Assignment 4 - Fundamentals of Biostatistics
1. A survey is to be run on BC high school students regarding substance use. Students will be offered a credit card in exchange for the completion of a survey of which the topic they do not know a priori.
a. Should voluntary bias be a concern here? (max 2 sentences)
b. If the true proportion of tobacco smokers is 31.7%, what is the probability that from a sample of size 100, less than 25 students are smokers? Assume smokers and non-smokers are equally likely to participate in the survey.
c. From a survey in 2005, a 95% confidence interval for the percentage of students using ADHD meds (Adderall, Ritalin, etc.) was found to be [0.081, 0.234]. If a confidence interval with margins of error that are at most 3% is to be obtained, what sample size should we use?
2. A random sample of 250 male autoworkers was tested for accumulation of methyl ketones. When these levels are above 500 ppm, they can lead to serious adverse events (dubbed dangerous levels). From the sample, the average concentration was found to be 75 ppm with a sample standard deviation of 157 ppm. In total 17 men were found to have concentrations in excess of 500.
a. Describe what the distribution should look like given these summary statistics (there are two scenarios that are possible here, please provide one).
b. We wish to study the mean concentration in autoworkers and the proportion that have dangerous levels of methyl ketones. What are the parameters and statistics here?
c. What are the estimators and the estimates?
d. Construct a 95% confidence interval for the proportion of people that have dangerous levels of methyl ketones.
e. It was hypothesized that levels are 50 ppm or lower. Test this hypothesis at the 0.05 significance level.
f. What type of error are we at risk of making in part e?
3. There is a theory that the anticipation of a birthday can prolong a person's life. In a study set up to examine that notion statistically, it was found that only 85 of 733 people whose obituaries were published in Salt Lake City in 1975 died in the three-month period preceding their birthday.
a. Construct a confidence interval for the probability of dying in the three months prior to your birthday (based on these data]. Include an interpretation of the interval.
b. If anticipating your birthday has no effect on the probability of dying, then what is the probability of dying in the three months prior to one's birthday?
c. Test this theory using the appropriate test. What do you conclude?
d. What is the p-value and how do you interpret it?
4. A researcher randomly sampled 350 Vancouver employees working at jobs that include no physical activity - desk jobs. Among other things, the researcher was interested in the mean number of sick days that people at these jobs took last year. The final report included the following statement: "A 95% confidence interval shows that there is a 19/20 probability that the desk job workers took an average of 5.33 to 7.25 sick days in 2010."
a. In your own words, explain the role of probability in confidence intervals and why the use of probability in this statement is wrong.
b. Provide the statement the researcher should have provided.
c. Provide the equation that was used to construct this interval.
d. Were the conditions for the use of this equation met? Give details.
e. The average number of sick days taken in physically active jobs is
5.02. Test whether the average in these non-active drugs is higher than 5.02 at the 0.01 significance level?
5. Time to re-incarceration in Vancouver among peoples who have multiple incarcerations is positively skewed with mean 236 days and standard deviation 62 days. It is skewed in a way that 8% of individuals take more than a year to get re-incarcerated. A study on this population sampled 150 individuals and measured the time between 1st release and 2nd incarcerations. Although it was a convenience sample, consider it to be a simple random
sample.
a. Describe the sampling distribution of the sample mean time to re- incarceration.
b. What is the probability that the sample mean is below 225 days?
c. What is the probability that less than 5% of the sample will take more than a year to be re-incarcerated?
6. An ophthalmological experiment randomly assigned one of two treatments to each eye (call these treatment A and B). This is what we call a matched pairs design. In total 74 individuals volunteered for the experiment. For each
individual the difference in healing time between the eye with treatment A and that with treatment B was recorded (TxA - TxB). The sample mean was found to be -2.3 days with a sample standard deviation of 5.2 days.
a. Construct a 95% confidence interval for the difference in healing time between the treatments.
b. Based on the above test, what would you conclude for the hypothesis test testing for a difference in healing time for both treatments? Keep in mind that a difference corresponds to mean difference that differs from 0.
c. Explain what a type I error would be in the context of this problem.