Question 1: 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.
Hypthesis \(H_0: p_1 = p_2\) \(H_a: p_1 > p_2\)
Where, \(p_1\) is the proportion of female students taking the biology exam \(p_2\) is the proportion of female students taking the calculus AB exam
Significance Level α = 0.05
p-value < 2.2e-16. 2.2e-16 < 0.05 Thus,statistically significant at 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 0.4846547We reject the null. There is 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.
Question 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.
conventional <- 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 <- 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)Where, \(\mu_1\) is the mean crying time for infants held using new methods (mothers holding them prior) \(\mu_2\) is the mean crying time for infants held using conventional methods
Hypothses \(H_0\): \(\mu_1 = \mu_2\) \(H_a\): \(\mu_1 < \mu_2\)
Signifcant Level α = 0.05
p-value = 0.4881. 0.4881 > 0.05, Thus,not statistically significant at α = 0.05.
t.test(new, conventional, alternative = "less", conf.level=0.95)##
## Welch Two Sample t-test
##
## data: new and conventional
## 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 35.30000We fail to reject the null. There is not enough evidence to conclude that infants cry less when using the new methods(mothers holding them prior).