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
\(H_0\): \(p_1\) = \(p_2\) \(H_a\): \(p_1\) > \(p_2\)
\(p_1\) = Proportion of students who were female taking the biology exam \(p_2\) = Proportion of students who were female taking the calculus AB exam
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. Statistically significant at α = 0.05. There is strong evidence that proportion of students taking biology exam is greater
95% CI for difference = (0.056, 1.000) The interval is entirely above 0 (the null value), showing Vaccine A has higher effectiveness than Vaccine B.
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.
\(H_0\): \(\mu_1\) = \(\mu_2\) \(H_a\): \(\mu_1\) < \(\mu_2\)
\(\mu_1\) = mean of infants who cried using new methods
\(\mu_2\) = mean of infants who cried using conventional methods
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)
mother_holding <- 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(mother_holding, conventional, conf.level = 0.95, alternative = "greater")
Welch Two Sample t-test
data: mother_holding and conventional
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 = 0.5119 Statistically significant at α = 0.05. Weak evidence that conventional methods work less than new methods.
95% CI = (-9.468, ∞). Since 0 is outside the interval, that means the difference in means is statistically significant, showing there is no difference.