Mat-255 probability and inferential statistics
1. A patient is classified as having gestational diabetes if their glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca’s doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca’s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ= 140+X mg/dl and σ = X+1 mg/dl, where X is the last digit of your GCU student ID number. What is the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured:
- X+1 times, where X is the last digit of your student ID?
- X+4 times, where X is the last digit of your student ID?
Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.
2. Suppose next that we have even less knowledge of our patient, and we are only given the accuracy of the blood test and prevalence of the disease in our population. We are told that the blood test is 9X percent reliable, gestational diabetes affects X+1 percent of the population in our patient’s age group, and that our test has a false positive rate of X+4 percent. Compute the following quantities based on this new information:
- If 100,000 people take the blood test, how many people that test positive will actually have gestational diabetes?
- What is the probability of having the disease given that you test positive?
- If 100,000 people take the blood test, how many people that test negative despite actually having gestational diabetes?
- What is the probability of having the disease given that you tested negative?
Comment on what you observe in the above computations. How does the prevalence of the disease affect whether the test can be trusted?
3. As we have seen in class, hypothesis testing and confidence intervals are the most common inferential tools used in statistics. Imagine that you have been tasked with designing an experiment to determine reliably if a patient should be diagnosed with diabetes based on their blood test results. Create a short outline of your experiment, including all of the following:
- A detailed discussion of your experimental design.
- How is randomization used in your sampling or assignment strategy?
- The type of inferential test utilized in your experiment.
- A formal statement of the null and alternative hypothesis for your test.
- A confidence interval for estimating the parameter in your test.
- An interpretation of your p-value and confidence interval, including what they mean in context of your experimental design