Question 01

\(H_0\): \(p_1\) = \(p_2\) \(H_a\): \(p_1\) > \(p_2\)

Where,

\(p_1\)= proportion of female students taking the biology exam \(p_2\) = Proportion of female students 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
  1. p-value = 2.2e-16. Statistically significant at α = 0.05.There is strong evidence that proportion of female students taking the biology exam is higher than the proportion of female students who taking the calculus AB exam.

Decision: Rejecting \(H_0\) at α = 0.05.

  1. 95% CI for difference: (0.094,1.000). The interval is entirely above 0, showing proportion of female students taking biology exam is higher than proportion of female students taking calculus AB exam.

Question 2:

mothers <- 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)

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)

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

Where,

\(\mu_1\) = average of infants crying when they are held by their mothers

\(\mu_2\) = average of infants crying when they held using conventional methods

t.test(mothers,conventional,alternative= "less")
## 
##  Welch Two Sample t-test
## 
## data:  mothers 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

We do not reject the null. However, We do not have enough evidence to prove that average of infants crying when they are held by their mothers is less than the average of infants crying when they held using conventional methods.