Include your answers in this document in the sections below the rubric where I have point values indicated (1 p).
Rubric
Answer the questions with the data examples.
Paired Comparisons of Two Sleep Remedies
The following data give the amount of sleep gained in hours from two sleep remedies, A and B, applied to 10 individuals who have trouble sleeping an adequate amount. Negative values imply sleep loss.
Let \(\mu_A\) = population mean sleep gain (among troubled sleepers) on remedy A, and \(\mu_B\) = population mean sleep gain (among troubled sleepers) on remedy B. Consider testing \(H_0: \mu_B - \mu_A = 0\) or equivalently \(\mu_D = 0\), where \(\mu_D = \mu_B - \mu_A\).
# Datadat_sleep <-read.table(text=" A B 0.7 1.9-1.6 0.8-0.2 1.1-1.2 0.1 0.1 -0.1 3.4 4.4 3.7 5.5 0.8 1.6 0.0 4.6 2.0 3.0", header =TRUE)# calculate paired difference bewteen two remedies, D = B - Adat_sleep <- dat_sleep %>%mutate(D = B - A )summary(dat_sleep)
A B D
Min. :-1.60 Min. :-0.100 Min. :-0.200
1st Qu.:-0.15 1st Qu.: 0.875 1st Qu.: 1.000
Median : 0.40 Median : 1.750 Median : 1.250
Mean : 0.77 Mean : 2.290 Mean : 1.520
3rd Qu.: 1.70 3rd Qu.: 4.050 3rd Qu.: 1.675
Max. : 3.70 Max. : 5.500 Max. : 4.600
A plot of both sleep remedies, with a 1-to-1 reference line to indicate which remedy is better for which people.
# determine range of each axis to use most extreme of each for square plot belowaxis_lim <-range(c(dat_sleep$A, dat_sleep$B))library(ggplot2)# scatterplot of A and B sleep times, with 1:1 linep <-ggplot(dat_sleep, aes(x = A, y = B))# draw a 1:1 line, dots above line indicate "B > A"p <- p +geom_abline(intercept =0, slope =1, alpha =0.4)p <- p +geom_vline(xintercept =0, alpha =0.2)p <- p +geom_hline(yintercept =0, alpha =0.2)p <- p +geom_point()p <- p +geom_rug()# make the axes square so it's a fair visual comparisonp <- p +coord_equal()p <- p +scale_x_continuous(limits = axis_lim)p <- p +scale_y_continuous(limits = axis_lim)p <- p +labs(title ="Sleep hours gained on two sleep remedies: A vs B")print(p)
Let \(\alpha = 0.05\), the significance level of the test and the Type-I error probability if the null hypothesis is true.
# one-sample t-test of differences (paired t-test)# t_summary <- t.test(dat_sleep$D) # this gives the same resultt_summary <-t.test( dat_sleep$B , dat_sleep$A , paired =TRUE )t_summary
Paired t-test
data: dat_sleep$B and dat_sleep$A
t = 3.7796, df = 9, p-value = 0.004352
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
0.610249 2.429751
sample estimates:
mean difference
1.52
e_plot_ttest_pval(t_summary)
(1 p) \(t_{s} = 99\).
(1 p) \(p = 99\), this is the observed significance of the test.
(2 p) Because \(p = 99\), …
Test assumptions
For one- and two-sample \(t\)-tests, our assumptions are that we have a representative (random) sample from the population (not something that can be tested very well), the the sampling distributions of our test statistic \(t\) is normal.
In the erikmisc package are functions for one-sample and two-sample boostrap appoximations of the sampling distributions of the mean to compare with a normal distribution for the normality assumption.
Paired Comparisons of Two Sleep Remedies
Check assumptions of the test.
Description of plots: The top plot is of the data, a histogram with smoothed density curve. The bottom plot is a comparison of the boostrap appoximation of the sampling distributions of the mean. If the histogram and smooth density curve are very close (basically one on top of the other) the red normal distribution, then we can say the sampling distribution is consistent with normality. If there is systematic bias (as below) where one side is over and the other side is under the red curve, then that’s evidence against normality, in which case we would pursue a nonparametric test (next week).
e_plot_bs_one_samp_dist(dat_sleep$D)
(2 p) Interpret the plots above.
African countries in the UN example
Check the assumptions of the two-sample \(t\)-test for this example (explained in a previous worksheet).
# UN Africa surveydat_UN_Africa <-read_csv("ADA1_CL_18_UN_MembershipSurvey.csv") %>%na.omit() %>%mutate(PrimingNumber =factor(PrimingNumber) , HighLow =factor(HighLow) ) %>%filter( Class =="F19" )
Rows: 198 Columns: 4
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): HighLow, Class
dbl (2): PrimingNumber, UN_Percentage
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
## If we create a summary data.frame with a similar structure as our data, then we## can annotate our plot with those summaries.mean_UN_Africa <- dat_UN_Africa %>%group_by( PrimingNumber ) %>%summarize(UN_Percentage =mean(UN_Percentage) , .groups ="drop_last" ) %>%ungroup()# histogram using ggplotp1 <-ggplot(dat_UN_Africa, aes(x = UN_Percentage))p1 <- p1 +geom_histogram(binwidth =4)p1 <- p1 +geom_rug()p1 <- p1 +geom_vline(data = mean_UN_Africa, aes(xintercept = UN_Percentage), colour ="red")p1 <- p1 +facet_grid(PrimingNumber ~ .)print(p1)
Questions to answer
A priori, before we observed the data, we hypothesized that those who were primed with a larger number (65) would provide a higher percentage (UN_Percentage) than those with the lower number (10). Therefore, this is a one-sided test.
# two-sample t-testt_summary_UN <-t.test( UN_Percentage ~ PrimingNumber , data = dat_UN_Africa , alternative ="less" )t_summary_UN
Welch Two Sample t-test
data: UN_Percentage by PrimingNumber
t = -2.9877, df = 47.438, p-value = 0.00222
alternative hypothesis: true difference in means between group 10 and group 65 is less than 0
95 percent confidence interval:
-Inf -4.463638
sample estimates:
mean in group 10 mean in group 65
17.97222 28.15152
