###Problem 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.
\(p1\) is the proportion of female students taking the biology exam, \(p2\) is the proportion of female students taking the calculus AB exam.
\(Ho: p1 > p2\)
\(Ha: p1 < p2\)
\(α\): 5% or .05
#proportion test
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 Conclusion: The p-value is 2.2e-16. Since this is greater than the significance level of .05, there is strong evidence that the proportion of female students taking the biology exam is greater than the proportion of female students taking the calculus AB exam. (fail to reject the null).
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.
\(µ1\) is the mean crying time in infants given shots when their mothers held them, and \(µ2\) is the mean crying time in infants given shots using conventional methods.
\(Ho: µ1 < µ2\)
\(Ha: µ1 > µ2\)
\(α\): 5% or .05
#create vectors
held_by_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)
conventional_methods <- 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)#perform t-test
t.test(held_by_mother, conventional_methods, conf.level = 0.95, alternative = "less")
Welch Two Sample t-test
data: held_by_mother and conventional_methods
t = 0.029953, df = 57.707, p-value = 0.5119
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
-Inf 9.468337
sample estimates:
mean of x mean of y
35.30000 35.13333 Conclusion: The p-value is .0.512, which is more than the alpha of .05, so we fail to reject the null. There is not enough evidence to suggest that infants cried less on average when they are held by their mothers than if held using conventional methods.