Downloaded from here via All States (CSV)
Codebook is available here
This dataset is messy. I've tidied it and focused on only Total Population and Black or African American alone.
# Remember to load the needed packages library(dplyr) library(mosaic) library(ggplot2) library(readr)
Suppose we were interested in testing the mean percentage of black residents in two different states. Let's choose Oregon and Washington.
Washington has 39 counties and Oregon has 36.
Assume we only have access to 20 randomly selected counties from each state.
wash_or <- read_csv("/shared/isma5720@pacificu.edu/wash-or_black.csv")
Do we have evidence that the mean percentage of black residents in Washington and Oregon is statistically different?
What is the null hypothesis?
What does this mean?
We are assuming that the mean percentage of black residents in Washington and Oregon is THE SAME.
We will use hypothesis testing to look to see if we have evidence to go against this assumption.
What is the observational unit?
DATA:
TEST STATISTIC:
(state_means <- wash_or %>% group_by(state) %>% summarize(mean_perc = mean(black_perc)))
## # A tibble: 2 × 2 ## state mean_perc ## <chr> <dbl> ## 1 Oregon 0.8817059 ## 2 Washington 1.7922830
(obs_diff <- diff(state_means$mean_perc))
## [1] 0.9105771
\[\verb+obs_diff+ = \bar{x}_{wa} - \bar{x}_{or} = 0.9105771\]
We assume that the two POPULATION mean percentages are the same:
\(H_0: \mu_{wa} = \mu_{or}\) or \(H_0: \mu_{wa} - \mu_{or} = 0\)
We want to use the data we have collected to simulate a process following the model for \(H_0\).
This is similar to bootstrapping except in this case we have two groups instead of just one.
BIG STEP: Assuming that \(H_0: \mu_{wa} = \mu_{or}\), any one of the percentages we saw in our sample data could have occurred (equally) in either Washington or in Oregon.
We can use the shuffle function in the mosaic package to shuffle the percentages between Oregon and Washington.
set.seed(2016)
(shuffled_state_means <- wash_or %>%
mutate(state = shuffle(state)) %>%
group_by(state) %>%
summarize(mean_shuf_perc = mean(black_perc)))
## # A tibble: 2 × 2 ## state mean_shuf_perc ## <chr> <dbl> ## 1 Oregon 1.174869 ## 2 Washington 1.499120
(obs_shuf_diff <- diff(shuffled_state_means$mean_shuf_perc))
## [1] 0.3242503
many_shuffles <- do(10000) *
(shuffled_state_means <- wash_or %>%
mutate(state = shuffle(state)) %>%
group_by(state) %>%
summarize(mean_shuf_perc = mean(black_perc)))
rand_distn <- many_shuffles %>%
group_by(.index) %>%
summarize(diffmean = diff(mean_shuf_perc))
rand_distn %>% ggplot(aes(x = diffmean)) + geom_histogram(color = "white", bins = 20)
Where will the center of this distribution always be?
We want to see where our observed effect (the original different in sample means) falls on our NULL DISTRIBUTION.
rand_distn %>% ggplot(aes(x = diffmean,
fill = (abs(diffmean) >= obs_diff))) +
geom_histogram(color = "white", bins = 20)
rand_distn %>% ggplot(aes(x = diffmean)) + geom_histogram(color = "white", bins = 20) + geom_vline(xintercept = obs_diff, color = "red") + geom_vline(xintercept = -obs_diff, color = "red")
rand_distn %>% filter(abs(diffmean) >= obs_diff) %>% nrow() / nrow(rand_distn)
## [1] 0.0439
So there is around a 4% chance we could have seen results such as this if the true population mean percentages were the same. What does that mean?
The "beyond a reasonable doubt" in statistics corresponds to setting the \(\alpha\) value before you conduct your test. Most often this is set to 5%.
So with our test, since 0.0439 < 0.05, we reject \(H_0\) in favor of \(H_A\). We have evidence to support the claim that the mean percentages of black residents per county are different for Washington and Oregon.
cc_full <- read_csv("/shared/isma5720@pacificu.edu/cc_full.csv")
Practice: Determine what the actual difference is in mean percentage of black residents for ALL Washington and Oregon counties. Is the difference practically large?