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

The variables: Type of exam and number of females

Test: Two Proportions Z-Test

\(p_1\) = Proportion of females taking the Biology Exam

\(p_2\) = Proportion of females taking the Calculus AB Exam

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

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

Significance Level: 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 < 2.2e-16, which is very significant as it is less than α = 0.05, therefore we can reject the null.

  1. 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

The variables: Method Type and Crying Time

Test: Two Sample T Test

\(\mu_1\) = Average crying time of infants given shots using new methods

\(\mu_2\) = Average crying time of infants given shots using conventional methods

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

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

Significance Level: α = 0.05

conventional_method <- 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_method <- 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_method, conventional_method, conf.level = 0.95, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  new_method and conventional_method
## 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 greater than α = 0.05, so we fail to reject the null.