From debb8396f68285dde7bfcff6cc278498cf648a6e Mon Sep 17 00:00:00 2001 From: Jeremy Foote Date: Mon, 15 Apr 2019 14:31:23 -0500 Subject: [PATCH] First draft of worked solutions for week 6 --- problem_sets/week_06/ps6-worked-solution.Rmd | 220 +++++++ problem_sets/week_06/ps6-worked-solution.html | 588 ++++++++++++++++++ 2 files changed, 808 insertions(+) create mode 100644 problem_sets/week_06/ps6-worked-solution.Rmd create mode 100644 problem_sets/week_06/ps6-worked-solution.html diff --git a/problem_sets/week_06/ps6-worked-solution.Rmd b/problem_sets/week_06/ps6-worked-solution.Rmd new file mode 100644 index 0000000..f5791b0 --- /dev/null +++ b/problem_sets/week_06/ps6-worked-solution.Rmd @@ -0,0 +1,220 @@ +--- +title: "Week 6 Worked Examples" +author: "Jeremy Foote" +date: "April 11, 2019" +output: html_document +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, messages = F) +``` + +## Programming Questions + +PC0. First we import the data. + +```{r} +raw_df = read.csv("~/Desktop/DeleteMe/Teaching/owan03.csv") # Note that I saved the file as a CSV for importing to R +head(raw_df) +``` + +PC1. Let's reshape the data + + +```{r} +library(tidyverse) + +# Rename the columns +colnames(raw_df) <- c("None", "Low", "Medium","High") + +# Gather gets all of the columns from the dataframe and +# use them as keys, with the value as the value from that cell. +# This gives us a "long" dataframe +df <- gather(raw_df, dose, weeks_alive) + +# We want to treat dose as a factor, so we can group by it easier +df$dose <- as.factor(df$dose) + +# Finally, when we look at it, we see some missing data (NA); this is because not all group sizes were the same. +# We can safely remove these. +df <- df[complete.cases(df),] +``` + +PC2: Now we're goint to get statistics and create some visualizations + +```{r} + +tapply(df$weeks_alive, df$dose, summary) + +# Alternative way to do this using tidyverse + +df %>% group_by(dose) %>% summarize_all(c('min','max','mean', 'IQR')) + +``` + +When it comes to visualizations, we definitely want to use ggplot. We have lots of options for what to do. + +```{r} + +# Histograms + +h_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize + fill = dose # How to group by color + )) + geom_histogram(position = 'dodge', bins = 5) +h_plot + +# In this case, faceted histograms is probably better + +h_facet = df %>% ggplot(aes(x=weeks_alive, # What to summarize + )) + geom_histogram(bins = 5) + facet_grid(~dose) + +h_facet + +# Density plots +d_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize + fill = dose # How to group by color + )) + geom_density(alpha = .2) +d_plot + + +# Boxplots +box_plot = df %>% ggplot(aes(y=weeks_alive, + x = dose + )) + geom_boxplot() +box_plot + +# My favorite - ridgeline plots + +# install.packages('ggridges') + +library(ggridges) + +ridge_plot = df %>% ggplot(aes(x=weeks_alive, y = dose)) + + geom_density_ridges(jittered_points = T) +ridge_plot + +``` + +It's a bit tough to tell, but the overall assumptions of normality and equal variance seem reasonable. + +The global mean is + +```{r} +mean(df$weeks_alive) +``` + +PC3. T-test between None and Any, and between None and High. + +```{r} + + +t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose + df[df$dose != 'None','weeks_alive'] # Samples with any dose + ) + +# Or, using formula notation +t.test(weeks_alive ~ dose == 'None', data = df) + +# T-test between None and High + +t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose + df[df$dose == 'High','weeks_alive'] # Samples with high dose + ) + +# Formula notation is a bit tricker. I would probably create a temprorary dataframe + +tmp = df %>% filter(dose %in% c('None', 'High')) + +t.test(weeks_alive ~ dose, data = tmp) + +``` + +The t-test supports the idea that receiving a dose of RD40 reduces lifespan + +PC4. Anova + +```{r} +summary(aov(weeks_alive ~ dose, data = df)) + +``` + +This provides evidence that the group means are different. + +## Statistical Questions + +Q1. +a) It is a sample statistic, because it comes from a sample. +b) Confidence intervals for proportions are equal to + +$$p \pm z * \sqrt{ \frac{p*(1-p)}{n}}$$ + +For a 95% confidence interval, $z = 1.96$, so we can calculate it like this: + + +```{r} + +lower = .48 - 1.96 * sqrt(.48 * .52 / 1259) + +upper = .48 + 1.96 * sqrt(.48 * .52 / 1259) + +ci = c(lower, upper) + +print(ci) + +``` + +This means that we are 95% confident that the true proportion of Americans who support legalizing marijuana is between ~45% and ~51%. + +c) We have a large enough sample, which is collected randomly, to assume that the distribution is normal. +d) The statement isn't justified, since our confidence interval include 50% + +6.20 +We can use the point estimate of the poll to estimate how large a sample we would need to have a confidence interval of a given width. + +Basically, we want each half of the confidence interval to be 1%, i.e., $1.96 * \sqrt{\frac{.48 * .52}{n}} = .01$ + +We can solve for $n$: + +$$\sqrt{\frac{.48 * .52}{n}} = .01/1.96$$ +$$\frac{.48 * .52}{n} = (.01/1.96)^2$$ +$$n = (.48 * .52)/(.01/1.96)^2$$ +```{r} +(.48 * .52)/(.01/1.96)^2 +``` +So, we need a sample of approximately 9,589 + +6.38 + +The question is whether there has been a change in the proportion over time. While the tools we have learned could allow you to answer that question, they assume that responses are independent. In this case, they are obviously not independent as they come from the same students. + +6.50 + +a) We can test this with a $\chi^2$ test. + +```{r} +chisq.test(x = c(83,121,193,103), p = c(.18,.22,.37,.23)) + +# Or manually + +# Calculate expected values +500 * c(.18,.22,.37,.23) + +# Use the formula for chi-squared +chisq = (83-90)^2/90 + (121-110)^2/110 + (193-185)^2/185 + (103-115)^2/115 + + +``` + +The p-value for this is large, meaning that we don't have evidence that the sample differs from the census distribution. + +b) +i) Opinion is the response and location is the explanatory variable, since it's unlikely that people move to a region based on their opinion. +ii) One hypothesis is that opinions differ by region. The null hypothesis is that opinion is independent of region, while the alternative hypothesis is that there is a relationship. +iii) We can again use a $\chi^2$ test. + +```{r} + +x <- matrix(c(29,54,44,77,62,131,36,67), nrow = 4, # this makes a matrix with 4 rows + byrow=T) # And this says that we've entered it row by row +chisq.test(x) +``` diff --git a/problem_sets/week_06/ps6-worked-solution.html b/problem_sets/week_06/ps6-worked-solution.html new file mode 100644 index 0000000..7a80c67 --- /dev/null +++ b/problem_sets/week_06/ps6-worked-solution.html @@ -0,0 +1,588 @@ + + + + + + + + + + + + + + + +Week 6 Worked Examples + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+

