For each question: Write the hypothesis tests. State the significance level () p-value State your decision.

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.

\(H_0\): \(P_f\) = \(P_m\)

\(H_a\): \(P_f\) > \(P_m\)

α = 0.05

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

P value is 2.2e-16, which is significantly less than our α, which is 0.05. This means that we reject the null hypothesis. Which means 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.

\(H_0\): \(μ_1\) = \(μ_2\)

\(H_a\): \(μ_1\) < \(μ_2\)

\(μ_1\) = The average time the baby cried after the shot with the mother holding them.

\(μ_2\) = The average time the baby cried after the shot using conventional methods

α = 0.05

infant_with_mother <- 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)
infant_without_mother <- 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)
t.test(infant_with_mother, infant_without_mother, alternative = "greater")
## 
##  Welch Two Sample t-test
## 
## data:  infant_with_mother and infant_without_mother
## 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

P value is 0.5119, which is more than our α, 0.05. This means that we fail to reject the null hypothesis. Which means that average time the baby cried after their shot with the mother holding them is equal to the average time the baby cried after their shot without the mother holding them.