For each question:
- Write the hypothesis tests.
- State the significance level ()
- p-value
- State your decision.
Write your answers on R markdown and submit your answers as a PDF. Make sure you
include your name and the date.
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 = proportion of females in biology exam
p2 = proportion of females in calculus AB exam
\(H_0\): \(p_1\) = \(p_2\)
\(H_a\): \(p_1\) > \(p_2\)
significance level: 0.05 (5%)
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 = 2.2 e -16 <- extremely significant
Yes, 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
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)
newer <-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,
\(\mu_1\) = babies crying using newer methods
\(\mu_2\) = babies crying using conventional methods
t.test(newer, conventional, conf.level = 0.95, alternative = "less")
Welch Two Sample t-test
data: newer 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 35.30000 No, there is not enough evidence to suggest that the babies cried less using newer methods than conventional ones. The p-value was 0.448, meaning we fail to reject the null. Additionally, the confidence interval included 0 in its bounds, indicating that there is no meaningful difference between the mean crying duration for each group.