Homework #3 (optional practice problems) DPHS 568 Biostatistics in Dentistry Summer 2007

  1. Dentistry problem from Rosner. .

In a study, 28 adults with mild periodontal disease are assessed before and 6 months after implementation of a dental-education program intended to promote better oral hygiene. After 6 months, the changes in periodontal status were graded on a 7-point scale, with +3 indicating the greatest improvement, 0 indicating no change and –3 indicating the greatest decline. Data are available online at under “perio status” data.

  1. Assess the impact of the program statistically using a two-sided sign test. Report a p-value.
  2. What non-parametric test (other than the sign test) could be used to determine whether a significant change in periodontal status has occurred over time that would not throw out as much information as the sign test?
  3. (optional) Perform the test in part b and report a p-value.
  1. Microbiology problem from Rosner

A study sought to demonstrate that soybeans inoculated with nitrogen-fixing bacteria yield more and grow adequately without the use of expensive environmentally deleterious synthesized fertilizers. The initial hypothesis was that inoculated plants would outperform their uninoculated counterparts. There were 8 inoculated plants (I) and 8 uninoculated plants (U). The plant yield as measured by pod weight for each plant is available in the “pod weight” data, which can be found at

  1. Suppose one can assume that the pod weights are normally distributed. Perform an appropriate t-test to assess whether the average yield of the inoculated plants are statistically significantly different from the average yield of the uninoculated plants. Report a p-value (approximate is OK).
  2. Now suppose that one is not comfortable with the assumption of Normality. What would be the appropriate non-parametric test to assess whether the average yields differ by group?
  3. (optional) Perform the test in part b and report a p-value.
  1. Calculate the power for a one-sided sign test with α = 0.05 significance level. Assume that we will have 20 (non-tied) observations, and that, in actuality, we expect the true proportion of positive differences in our population to be 75%.
  2. Step 1, determine the rejection region: Remember that in a sign test we are only concerned whether the observations are negative or positive. Under the null hypothesis the probability of an observation being positive is 0.5. Thus, the number of positive observations we would see out of 20 will have a binomial distribution with n=20 and p=0.5. To find the rejection region, assume that we will be rejecting when the number of positives is high (remember, a one-sided test only rejects for values of the statistic on one particular side of the null). Find the smallest number of observations for which the probability of that many or more positives is less than 0.05. That number will be the boundary of your rejection region. If the number of positives out of 20 is equal to or greater than this boundary number, we will reject the null hypothesis.
  3. Step 2, calculate the power: Now assume that 75% of the differences are positive (what we are expecting in real life). Note that this assumption implies that the null hypothesis is not true. Thus, the power of our test will be the probability that we will reject the null hypothesis. We will reject if the number of positives in our 20 observations is in the rejection region determined in step 1 above.Calculate this probability the binomial distribution where n is 20 and the probability of positive is 75%. This will be the power of the test under the assumptions given at the start of the problem.
  4. Extra special bonus question: Because of the discrete nature of the data here, the significance level of our test will probably not be exactly equal to 0.05, but should be less than 0.05. What is the true significance level of the sign test in the scenario detailed above?
  1. In a study[i] to determine the prevalence of oral hygiene practices and halitosis among undergraduate students, 200 males and 100 females were assessed. A survey question found that 156 (78%) of the males felt they had bad breath upon waking, while 63 (63%) of the females felt they had bad breath upon waking.
  2. Perform a hypothesis test to assess whether these proportions are statistically significantly different. Report a p-value and a conclusion.
  3. Calculate a 95% confidence interval for the true difference in proportion of self-perceived morning breath between males and females.
  4. The participants were assessed by a dentist for halitosis. There were 15 (7.5%) males and 3 (3.0%) females diagnosed with bad breath. Of the hypothesis tests we have discussed in class, which would be most appropriate to determine whether these proportions are evidence of a difference in bad-breath prevalence between gender?
  1. In a study to compare two types of implants 100 patients are chosen who require at least two implants. For these patients two sites are assigned to be experimental sites (the ones that seem most similar within the mouth) and for these sites the two competing implants (A and B) are placed. Which site gets which type of implant is determined by a random selection procedure. The patients are followed for 1 year. In 77 of the patients both implants are still viable after one year, and in 4 of the patients both implants failed. In 13 patients implant A was still viable whereas implant B needed to be removed, and in 6 patients implant B was still viable and implant A needed to be removed. Perform a significance test to assess whether or not there is evidence to show the implant types are better or not in terms of one-year survival.
  2. Clearly state the null and alternative hypotheses
  3. Report the p-value.
  4. State the conclusion of the hypothesis test.
  5. Suppose implant type B is the standard while A is an experimental prototype that is expected to be much more expensive than B. In this case a one-sided hypothesis test may be appropriate. Report the p-value for a one-sided test where the alternative hypothesis is H1: implants of type A have a better one year survival rate than implants of type B.
  1. Revisit the Mercury Vapor data available from Homework #1 handout and also at . Consider specifically the number of occlusal surfaces and the urinary mercury levels in the control subjects.
  2. Draw a scatterplot depicting the association between number of occlusal surfaces and urinary mercury levels in the controls. Which variable should you put on the horizontal axis?
  3. Compute the correlation coefficient for these two variables in the controls. The Excel function CORREL can be used to do this.
  4. Compute the regression coefficients describing the relationship between these two variables (use urinary mercury as the dependent variable). Draw the “line of best fit” into the scatterplot in part a.
  5. Perform an hypothesis test to see if there is evidence to say these two variables are associated. Report a p-value (approximate is OK) and a conclusion.
  1. Use the SPSS output presented in class relating cigarettes smoked per day to mean attachment level to answer the following questions. The output can be found at .
  2. What would the best guess for the attachment level of someone who smokes one pack of cigarettes (20 cigarettes) per day?
  3. Compute the 95% confidence interval estimating the average attachment level of people who smoke 20 cigarettes per day.
  4. Compute a range of values that should contain the attachment levels of 95% of the people who smoke 20 cigarettes per day (confidence interval for prediction of a single value).
  5. How are the confidence intervals from parts b and c different? Why do we need different confidence intervals?

[i]Almas K, Al-Hawish A, Al-Khamis W. Oral Hygiene Practices, Smoking Habits, and Self-Perceived Oral Malodor Among Dental Students. J Contemp Dent Pract 2003 November;(4)4:077-090.