Question 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.

Hypothesis

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

Where: \(p_1\)= proportion of female students taking the Biology AP exam \(p_2\)= proportion of females taking Calculus AB exam

Significance Level

Is the result statistically significant at α = 0.05?

# Right tail test for two 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 0.4846547

Results

p-value < 2.2e-16. Statistically significat at α = 0.05 since p-value is a very small number.There is strong evidence that proportion of female students taking the Biology exam is higher than the proportion taking the Calculus exam.

Question 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(26, 0, 2 ,46, 33, 33, 39, 23, 11, 12, 48, 15, 33, 14, 51, 37, 24, 70, 63, 0, 73, 39, 54, 52, 39, 34, 30, 55, 58,18)

new_method <- c(0, 32, 20, 23, 14, 19, 60, 59, 64, 64, 72, 50, 44, 14, 10, 58, 19, 41, 17, 5, 38, 73, 19, 46, 9, 43, 73, 27, 25, 18)

Hypothesis

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

Where: \(\mu_1\)= crying time for infants using the new method \(\mu_2\)= crying time for infants using the conventional method

Significance Level The problem says to test at the 5% level.

t.test(new_method, conventional, alternative = "less")
## 
##  Welch Two Sample t-test
## 
## data:  new_method and conventional
## t = 0.14575, df = 57.41, p-value = 0.5577
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##      -Inf 9.976626
## sample estimates:
## mean of x mean of y 
##      35.2      34.4

Results

p-value = 0.5577. Meaning that there is not enough evidence at the 5% significance level to conclude that infants cry less when held by their mothers compared to the conventional method. In fact, the mean crying time for the new method is higher than the mean crying time for the conventional time, showing that babies cry slightly more with the new method.