{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Stat 140-03/05 Fall 2018\n", "# Homework 7R (Ch 18 & 19)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\n", "Attaching package: ‘dplyr’\n", "\n", "The following objects are masked from ‘package:stats’:\n", "\n", " filter, lag\n", "\n", "The following objects are masked from ‘package:base’:\n", "\n", " intersect, setdiff, setequal, union\n", "\n", "Loading required package: lattice\n", "Loading required package: ggformula\n", "Loading required package: ggstance\n", "\n", "Attaching package: ‘ggstance’\n", "\n", "The following objects are masked from ‘package:ggplot2’:\n", "\n", " geom_errorbarh, GeomErrorbarh\n", "\n", "\n", "New to ggformula? Try the tutorials: \n", "\tlearnr::run_tutorial(\"introduction\", package = \"ggformula\")\n", "\tlearnr::run_tutorial(\"refining\", package = \"ggformula\")\n", "Loading required package: mosaicData\n", "Loading required package: Matrix\n", "\n", "The 'mosaic' package masks several functions from core packages in order to add \n", "additional features. The original behavior of these functions should not be affected by this.\n", "\n", "Note: If you use the Matrix package, be sure to load it BEFORE loading mosaic.\n", "\n", "Attaching package: ‘mosaic’\n", "\n", "The following object is masked from ‘package:Matrix’:\n", "\n", " mean\n", "\n", "The following objects are masked from ‘package:dplyr’:\n", "\n", " count, do, tally\n", "\n", "The following object is masked from ‘package:ggplot2’:\n", "\n", " stat\n", "\n", "The following objects are masked from ‘package:stats’:\n", "\n", " binom.test, cor, cor.test, cov, fivenum, IQR, median, prop.test,\n", " quantile, sd, t.test, var\n", "\n", "The following objects are masked from ‘package:base’:\n", "\n", " max, mean, min, prod, range, sample, sum\n", "\n" ] } ], "source": [ "library(ggplot2)\n", "library(dplyr)\n", "library(readr)\n", "library(mosaic)\n", "library(repr)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 1: SDM4 19.22, modified\n", "A congenital disorder is a health condition present at birth. Congenital disorders may result in health problems such as physical disability, intellectual disability, or developmental disability. In the 1980s, it was generally believed that congenital disorders affected about 5% of children in the US. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of congenital disorders." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1a) What is the population the researchers are interested in?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Children in the United States." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1b) What is the population parameter being studied? What symbol would we use for this parameter?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The population parameter is percentage of children affected by congenital disorders." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1c) What is the sample statistic? Describe it in a sentence and state the symbol we will use to represent it." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The sample statistic is the 5% of children affected by congenital disorders in the 1980s." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1d) Write down the appropriate hypotheses for a hypothesis test to address the researchers' question." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The null hypothesis should be p = 0.05, or 5% of children are affected by congenital disorders. The alternative is p =/= 0.05." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1e) A recent study examined 384 randomly selected children from across the US and found that 46 of them showed signs of a disorder. Check the conditions for conducting inference (that is, performing hypothesis tests and calculating confidence intervals) with these data." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We know the sample is random, we assume independence, and this is less than 10% of the population with at least 10 successes and 10 failures.\n", "46 out of 384 is the same as 11.979%." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1f) Run the R code necessary to conduct the hypothesis test." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With a sample of 384, we can calculate a hypothesis test." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0111219565125326" ], "text/latex": [ "0.0111219565125326" ], "text/markdown": [ "0.0111219565125326" ], "text/plain": [ "[1] 0.01112196" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sqrt((0.05 * 0.95)/384)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "6.27512493759739" ], "text/latex": [ "6.27512493759739" ], "text/markdown": [ "6.27512493759739" ], "text/plain": [ "[1] 6.275125" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "(0.119791666666667 - 0.05)/0.0111219565125326" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.999999999825323" ], "text/latex": [ "0.999999999825323" ], "text/markdown": [ "0.999999999825323" ], "text/plain": [ "[1] 1" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pnorm(6.27512493759739)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "1.74677050601701e-10" ], "text/latex": [ "1.74677050601701e-10" ], "text/markdown": [ "1.74677050601701e-10" ], "text/plain": [ "[1] 1.746771e-10" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "1 - 0.999999999825323" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1g) Draw a conclusion for the test in the context of this problem using a significance level threshold of $\\alpha = 0.01$. State the P-value for the hypothesis test and say what it means." