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* commit 'eaed53d96e44a7c16145d1ac9701827607ff819e':
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06_StatisticalInference/homework/hw3.Rmd

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+---
+title       : Homework 3 for Stat Inference
+subtitle    : Extra problems for Stat Inference
+author      : Brian Caffo
+job         : Johns Hopkins Bloomberg School of Public Health
+framework   : io2012
+highlighter : highlight.js  
+hitheme     : tomorrow       
+#url:
+#    lib: ../../librariesNew #Remove new if using old slidify
+#    assets: ../../assets
+widgets     : [mathjax, quiz, bootstrap]
+mode        : selfcontained # {standalone, draft}
+---
+```{r setup, cache = F, echo = F, message = F, warning = F, tidy = F, results='hide'}
+# make this an external chunk that can be included in any file
+library(knitr)
+options(width = 100)
+opts_chunk$set(message = F, error = F, warning = F, comment = NA, fig.align = 'center', dpi = 100, tidy = F, cache.path = '.cache/', fig.path = 'fig/')
+
+options(xtable.type = 'html')
+knit_hooks$set(inline = function(x) {
+  if(is.numeric(x)) {
+    round(x, getOption('digits'))
+  } else {
+    paste(as.character(x), collapse = ', ')
+  }
+})
+knit_hooks$set(plot = knitr:::hook_plot_html)
+```
+
+## About these slides
+- These are some practice problems for Statistical Inference Quiz 3
+- They were created using slidify interactive which you will learn in 
+Creating Data Products
+- Please help improve this with pull requests here
+(https://github.com/bcaffo/courses)
+
+
+
+--- &multitext
+Load the data set `mtcars` in the `datasets` R package. Calculate a 
+95% confidence interval to the nearest MPG.
+
+1. What is the lower endpoint of the interval?
+2. What is the upper endpoint of the interval?
+
+*** .hint
+Do `library(datasets)` and then `data(mtcars)` to get the data.
+Consider `t.test` for calculations. You may have to install
+the datasets package.
+
+
+*** .explanation
+```{r}
+library(datasets); data(mtcars)
+round(t.test(mtcars$mpg)$conf.int)
+```
+
+<span class="answer">`r round(min(t.test(mtcars$mpg)$conf.int))`</span>
+<span class="answer">`r round(max(t.test(mtcars$mpg)$conf.int))`</span>
+
+--- &multitext
+Suppose that data of 9 paired differences has a standard error of $1$, what value would the average difference have to be to have the lower endpoint of a 95%
+students t confidence interval touch zero?
+
+1. Give the number here to two decimal places
+
+*** .hint
+The t interval is $\bar x t_{.95, 8}\pm s /sqrt{n}$
+
+*** .explanation
+<span class="answer">`r round(qt(.95, df = 3) * 1 / 3, 2)`</span>
+
+We want $\bar x = t_{.95} s / sqrt{n}$
+```{r}
+round(qt(.95, df = 3) * 1 / 3, 2)
+```
+
+
+--- &radio
+An independent group Student's T interval is used over
+a paired T interval when:
+
+1. The observations are paired between the groups.
+2. _The observations within the groups are natually assumed to be statistically independent_
+3. As long as you do it correctly, either is fine.
+4. More details are needed to answer this question
+
+*** .hint
+A paired interval is for paired observations.
+
+*** .explanation
+If the groups are independent is the correct interval.
+
+
+--- &multitext
+Consider the `mtcars` dataset. Construct a 95% T interval for MPG comparing
+4 to 6 cylinder cars (subtracting in the order of 4 - 6) 
+assume a constant variance.
+
+1. What is the lower endpoint of the interval to 1 decimal place?
+2. What is the upper endpoint of the interval to 1 decimal place?
+
+*** .hint
+Use `t.test` with `var.equal=TRUE`
+
+*** .explanation
+
+```{r}
+m4 <- mtcars$mpg[mtcars$cyl == 4]
+m6 <- mtcars$mpg[mtcars$cyl == 6]
+#this does 4 - 6
+confint <- as.vector(t.test(m4, m6, var.equal = TRUE)$conf.int)
+```
+
+<span class="answer">`r round(min(confint), 1)`</span>
+<span class="answer">`r round(max(confint), 1)`</span>
+
+
+--- &radio
+If someone put a gun to your head and said "Your confidence interval
+must contain what it's estimating or I'll pull the trigger", what would
+be the smart thing to do?
+
+1. _Make your interval as wide as possible_
+2. Make your interval as small as possible
+3. Call the authorities
+
+*** .hint
+C'mon. You don't need a hint
+
+*** .explanation
+This is just an example of what happens to confidence intervals as you
+increas the confidence level. You want to be quite sure in your interval (i.e.
+have a large confidence level) and so you would increase the interval's width
+
+--- &radio
+
+Refer back to comparing MPG for 4 versus 6 cylinders. What do you conclude?
+
+1. The interval is above zero, suggesting 6 is better than 4 in the terms of MPG
+2. _The interval is above zero, suggesting 4 is better than 6 in the terms of MPG_
+3. The interval does not tell you anything about the hypothesis test; you have to do the test.
+4. The interval contains 0 suggesting no difference.
+
+*** .hint
+Refer back to the problem, consider the implications of the interval being
+larger than 0, double check the order in which things were subtracted and
+make sure the results make sense in the context of the problem.
+
+*** .explanation
+The interval was conducted subtracting 4 - 6 and was entirely above zero.
+
+--- &multitext
+Suppose that 18 obese subjects were randomized, 9 each, to a new diet pill and a placebo. Subjects' body mass indices (BMIs) were measured at a baseline and again after having received the treatment or placebo for four weeks. The average difference from follow-up to the baseline (followup - baseline) was ???3 kg/m2 for the treated group and 1 kg/m2 for the placebo group. The corresponding standard deviations of the differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for the placebo group. Does the change in BMI over the four week period appear to differ between the treated and placebo groups?  
+
+1. Calculate the pooled variance estimate to 2 decimal places
+
+
+*** .hint
+The sample sizes are equal, so the pooled variance is the average of the 
+individual variances
+
+
+*** .explanation
+<span class="answer">`r round(min(confint), 1)`</span>
+```{r}
+n1 <- n2 <- 9
+x1 <- -3  ##treated
+x2 <- 1  ##placebo
+s1 <- 1.5  ##treated
+s2 <- 1.8  ##placebo
+spsq <- ( (n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)
+```
+<span class="answer">`r round(spsq, 2)`</span>
+
+
+--- &radio
+
+For Binomial data the maximum likelihood estimate for the probability of 
+a success is
+
+1. _The proportion of successes_
+2. The proportion of failures
+3. A shrunken version of the proportion of successes
+4. A shrunken version of the proportion of failures
+
+*** .hint
+Look back at the notes about likelihood.
+
+*** .explanation
+The MLE for binomial data is always the proportion of successes.
+
+--- &radio
+
+Bayesian inference requires
+
+1. A type I error rate
+2. Setting your confidence level
+3. _Assigning a prior probability distribution_
+4. Evaluating frequency error rates
+
+*** .explanation
+All of the other answers discuss frequentist concepts. All Bayesian analyses requiring setting a prior.
+
+