Lab 10. Confidently dealing with proportions.¶

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Prompt: In the group of subjects that had the highest markers of stress (based on cortisol), there were 14 births to males out of a total of 38.

Constructing four confidence intervals.¶

Large sample confidence interval¶

p_hat    <- 14/38
se       <- sqrt(p_hat*(1-p_hat)/38)

# * REPORT FINAL CI
100*c(estimate=p_hat, lower=p_hat - 1.96*se, upper=p_hat + 1.96*se)

Wilson score method¶

prop.test(x=14, n=38, p=0.5)

	1-sample proportions test with continuity correction

data:  14 out of 38, null probability 0.5
X-squared = 2.1316, df = 1, p-value = 0.1443
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.2229295 0.5400424
sample estimates:
        p 
0.3684211

Plus 4 method¶

p_tilde  <- (14 + 2)/(38 + 4)
se       <- sqrt(p_tilde*(1 - p_tilde)/42)

100*c(estimate=p_hat, lower=p_tilde - 1.96*se, upper=p_tilde + 1.96*se)

Clopper-Pearson method (Exact)¶

binom.test(x=14, n=38, p=0.5)

	Exact binomial test

data:  14 and 38
number of successes = 14, number of trials = 38, p-value = 0.1433
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
 0.2181250 0.5400572
sample estimates:
probability of success 
             0.3684211

Comparison Table¶

Method	95% Confidence Interval
Large sample	21.5% to 52.18%
Wilson Score*	22.29% to 54%
Plus four	23.41% to 52.78%
Exact	21.81% to 54%

Visualizing¶

library(ggplot2)
library(tibble)

sex_CIs <- tibble(method   = c("Large sample", "Exact", "Wilson", "Plus 4"),
                  lower_CI = c(21.5          , 21.81   ,  22.29   ,  23.41),
                  upper_CI = c(52.18          , 54.0   ,  54.0   ,  52.78),
                  estimate = c(36.84          , 36.84   ,  36.84   ,  36.84)
                  )

# Step 2:
ggplot(data = sex_CIs, aes(x = method, y = estimate)) + 
  geom_point() +
  geom_hline(aes(yintercept = 50), lwd=2, col="cornflowerblue", alpha=0.75) +
  geom_segment(aes(x = method, xend = method, y = lower_CI, yend = upper_CI)) +
  labs(y = "Estimate with 95% CI")

What are our conclusions?¶

50%, the null value, is within our confidence interval. There are no statistically signficant rejections for any of the confidence intervals. However, there is scientific significance.