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The p-value is much smaller than alpha (0.01) and so we reject the null hypothesis. Thus we reject p = 0.05." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1h) Run the R code necessary to find a 99% confidence interval for the population parameter." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "2.5758293035489" ], "text/latex": [ "2.5758293035489" ], "text/markdown": [ "2.5758293035489" ], "text/plain": [ "[1] 2.575829" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "qnorm(.995)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0165706837661896" ], "text/latex": [ "0.0165706837661896" ], "text/markdown": [ "0.0165706837661896" ], "text/plain": [ "[1] 0.01657068" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "SE = sqrt((0.119791666666667 * 0.880208333) / 384)\n", "SE" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0771084138418738" ], "text/latex": [ "0.0771084138418738" ], "text/markdown": [ "0.0771084138418738" ], "text/plain": [ "[1] 0.07710841" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "0.16247491949146" ], "text/latex": [ "0.16247491949146" ], "text/markdown": [ "0.16247491949146" ], "text/plain": [ "[1] 0.1624749" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "0.119791666666667 - (2.5758293035489 * 0.0165706837661896)\n", "0.119791666666667 + (2.5758293035489 * 0.0165706837661896)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0111219565125326" ], "text/latex": [ "0.0111219565125326" ], "text/markdown": [ "0.0111219565125326" ], "text/plain": [ "[1] 0.01112196" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "SD = sqrt((0.05 * 0.95)/384)\n", "SD" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.021351738502222" ], "text/latex": [ "0.021351738502222" ], "text/markdown": [ "0.021351738502222" ], "text/plain": [ "[1] 0.02135174" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "0.078648261497778" ], "text/latex": [ "0.078648261497778" ], "text/markdown": [ "0.078648261497778" ], "text/plain": [ "[1] 0.07864826" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "0.05 - (2.5758293035489 * 0.0111219565125326)\n", "0.05 + (2.5758293035489 * 0.0111219565125326)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1i) State the confidence interval for the population parameter and its precise interpretation in the context of this problem." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 99% confidence interval is between 7.711% and 16.25% for the sample." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the p = 0.05, the 99% confidence interval would be between 2.135% and 7.865%." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In both of these, the 99% confidence interval means 99% of samples will capture the true value." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1j) Do your results above prove that environmental chemicals cause congenital disorders?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "No, we can't say anything about the causation of the higher statistic of children with congenital disorders." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 2: Atheism in the US\n", "\n", "In August of 2012, news outlets ranging from the [Washington Post](http://www.washingtonpost.com/national/on-faith/poll-shows-atheism-on-the-rise-in-the-us/2012/08/13/90020fd6-e57d-11e1-9739-eef99c5fb285_story.html) to the [Huffington Post](http://www.huffingtonpost.com/2012/08/14/atheism-rise-religiosity-decline-in-america_n_1777031.html)\n", "ran a story about the rise of atheism in America. The source for the story was a poll that asked people, \"Irrespective of whether you attend a place of worship or not, would you say you are a religious person, not a religious person or a convinced atheist?\"\n", "\n", "### 2a) In the first paragraph of the press release below, several key findings are reported. Do these percentages appear to be sample statistics or population parameters?\n", "\n", "> We have no opinion polling data prior to the 20th century, but according to the latest global poll released by WIN-Gallup International, a world-wide network of leading opinion pollsters, atheists are a small minority in the early years of the 21st century. Only 13% say they are atheists. They are mostly concentrated in East Asia where 47% of the Chinese and 31% of the Japanese say they are atheists; and Western Europe, 14% of whose population says the same. Elsewhere sizeable majorities claim to be religious; the global average is 59%. Another sizeable group claims to be \"not-religious\" but stops short of defining itself as \"atheist\". Their global average is 23%." