# Homework 4

Due 2017/10/27

Note this homeowrk is due Friday at midnight.

# 1. Exact Binomial test

As we saw in lecture binom.test() performs an exact binomial test.

1. Use binom.test() to perform a two-sided test of the null hypothesis that $$p = 0.6$$ when you observe $$X = 8$$ successes in $$n = 15$$ trials. What is the reported p-value?

2. Confirm that the p-value from (a) agrees with the method discussed in class: use dbinom() to find the probabilities of each outcome $$X = 0, 1, 2,\ldots, 15$$ and add up the probabilities of outcomes as likely as or less likely than $$X = 8$$ when $$p = 0.6$$.

3. Based on the p-value from part (a), would you reject the null hypothesis $$H_0: p = 0.6$$ vs. $$H_A: p \ne 0.6$$ at level $$\alpha = 0.05$$?

4. In class we found a rough confidence interval by inverting the exact test (Q8 from the worksheet on Oct 13th). Use binom.test() and a sequence of $$p_0$$ to emulate this procedure in R to find a 95% confidence interval for $$p$$ up to 2 decimal places of precision.

# 2. Data Analysis

Using the same brfss data as in HW #3.

1. Estimate the proportion of US residents who are in the overweight or obese categories based on body mass index (BMI $$= \frac{\text{weight in kg}} {(\text{height in m})^2} > 25$$). Write a statistical summary of your findings

2. Is the median desired weight loss greater than zero for US females? What is a likely range for the median desired weight loss. Conduct the appropriate analyses and write a statistical summary of your findings (Hint: if observed values take the exact value of the hypothesized median, you will need to drop them before proceeding with the sign test.)

3. A colleague suggests testing that median desired loss is zero with a Wilcoxon Signed Rank test. Write a two to three sentence argument for why this is not appropriate in this case.

# 3. The Approximate Binomial test

In lecture we saw both the “Exact Binomial Test” and the “Approximate Binomial Test”. Recall you can get the approximate test p-value in R with prop.test() and the exact p-value from binom.test().

We also saw a rough guideline, that if $$np > 5$$ and $$n(1-p) > 5$$, the approximate test should be a good approximation the the exact test.

Your task for this question is to investigate this guideline through simulation. Perform simulations to assess the Type I error rate of the two procedures for a range of $$np$$ (i.e. set $$p$$ and vary $$n$$, or set $$n$$ and vary $$p$$). Does the guideline seem reasonable?