Problem #1:

Many highschool 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.

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

Where: \(p_1\)= proportion of female students taking the biology exam \(p_2\) = proportion of female students taking the 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
  1. The significance level is α = 0.05. The p-value is less than 2.2e-16 which is also less than the significance level of α = 0.05. Therefore, we reject the null hypothesis since there is an overwhelming statistical evidence that the proportion of female students who took the biology exam is higher than the proportion of female who took the calculus AB exam.

Problem #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.

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

Where: \(mu_1\) = mean of infants that cried that was given the shot where the mother held the infant prior and during the shot \(mu_2\) = mean of infants that cried after given the shot using conventional methods

conventional_methods <- 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_methods, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  new_methods and conventional_methods
## 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
  1. The significance level is α = 0.05. The p-value is 0.4881 which is greater than the significance level. Therefore, there is not enough evidence to reject the null and that the proportion of infants that cried after getting the shots through conventional versus new methods are not that different from each other.