Introductory Biostatistics - Practice Problems with Solutions | PUBHLTH 640, Study notes of Community Health

Download Introductory Biostatistics - Practice Problems with Solutions | PUBHLTH 640 and more Study notes Community Health in PDF only on Docsity! PubHlth 640 Intermediate Biostatistics 1 Unit 1 – Review of PubHlth 540, Introductory Biostatistics Practice Problems - Week #1 SOLUTIONS 1. Recall that variables can be of different types. We learned in introductory biostatistics that appropriate methods of summarization depend on the variable type. And we noticed that, sometimes, a method of summarization is not appropriate. For example, it is not appropriate to construct a cumulative frequency graph summary of nominal data. Using whatever sources you have for introductory biostatistics, complete the following table. Random Variable Discrete Continuous Nominal Ordinal Interval Ratio Descriptive Methods Bar chart Pie chart - - Bar chart Pie chart - - - - Dot diagram Scatter plot (2 variables) Stem-Leaf Histogram Box Plot Quantile-Quantile Plot - - Dot diagram Scatter plot (2 variables) Stem-Leaf Histogram Box Plot Quantile-Quantile Plot Numerical Summaries Freq Table Rel Freq Table Freq Table Rel Freq Table Cum Freq Table Cum Rel Freq Table - - - - - means, variances, percentiles - - - - - means, variances, percentiles wk1_solutions.doc PubHlth 640 Intermediate Biostatistics 2 2. Try your hand at the following probability exercises. a) Divide a line segment into three parts such that one portion is half the length of original line and the other two portions are each one quarter then length of the original line. Choose a point at random. What is the probability that this point is in the ½ length portion? Answer: .5 Solution: As all points along the line segment are equally likely, length is proportional to probability. Thus, ½ length corresponds to ½ of the total probability. b) If there is a 14% chance that any person selected at random was born on a Monday, what is the probability that, of any seven people selected at random, exactly one was born on a Monday? Answer: .40 Solution: This is a binomial probability calculation. X = Realized count of number of Monday birthdays in sample of 7 N = # trials = 7 Event of interest is “birthday on a Monday” Calculate Pr [ X = 1 ] = .3965 π = Probability[event] = .14 http://faculty.vassar.edu/lowry/binomialX.html c) What are the odds of getting exactly one pair in five card stud poker using a 52 card deck? Answer: Odds are 42 to 58 Solution: This is a combinatorial calculation that assumes all five card hands are equally likely. # hands that are exactly one pairprobability[exactly one pair] = Total # hands possible 52 52! (52)(51)(50)(49)(48)total # hands possible = 2,598,960 5 5!47! (5)(4)(3)(2)(1) ⎛ ⎞ = = =⎜ ⎟ ⎝ ⎠ To solve for the # hands that are exactly one pair, the idea is to think in steps. 1. # choices of a rank ( ace or 2 or 3 or … or queen or king) = 13 wk1_solutions.doc PubHlth 640 Intermediate Biostatistics 5 4. See if you can recall some of the important concepts that are discussed in introductory biostatistics. (a) What is a sampling distribution? Answer: A sampling distribution is a probability distribution for a random variable that is itself a statistic. Thus, sampling distributions refer to the probability distributions of such things as the sample mean, the sample variance etc. These “probability distributions” are the result of the idea of replicate sampling infinitely many times. (b) What does the central limit theorem tell us? Why is it so useful to us? What is a z-score? What is a t-score? Answer: The central limit theorem tells us that as the choice of sample size increases, the sampling distribution of the average approaches normality. This is very useful to us because it allows us, even for moderate sample size, to regard the average as a realization of a Normal distribution. random variable - E[random variable]z-score = SE[random variable] random variable - E[random variable]t-score = Estimated SE[random variable] (c) In a sentence or two, explain the meaning of a 95% confidence interval for a population mean that has lower limit 35.6 and upper limit 52.8 Answer: This interval allows the investigator to say that he/she is 95% confident that the unknown true population mean is between 35.6 and 52.8. Of course, the true mean is either in this interval or not; we don’t actually know. Recall that it is NOT CORRECT to say that the probability is .95 that the population mean is between 35.6 and 52.8. wk1_solutions.doc PubHlth 640 Intermediate Biostatistics 6 (d) Define p-value. Interpret p<.05 and p<.01. Given identical study conditions, which gives stronger evidence against the null hypothesis? Answer: A p-value is a chance statement. A p-value is the probability that the test statistic of interest attains a value as extreme, or more extreme (relative to the null hypothesis), under the null hypothesis probability model. All other things being equal, a p < .01 gives stronger evidence against the null hypothesis. (e) Suppose a two sided hypothesis test of treatment benefit in a randomized controlled trial of placebo versus active treatment yields a p-value of 0.045. What are the possible explanations for this result? Answer: There are several possible explanations and, ultimately, we do not know which one is the correct explanation. Among them are - The active treatment is truly different than the placebo. - The active treatment is equivalent to the placebo but an event of low probability has occurred. - The active treatment is equivalent to the placebo but one or more biases have biased the data in the direction of statistical significance (albeit marginal) wk1_solutions.doc