Question 1

Hypothesis

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

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

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

p-value = < 2.2e-16 this is far smaller than α = 0.05, meaning it’s statistically significant at α = 0.05. There is strong evidence to suggest that proportion of females taking the biology exam is higher than the proportion of female students who are taking the AB calculus exam.

My decision is to reject the null which is \(H_0\): \(p_1\) = \(p_2\) at α = 0.05.

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

Question 2

new_method <- 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 crying time of infants held by their mothers (new method).

\(\mu_2\) = average crying time of infants when they are held using the conventional method.

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

p-value = 0.4881 this is higher than α = 0.05, meaning it’s not statistically significant at α = 0.05. Therefore we do not reject the null hypothesis. We do not have enough evidence to prove that the average crying time of infants held by mothers is less than to the average crying time of infants held while crying using the conventional method.