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. Test at the 5% level

Hypothesis Test:

Two Proportion Z-Test \(H_0\): \(p_1\) = \(p_2\) \(H_a\): \(p_1\) > \(p_2\)

Where: \(p_1\) = proportion of females taking biology exam \(p_2\) = proportion of females taking calculus AB exam

Significance Level:

α = 0.05

P-value:

prop.test(c(84200, 144790), c(102598, 211693), alternative = "greater")
## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  c(84200, 144790) out of c(102598, 211693)
## X-squared = 6531.4, df = 1, p-value < 2.2e-16
## alternative hypothesis: greater
## 95 percent confidence interval:
##  0.1341319 1.0000000
## sample estimates:
##    prop 1    prop 2 
## 0.8206788 0.6839622

**Result: p-value: 2.2e-16.

Decision

The p-value: 2.2e-16 < α = 0.05. This is strong evidence that the proportion of females students taking the biology exam is higher than the proportion of females 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.

Hypothesis Test:

Two Sample T-Test \(H_0\): \(\mu_1\) = \(\mu_2\) \(H_a\): \(\mu_1\) < \(\mu_2\)

Where: \(\mu_1\): infants that cried less on average held by their mothers \(\mu_2\): infants that cried less on average held using conventional methods

Significance Level:

α = 0.05

P-value:

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


t.test(mothers, conventional_methods, conf.level = 0.95, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  mothers and conventional_methods
## t = -0.088794, df = 57.878, p-value = 0.4648
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##      -Inf 8.912894
## sample estimates:
## mean of x mean of y 
##  35.13333  35.63333

Result: p-value = 0.4648

Decision

The p-value = 0.4648 > α = 0.05. This is not enough evidence to show that infants cried less on average when they are held by their mothers than if held using conventional methods.