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The percentages in these findings are the sample statistics. The population parameter is religion/non-religion or atheism in the population." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2b) The title of the report is \"Global Index of Religiosity and Atheism\". To generalize the report's findings to the global human population, what must we assume about the sampling method? Does that seem like a reasonable assumption?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We must assume it is random and independent, and less than 10 percent of the population. I think this does seem like a reasonable assumption." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2c) The data containing individual responses to the survey are in the file \"atheism.csv\" in this directory. Read these data in to a data frame and look at the first few rows of the data." ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Parsed with column specification:\n", "cols(\n", " nationality = col_character(),\n", " response = col_character(),\n", " year = col_integer()\n", ")\n" ] }, { "data": { "text/html": [ "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "

nationality	response	year
Afghanistan	non-atheist	2012
Afghanistan	non-atheist	2012
Afghanistan	non-atheist	2012
Afghanistan	non-atheist	2012
Afghanistan	non-atheist	2012
Afghanistan	non-atheist	2012

\n" ], "text/latex": [ "\\begin{tabular}{r|lll}\n", " nationality & response & year\\\\\n", "\\hline\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\t Afghanistan & non-atheist & 2012 \\\\\n", "\\end{tabular}\n" ], "text/markdown": [ "\n", "nationality | response | year | \n", "|---|---|---|---|---|---|\n", "| Afghanistan | non-atheist | 2012 | \n", "| Afghanistan | non-atheist | 2012 | \n", "| Afghanistan | non-atheist | 2012 | \n", "| Afghanistan | non-atheist | 2012 | \n", "| Afghanistan | non-atheist | 2012 | \n", "| Afghanistan | non-atheist | 2012 | \n", "\n", "\n" ], "text/plain": [ " nationality response year\n", "1 Afghanistan non-atheist 2012\n", "2 Afghanistan non-atheist 2012\n", "3 Afghanistan non-atheist 2012\n", "4 Afghanistan non-atheist 2012\n", "5 Afghanistan non-atheist 2012\n", "6 Afghanistan non-atheist 2012" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "atheism <- read_csv(\"atheism.csv\")\n", "head(atheism)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2d) What are the observational units and variables in this data set?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The observational units are the people being surveyed. The variables are the nationality, response and year." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2e) The news articles we referred to at the beginning of this problem were talking about the respondents who were from the United States and answered the survey in 2012. Create a new data frame that includes only those respondents." ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\n", "

nationality	response	year
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
?	?	?
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012
United States	non-atheist	2012

\n" ], "text/latex": [ "\\begin{tabular}{r|lll}\n", " nationality & response & year\\\\\n", "\\hline\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t ? & ? & ?\\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\t United States & non-atheist & 2012 \\\\\n", "\\end{tabular}\n" ], "text/markdown": [ "\n", "nationality | response | year | \n", "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| ? | ? | ? | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "| United States | non-atheist | 2012 | \n", "\n", "\n" ], "text/plain": [ " nationality response year\n", "1 United States non-atheist 2012\n", "2 United States non-atheist 2012\n", "3 United States non-atheist 2012\n", "4 United States non-atheist 2012\n", "5 United States non-atheist 2012\n", "6 United States non-atheist 2012\n", "7 United States non-atheist 2012\n", "8 United States non-atheist 2012\n", "9 United States non-atheist 2012\n", "10 United States non-atheist 2012\n", "11 United States non-atheist 2012\n", "12 United States non-atheist 2012\n", "13 United States atheist 2012\n", "14 United States non-atheist 2012\n", "15 United States non-atheist 2012\n", "16 United States non-atheist 2012\n", "17 United States non-atheist 2012\n", "18 United States non-atheist 2012\n", "19 United States non-atheist 2012\n", "20 United States atheist 2012\n", "21 United States non-atheist 2012\n", "22 United States non-atheist 2012\n", "23 United States non-atheist 2012\n", "24 United States non-atheist 2012\n", "25 United States atheist 2012\n", "26 United States non-atheist 2012\n", "27 United States non-atheist 2012\n", "28 United States non-atheist 2012\n", "29 United States non-atheist 2012\n", "30 United States non-atheist 2012\n", "? ? ? ? \n", "973 United States non-atheist 2012\n", "974 United States non-atheist 2012\n", "975 United States non-atheist 2012\n", "976 United States non-atheist 2012\n", "977 United States non-atheist 2012\n", "978 United States non-atheist 2012\n", "979 