1- Many high school students take the AP tests in different subject areas. In 2017, of the 144,790 students who took the biology exam 84,200 of them were female. In that same year, of the 211,693 students who took the calculus AB exam 102,598 of them were female. Is there enough evidence to show that the proportion of female students taking the biology exam is higher than the proportion of female students taking the calculus AB exam? Test at the 5% level.

\(H_o\) = \(P_B\) = \(P_C\), where \(P_B\) is the proportion of students who took the biology exam, and \(P_C\) is the proportion of students who took the calculus AB exam.

\(H_a\) = \(P_B\) > \(P_C\)

α = 0.05

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 very small (less than 0.05), therefore, we reject the null. We have very strong evidence that the proportion of female students taking the biology exam is higher than the proportion of female students taking the calculus AB exam.

2- A vitamin K shot is given to infants soon after birth. The study is to see if how they handle the infants could reduce the pain the infants feel. One of the measurements taken was how long, in seconds, the infant cried after being given the shot. A random sample was taken from the group that was given the shot using conventional methods, and a random sample was taken from the group that was given the shot where the mother held the infant prior to and during the shot. Is there enough evidence to show that infants cried less on average when they are held by their mothers than if held using conventional methods? Test at the 5% level.

\(H_o\) = \(μ_m\) = \(μ_c\), where \(μ_c\) is the average time an infant cried during conventional methods and \(μ_m\) is the average time an infant cried while being held by their mother.

\(H_a\) = \(μ_m\) < \(μ_c\)

α = 0.05

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)

new_methods <- 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)
t.test(new_methods, conventional, alternative = "greater")
## 
##  Welch Two Sample t-test
## 
## data:  new_methods and conventional
## t = -0.029953, df = 57.707, p-value = 0.5119
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  -9.468337       Inf
## sample estimates:
## mean of x mean of y 
##  35.13333  35.30000

The p-value is greater than 0.05 (0.5119). We fail to reject the null. There is no evidence that suggests infants cried less on average when they were held by their mothers than conventional methods.