For each question:

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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.

Write the hypothesis tests.

\(p_1\) = proportion of female students that took biology exam

\(p_2\) = proportion of female students that Calculus AB Exam

\(H_0\): \(p_1\) \(<=\) \(p_2\)

\(H_a\):\(p_1\) \(>\) \(p_2\)

State the significance level

\(\alpha\) = 0.05

p-value

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 * 10^-16

State your decision.

Based on the results, we can see that we have enough evidence to show that the proportion of females who took the biology exam is greater than the proportion of females that took the Calculus AB Exam. This is because our p-value in this question is less than the significance level(0.5).

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.

Write the hypothesis tests.

\(\mu_1\) = mean of a babies crying time(in seconds), using conventional method.

\(\mu_2\) = mean of a babies crying time(in seconds), being held by their mothers

\(H_0\): \(\mu_1\) < \(\mu_2\)

\(H_a\): \(\mu_1\) > \(\mu_2\)

State the significance level

\(\alpha\) = 0.05

p-value

conventional_Cry_Time <- 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)


with_Mother_Cry_Time  <- 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(conventional_Cry_Time, with_Mother_Cry_Time, paired = TRUE)
## 
##  Paired t-test
## 
## data:  conventional_Cry_Time and with_Mother_Cry_Time
## t = 0.028519, df = 29, p-value = 0.9774
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -11.78558  12.11892
## sample estimates:
## mean difference 
##       0.1666667

p-value = 0.9774

State your decision.

Based on the test that I did, we can see that we do not have enough evidence to support that the cry time(in seconds) from a baby being held by their mother while getting a shot is less than when a baby get’s a shot using the conventional method. This is because our p-value is greater than the significance level.