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

Hypotheses

\(H_0\): p1 <= p2

\(H_a\): p1 > p2

\(p_1\) = Is the proportion of female students that took the Biology Exam.

\(p_2\) = Is the proportion of female students that took Calculus AB Exam.

Significance Level

α = 0.05

p-value

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

p-value = 2.2e-16

Decision Statement:

Since the p-value < α = 0.05

There is strong evidence at the 5% significance level that the proportion of female students taking the Biology exam is higher than the proportion of female 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.

\(\mu1\) = The mean of babies crying time while using conventional methods.

\(\mu2\) = The mean of babies crying time while being with their mothers.

\(H_0\): \(\mu2\) = \(\mu1\)

\(H_a\): \(\mu2\) < \(\mu1\)

Significance level

α = 0.05

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)

with_mother_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(with_mother_method, conventional_methods, alternative = "less", paired = FALSE)
## 
##  Welch Two Sample t-test
## 
## data:  with_mother_method 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

p-value = 0.4881

Decision Statement:

We fail to reject the null hypothesis as there is not enough evidence at 5% significance to conclude that infants cry less when they are being held by their mothers in comparison to when they receive shots using conventional methods.