428 / 2 / 7 / 5
1,002 / 8 / 8 / 0 / 430 / 7 / 5 / -2
434 / 4 / 10 / 6
455 / 7 / 8 / 1
456 / 0 / 0 / 0
457 / 4 / 12 / 8
460 / 1 / 0 / -1
Table 2: Mercury Vapor Data
ID / GROUP* / age / occlusal amalgam surfaces / total amalgam surfaces / daily inhaled Hg vapor (g Hg/24 hr) / Urinary Hg (nmol Hg/l creatinine) / Fish consumption (meals/month)
1 / CASE / 44 / 13 / 43 / 1.6 / 0.8 / 8.0
2 / CASE / 42 / 13 / 46 / 2.1 / 1.0 / 2.0
3 / CASE / 43 / 10 / 30 / 2.6 / 2.2 / 4.0
4 / CASE / 43 / 19 / 68 / 4.1 / 3.6 / 10.0
5 / CASE / 40 / 20 / 48 / 4.6 / 1.4 / 6.0
6 / CASE / 36 / 11 / 24 / 0.3 / 1.7 / 4.0
7 / CASE / 49 / 13 / 39 / 1.4 / 1.7 / 4.0
8 / CASE / 62 / 4 / 34 / 0.6 / 2.0 / 7.0
9 / CASE / 47 / 14 / 62 / 1.7 / 0.9 / 5.0
10 / CASE / 40 / 13 / 50 / 1.2 / 1.0 / 4.0
11 / CONTROL / 45 / 16 / 63 / 6.1 / 7.1 / 3.0
12 / CONTROL / 43 / 11 / 43 / 3.0 / 2.4 / 3.0
13 / CONTROL / 47 / 12 / 31 / 2.4 / 3.3 / 4.0
14 / CONTROL / 34 / 11 / 25 / 0.7 / 1.5 / 8.0
15 / CONTROL / 41 / 10 / 24 / 0.4 / 1.4 / 5.0
16 / CONTROL / 40 / 15 / 46 / 0.6 / 4.7 / 1.0
17 / CONTROL / 42 / 12 / 45 / 2.5 / 3.8 / 3.0
18 / CONTROL / 48 / 16 / 50 / 3.8 / 3.3 / 2.0
Data from: / Berglund A, Molin M: "Mercury vapor release from dental amalgam in patients with symptoms allegedly caused by amalgam fillings." European Journal of Oral Sciences (1996): 104: 56-63
* / "Cases" are participants complaining of symptoms possibly associated with mercury.