Programming Questions

+

PC0. First we import the data.

+
raw_df = read.csv("~/Desktop/DeleteMe/Teaching/owan03.csv") # Note that I saved the file as a CSV for importing to R
+head(raw_df)
+
##   X1 X2 X3 X4
+## 1 70 49 30 34
+## 2 77 60 37 36
+## 3 83 63 56 48
+## 4 87 67 65 48
+## 5 92 70 76 65
+## 6 93 74 83 91
+

PC1. Let’s reshape the data

+
library(tidyverse)
+
## ── Attaching packages ────────────────────────────────── tidyverse 1.2.1 ──
+
## ✔ ggplot2 3.1.0     ✔ purrr   0.2.5
+## ✔ tibble  2.1.1     ✔ dplyr   0.7.7
+## ✔ tidyr   0.8.2     ✔ stringr 1.3.1
+## ✔ readr   1.1.1     ✔ forcats 0.3.0
+
## ── Conflicts ───────────────────────────────────── tidyverse_conflicts() ──
+## ✖ dplyr::filter() masks stats::filter()
+## ✖ dplyr::lag()    masks stats::lag()
+
# Rename the columns
+colnames(raw_df) <- c("None", "Low", "Medium","High")
+
+# Gather gets all of the columns from the dataframe and 
+# use them as keys, with the value as the value from that cell.
+# This gives us a "long" dataframe
+df <- gather(raw_df, dose, weeks_alive)
+
+# We want to treat dose as a factor, so we can group by it easier
+df$dose <- as.factor(df$dose)
+
+# Finally, when we look at it, we see some missing data (NA); this is because not all group sizes were the same.
+# We can safely remove these.
+df <- df[complete.cases(df),]
+

