---
title: 'Normal-based Inference for Proportions: Examples'
date: "1/25/2022"
output:
  html_document: default
  pdf_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


## Problem 1

To investigate the flame resistance of a material used in children’s pajamas, 85 material specimens were subjected to high temperatures, and 21 of those specimens ignited. The material will only be safe for use on the market if the true proportion of ignitions is less than 20%. Use these data to determine if the material is safe to use on the market.

## Problem 2

Bozeman is known for having a large number of drivers who own Subaru vehicles, perhaps because these vehicles are known to perform well in winter driving conditions. But, is the proportion of residents in Bozeman who own Subarus higher than a place like Madison, Wisconsin? Researchers randomly selected 500 phone numbers from all local landlines in both Bozeman and Madison and found that 198 Bozeman residents owned a Subaru, while only 147 Madison residents owned a Subaru.


## Problem 3

Bozeman is known nationwide as one of the premier ski towns in the US. In fact, many students choose to attend MSU for that reason. With the hopes of increasing enrollment, MSU’s advertising team created two brochures. One of the brochures has a skier on the front, and the other has a snowboarder on the front. One hundred and fifty California students were chosen at random, half were randomly assigned to receive the skier brochure, and half were randomly assigned to receive the snowboarder brochure. The advertising team wants to know if the probability a California student enrolls at MSU differs based on the type of brochure they receive. The data are summarized below.
```{r}
dat <- matrix(c(17,14,58,61), nrow=2, ncol=2, byrow=TRUE,
              dimnames=list(c("Enrolled","Not Enrolled"),
                            c("Skier","Snowboarder")))
dat
```

## Problem 4

There is no longer any doubt that smoking cigarettes is hazardous to your health and the health of those around you. Yet for someone who is addicted to smoking, quitting is no simple matter. One promising technique is to apply a patch to the skin that dispenses nicotine into the blood. In fact, these nicotine patches have become one of the most frequently prescribed medications in the United States. 

To test the effectiveness of these patches on the cessation of smoking, Dr. Richard Hurt and his colleagues (1994) recruited 240 smokers at Mayo Clinics in Rochester, Minnesota; Jacksonville, Florida; and Scottsdale, Arizona. Volunteers were required to be between the ages of 20 and 65, have an expired carbon monoxide level of 10 ppm or greater (showing that they were indeed smokers), be in good health, have a history of smoking at least 20 cigarettes per day for the past year, and be motivated to quit. 

Volunteers were randomly assigned to receive either 22-mg nicotine patches or placebo patches for 8 weeks (120 in each group). They were also provided with an intervention program recommended by the National Cancer Institute, in which they received counseling before, during, and for many months after the 8-week period of wearing the patches.

After the 8-week period of nicotine or placebo patch use, almost half (46%) of the nicotine patch group had quit smoking, while only one-fifth (20%) of the control group had done so. Having quit was defined as “self-reported abstinence (not even a puff) since the last visit and an expired air carbon monoxide level of 8 ppm or less.” (Example quoted from Utts and Heckard, 2013, p. 201-202)


## Problem 5

About 10% of the general US population is left-handed. Researchers notice that professional tennis players often tend to be left-handed. To study whether professional tennis players tend to be left-handed more often than the general US population, they collected a simple random sample of 125 professional tennis players and observed that 16 of these tennis players were left handed.