United States non-atheist 2012\n", "980 United States non-atheist 2012\n", "981 United States non-atheist 2012\n", "982 United States non-atheist 2012\n", "983 United States non-atheist 2012\n", "984 United States non-atheist 2012\n", "985 United States non-atheist 2012\n", "986 United States non-atheist 2012\n", "987 United States non-atheist 2012\n", "988 United States non-atheist 2012\n", "989 United States non-atheist 2012\n", "990 United States non-atheist 2012\n", "991 United States non-atheist 2012\n", "992 United States non-atheist 2012\n", "993 United States non-atheist 2012\n", "994 United States non-atheist 2012\n", "995 United States non-atheist 2012\n", "996 United States non-atheist 2012\n", "997 United States non-atheist 2012\n", "998 United States non-atheist 2012\n", "999 United States non-atheist 2012\n", "1000 United States non-atheist 2012\n", "1001 United States non-atheist 2012\n", "1002 United States non-atheist 2012" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "1002" ], "text/latex": [ "1002" ], "text/markdown": [ "1002" ], "text/plain": [ "[1] 1002" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "americans <- atheism %>% filter(nationality == \"United States\") %>% filter(year == 2012)\n", "americans\n", "nrow(americans)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ " nationality response year \n", " Length:1002 non-atheist:952 Min. :2012 \n", " Class :character atheist : 50 1st Qu.:2012 \n", " Mode :character Median :2012 \n", " Mean :2012 \n", " 3rd Qu.:2012 \n", " Max. :2012 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "americans <- mutate(americans,\n", " response = factor(response, levels = c(\"non-atheist\", \"atheist\")))\n", "summary(americans)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0499001996007984" ], "text/latex": [ "0.0499001996007984" ], "text/markdown": [ "0.0499001996007984" ], "text/plain": [ "[1] 0.0499002" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p = 50/1002\n", "p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2f) According to Wikipedia, in 1991 2% of U.S. citizens identified as atheists. Conduct a hypothesis test to evaluate the newspaper article claims that there has been a rise in Atheism in America since 1991. Check all conditions and state your hypotheses." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Conditions: independent, random, less than 10% of the population.\n", "Our null hypothesis is p = 0.02. Our alternative hypothesis is p > 0.02." ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.00442276816524594" ], "text/latex": [ "0.00442276816524594" ], "text/markdown": [ "0.00442276816524594" ], "text/plain": [ "[1] 0.004422768" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "6.76051705259024" ], "text/latex": [ "6.76051705259024" ], "text/markdown": [ "6.76051705259024" ], "text/plain": [ "[1] 6.760517" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "SD = sqrt((0.02 * 0.98)/1002)\n", "SD\n", "(0.0499001996007984 - 0.02)/0.00442276816524594" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "6.87505607999128e-12" ], "text/latex": [ "6.87505607999128e-12" ], "text/markdown": [ "6.87505607999128e-12" ], "text/plain": [ "[1] 6.875056e-12" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "1 - pnorm(6.76051705259024)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have a very small p-value, much less than a typical alpha of 0.05 or 0.01. So we should reject out null hypothesis." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2g) Find and interpret a 95% confidence interval for the proportion of Americans who identified as atheists in 2012." ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.00687862912094493" ], "text/latex": [ "0.00687862912094493" ], "text/markdown": [ "0.00687862912094493" ], "text/plain": [ "[1] 0.006878629" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sqrt((0.0499001996007984 * 0.9500998) / 1002)" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/html": [ "1.95996398454005" ], "text/latex": [ "1.95996398454005" ], "text/markdown": [ "1.95996398454005" ], "text/plain": [ "[1] 1.959964" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "qnorm(.975)" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.036418334260738" ], "text/latex": [ "0.036418334260738" ], "text/markdown": [ "0.036418334260738" ], "text/plain": [ "[1] 0.03641833" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "0.0633820649408589" ], "text/latex": [ "0.0633820649408589" ], "text/markdown": [ "0.0633820649408589" ], "text/plain": [ "[1] 0.06338206" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "0.0499001996007984 - (1.95996398454005*0.00687862912094493)\n", "0.0499001996007984 + (1.95996398454005*0.00687862912094493)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 95 percent confidence interval is between 3.64% and 6.338%." ] } ], "metadata": { "kernelspec": { "display_name": "R", "language": "R", "name": "ir" }, "language_info": { "codemirror_mode": "r", "file_extension": ".r", "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", "version": "3.4.4" } }, "nbformat": 4, "nbformat_minor": 2 }

Homework 7R.ipynb