PC2: Now we’re goint to get statistics and create some visualizations

+
tapply(df$weeks_alive, df$dose, summary)
+
## $High
+##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
+##   34.00   45.00   56.50   65.25   92.75  102.00 
+## 
+## $Low
+##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
+##   49.00   63.00   70.00   69.89   77.00   89.00 
+## 
+## $Medium
+##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
+##   30.00   58.25   79.50   71.50   89.25   97.00 
+## 
+## $None
+##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
+##   70.00   85.00   93.00   91.36  101.00  103.00
+
# Alternative way to do this using tidyverse
+
+df %>% group_by(dose) %>% summarize_all(c('min','max','mean', 'IQR'))
+
## # A tibble: 4 x 5
+##   dose     min   max  mean   IQR
+##   <fct>  <dbl> <dbl> <dbl> <dbl>
+## 1 High      34   102  65.2  47.8
+## 2 Low       49    89  69.9  14  
+## 3 Medium    30    97  71.5  31  
+## 4 None      70   103  91.4  16
+

When it comes to visualizations, we definitely want to use ggplot. We have lots of options for what to do.

+
# Histograms
+
+h_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize
+                      fill = dose # How to group by color
+                      )) + geom_histogram(position = 'dodge', bins = 5)
+h_plot
+

+
# In this case, faceted histograms is probably better
+
+h_facet = df %>% ggplot(aes(x=weeks_alive, # What to summarize
+                      )) + geom_histogram(bins = 5) + facet_grid(~dose)
+
+h_facet
+

+
# Density plots
+d_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize
+                      fill = dose # How to group by color
+                      )) + geom_density(alpha = .2)
+d_plot
+

+
# Boxplots
+box_plot = df %>% ggplot(aes(y=weeks_alive,
+                      x = dose
+                      )) + geom_boxplot()
+box_plot
+

+
# My favorite - ridgeline plots
+
+# install.packages('ggridges')
+
+library(ggridges)
+
## 
+## Attaching package: 'ggridges'
+
## The following object is masked from 'package:ggplot2':
+## 
+##     scale_discrete_manual
+
ridge_plot = df %>% ggplot(aes(x=weeks_alive, y = dose)) + 
+  geom_density_ridges(jittered_points = T)
+ridge_plot
+
## Picking joint bandwidth of 10.5
+

+

It’s a bit tough to tell, but the overall assumptions of normality and equal variance seem reasonable.

+

The global mean is

+
mean(df$weeks_alive)
+
## [1] 75.55263
+

PC3. T-test between None and Any, and between None and High.

+
t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose
+       df[df$dose != 'None','weeks_alive'] # Samples with any dose
+       )
+
## 
+##  Welch Two Sample t-test
+## 
+## data:  df[df$dose == "None", "weeks_alive"] and df[df$dose != "None", "weeks_alive"]
+## t = 4.2065, df = 33.732, p-value = 0.0001806
+## alternative hypothesis: true difference in means is not equal to 0
+## 95 percent confidence interval:
+##  11.49879 33.00626
+## sample estimates:
+## mean of x mean of y 
+##  91.36364  69.11111
+
# Or, using formula notation
+t.test(weeks_alive ~ dose == 'None', data = df)
+
## 
+##  Welch Two Sample t-test
+## 
+## data:  weeks_alive by dose == "None"
+## t = -4.2065, df = 33.732, p-value = 0.0001806
+## alternative hypothesis: true difference in means is not equal to 0
+## 95 percent confidence interval:
+##  -33.00626 -11.49879
+## sample estimates:
+## mean in group FALSE  mean in group TRUE 
+##            69.11111            91.36364
+
# T-test between None and High
+
+t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose
+       df[df$dose == 'High','weeks_alive'] # Samples with high dose
+       )
+
## 
+##  Welch Two Sample t-test
+## 
+## data:  df[df$dose == "None", "weeks_alive"] and df[df$dose == "High", "weeks_alive"]
+## t = 2.4958, df = 8.5799, p-value = 0.0353
+## alternative hypothesis: true difference in means is not equal to 0
+## 95 percent confidence interval:
+##   2.266399 49.960874
+## sample estimates:
+## mean of x mean of y 
+##  91.36364  65.25000
+
# Formula notation is a bit tricker. I would probably create a temprorary dataframe
+
+tmp = df %>% filter(dose %in% c('None', 'High'))
+
+t.test(weeks_alive ~ dose, data = tmp)
+
## 
+##  Welch Two Sample t-test
+## 
+## data:  weeks_alive by dose
+## t = -2.4958, df = 8.5799, p-value = 0.0353
+## alternative hypothesis: true difference in means is not equal to 0
+## 95 percent confidence interval:
+##  -49.960874  -2.266399
+## sample estimates:
+## mean in group High mean in group None 
+##           65.25000           91.36364
+

