Week 3 Practice Problems

Author

Affiliation

Alex Kaizer

University of Colorado-Anschutz Medical Campus

This page includes optional practice problems, many of which are structured to assist you on the homework with Solutions provided on a separate page. Data sets, if needed, are provided on the BIOS 6618 Canvas page for students registered for the course.

This week’s extra practice exercises focus on subsetting data objects and identifying relationships between the various assumptions in power calculations.

Exercise 1

1a

Run the following code to combine a few state-related data sets that are part of R’s available data sets:

Code

states <- data.frame(state.x77, state.region, state.abb)

1b

Calculate the mean (SD) life expectancy by state region.

1c

Subset the four corner states (Utah, Colorado, Arizona, and New Mexico) by row name. Which state has the largest population? Which state has the lowest high school graduation rate?

1d

Subset states that either have an area greater than 90,000 miles$^2$ or a percent of high school graduates below 50%. How many states meet this criteria?

1e

Subset states that have an area greater than 90,000 miles$^2$ and a percent of high school graduates below 50%. How many states meet this criteria?

1f

Are there any states where the mean number of days with a minimum temperature below freezing is above 100, the murder rate is greater than 10 per 100,000 population, and the rate of illiteracy is below 1%?

Exercise 2

This exercise focuses on evaluating the relationship of what happens as we change different parts of a power calculation for the case of the known standard deviation. You may find the website https://rpsychologist.com/d3/nhst/ helpful to visualize the different scenarios.

For each of these scenarios, assume that all parameters not discussed are fixed at some value, then identify what happens to the different quantities.

As sample size $n$ increases:

Power:
Detectable Difference $|\mu_0 - \mu_1|$:

As the difference to be detected, $|\mu_0 - \mu_1|$, increases:

Power:
Required Sample Size:

As desired power increases:

Required Sample Size:
Detectable Difference $|\mu_0 - \mu_1|$:

As $\sigma$, the population SD, increases:

Power:
Detectable Difference $|\mu_0 - \mu_1|$:
Required Sample Size:

As $\alpha$, the significance level of the test, increases:

Power:
Detectable Difference $|\mu_0 - \mu_1|$:
Required Sample Size:

--- title: "Week 3 Practice Problems" author: name: Alex Kaizer roles: "Instructor" affiliation: University of Colorado-Anschutz Medical Campus toc: true toc_float: true toc-location: left format: html: code-fold: show code-overflow: wrap code-tools: true --- ```{r, echo=F, message=F, warning=F} library(kableExtra) library(dplyr) ``` This page includes optional practice problems, many of which are structured to assist you on the homework with [Solutions provided on a separate page](/labs/prac3s/index.qmd). Data sets, if needed, are provided on the BIOS 6618 Canvas page for students registered for the course. This week's extra practice exercises focus on subsetting data objects and identifying relationships between the various assumptions in power calculations. # Exercise 1 ## 1a Run the following code to combine a few state-related data sets that are part of R's available data sets: ```{r} states <- data.frame(state.x77, state.region, state.abb) ``` ## 1b Calculate the mean (SD) life expectancy by state region. ## 1c Subset the four corner states (Utah, Colorado, Arizona, and New Mexico) **by row name**. Which state has the largest population? Which state has the lowest high school graduation rate? ## 1d Subset states that either have an area greater than 90,000 miles$^2$ *or* a percent of high school graduates below 50\%. How many states meet this criteria? ## 1e Subset states that have an area greater than 90,000 miles$^2$ *and* a percent of high school graduates below 50\%. How many states meet this criteria? ## 1f Are there any states where the mean number of days with a minimum temperature below freezing is above 100, the murder rate is greater than 10 per 100,000 population, and the rate of illiteracy is below 1\%? # Exercise 2 This exercise focuses on evaluating the relationship of what happens as we change different parts of a power calculation for the case of the **known standard deviation**. You may find the website <https://rpsychologist.com/d3/nhst/> helpful to visualize the different scenarios. For each of these scenarios, assume that all parameters not discussed are fixed at some value, then identify what happens to the different quantities. **As sample size $n$ increases:** 1. Power: 2. Detectable Difference $|\mu_0 - \mu_1|$: **As the difference to be detected, $|\mu_0 - \mu_1|$, increases:** 1. Power: 2. Required Sample Size: **As desired power increases:** 1. Required Sample Size: 2. Detectable Difference $|\mu_0 - \mu_1|$: **As $\sigma$, the population SD, increases:** 1. Power: 2. Detectable Difference $|\mu_0 - \mu_1|$: 3. Required Sample Size: **As $\alpha$, the significance level of the test, increases:** 1. Power: 2. Detectable Difference $|\mu_0 - \mu_1|$: 3. Required Sample Size: