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
\(H_0\): \(p_1\) = \(p_2\) \(H_a\): \(p_1\) > \(p_2\)
where:
\(p_1\) = The proportion of females who took biology exam \(p_2\) = The proportion of females who took AP calculus exam
The significance level is 0.05.
#right tail test for 2 proportions
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 0.4846547P value < 2.2e-16 which is much smaller than the significance level of 0.05. Therefore we have to reject the null. There is strong evidence that the proportion of female students taking AP Biology is significantly higher than the proportion taking AP Calculus.
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_methods <- 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_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)Hypotheses
\(H_0\): \(\mu_1\) = \(\mu_2\) \(H_a\): \(\mu_1\) < \(\mu_2\)
where:
\(\mu_1\) = mean crying time when held by their mother \(\mu_2\) = mean crying time when held in conventional methods
The significance level is 0.05.
t.test(new_methods, conventional_methods, alternative = "less" )##
## Welch Two Sample t-test
##
## data: new_methods and conventional_methods
## 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 35.30000Since the p value(0.4881) is greater than the significance level of 0.05, We have to fail to reject the null hypotheses. There is not enough statistical evidence to conclude that infants cry less on average when they are held by their mothers compared to when they are held using conventional methods. The two sample means (35.13 seconds vs. 35.30 seconds) are almost the same, and the p-value of 0.4881 indicates that this difference could easily occur by random chance.