The t-test supports the idea that receiving a dose of RD40 reduces lifespan

+

PC4. Anova

+
summary(aov(weeks_alive ~ dose, data = df))
+
##             Df Sum Sq Mean Sq F value Pr(>F)  
+## dose         3   4052  1350.7    3.55 0.0245 *
+## Residuals   34  12937   380.5                 
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+

This provides evidence that the group means are different.

+
+
+

Statistical Questions

+

Q1. a) It is a sample statistic, because it comes from a sample. b) Confidence intervals for proportions are equal to

+

\[p \pm z * \sqrt{ \frac{p*(1-p)}{n}}\]

+

For a 95% confidence interval, \(z = 1.96\), so we can calculate it like this:

+
lower = .48 - 1.96 * sqrt(.48 * .52 / 1259)
+
+upper = .48 + 1.96 * sqrt(.48 * .52 / 1259)
+
+ci = c(lower, upper)
+
+print(ci)
+
## [1] 0.4524028 0.5075972
+

This means that we are 95% confident that the true proportion of Americans who support legalizing marijuana is between ~45% and ~51%.

+
    +
  1. We have a large enough sample, which is collected randomly, to assume that the distribution is normal.
    2. +
  2. The statement isn’t justified, since our confidence interval include 50%
    3. +
+

6.20 We can use the point estimate of the poll to estimate how large a sample we would need to have a confidence interval of a given width.

+

Basically, we want each half of the confidence interval to be 1%, i.e., \(1.96 * \sqrt{\frac{.48 * .52}{n}} = .01\)

+

We can solve for \(n\):

+

\[\sqrt{\frac{.48 * .52}{n}} = .01/1.96\] \[\frac{.48 * .52}{n} = (.01/1.96)^2\] \[n = (.48 * .52)/(.01/1.96)^2\]

+
(.48 * .52)/(.01/1.96)^2
+
## [1] 9588.634
+

So, we need a sample of approximately 9,589

+

6.38

+

The question is whether there has been a change in the proportion over time. While the tools we have learned could allow you to answer that question, they assume that responses are independent. In this case, they are obviously not independent as they come from the same students.

+

6.50

+
    +
  1. We can test this with a \(\chi^2\) test.
    2. +
+
chisq.test(x = c(83,121,193,103), p = c(.18,.22,.37,.23))
+
## 
+##  Chi-squared test for given probabilities
+## 
+## data:  c(83, 121, 193, 103)
+## X-squared = 3.2426, df = 3, p-value = 0.3557
+
# Or manually
+
+# Calculate expected values
+500 * c(.18,.22,.37,.23)
+
## [1]  90 110 185 115
+
# Use the formula for chi-squared
+chisq = (83-90)^2/90 + (121-110)^2/110 + (193-185)^2/185 + (103-115)^2/115
+

The p-value for this is large, meaning that we don’t have evidence that the sample differs from the census distribution.

+
    +
  1. +
  2. Opinion is the response and location is the explanatory variable, since it’s unlikely that people move to a region based on their opinion.
    3. +
+
    +
  1. One hypothesis is that opinions differ by region. The null hypothesis is that opinion is independent of region, while the alternative hypothesis is that there is a relationship.
    2. +
  2. We can again use a \(\chi^2\) test.
    3. +
+
x <- matrix(c(29,54,44,77,62,131,36,67), nrow = 4, # this makes a matrix with 4 rows
+            byrow=T) # And this says that we've entered it row by row
+chisq.test(x)
+
## 
+##  Pearson's Chi-squared test
+## 
+## data:  x
+## X-squared = 0.66724, df = 3, p-value = 0.8809
+
+ + + + +
+ + + + + + + + -- 2.39.5