Problem 1

Hypotheses

\(H_0\): \(p_1\) = \(p_2\)
\(H_a\): \(p_1\) > \(p_2\)

Where:
\(p_1\) = proportion of female students taking the AP Biology exam
\(p_2\) = proportion of female students taking the AP Calculus AB exam

prop.test(c(84200, 102598), c(144790, 211693), alternative = "greater")
## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  c(84200, 102598) out of c(144790, 211693)
## X-squared = 3234.9, df = 1, p-value < 2.2e-16
## alternative hypothesis: greater
## 95 percent confidence interval:
##  0.09408942 1.00000000
## sample estimates:
##    prop 1    prop 2 
## 0.5815319 0.4846547


The p-value is 2.2e-16, which is statistically significant at α = 0.05. Since the p value is less than α, we reject the null. There is strong evidence at the 5% significance level that the proportion of female students taking the AP Biology exam is higher than the proportion of female students taking the AP Calculus AB exam.


Problem 2

mothers <- c(0, 32, 20, 23, 14, 19, 60, 59, 64, 64, 72, 50, 44, 14, 10, 58, 19, 41, 17, 5, 36, 73, 19, 46, 9, 43, 73, 27, 25, 18)

conventional <- c(63, 0, 2, 46, 33, 33, 29, 23, 11, 12, 48, 15, 33, 14, 51, 37, 24, 70, 63, 0, 73, 39, 54, 52, 39, 34, 30, 55, 58, 18)

Hypotheses

\(H_0\): \(\mu_1\) = \(\mu_2\)
\(H_a\): \(\mu_1\) < \(\mu_2\)

Where:
\(\mu_1\) = mean crying time for infants held by their mothers
\(\mu_2\) = mean crying time for infants given the shot using conventional methods

t.test(mothers, conventional, conf.level = 0.95, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  mothers and conventional
## t = -0.029953, df = 57.707, p-value = 0.4881
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##      -Inf 9.135003
## sample estimates:
## mean of x mean of y 
##  35.13333  35.30000


The p-value is 0.4881, which is not statistically significant at α = 0.05. Since the p value is greater than α, we do not reject the null. There is not evidence at the 5% significance level that the mean crying time for infants held by their mothers is less than the mean crying time for infants using conventional methods.