This homework has two parts. Part 1 asks you to write your own functions. Part 2 applies hypothesis tests to two real scenarios.

Part 1 — Functions

# Q1. Write a function called calculate_area_of_rectangle that takes two parameters
#     (length, width) and returns the area (area = length * width).
#     Test it with 2 different inputs.
calculate_area_of_rectangle <- function(length, width){
  area = length * width
  return(area)}

calculate_area_of_rectangle(5, 10)
## [1] 50
calculate_area_of_rectangle(8, 20)
## [1] 160
# Q2. Write a function called calculate_average that takes a numeric vector and returns
#     its average. Handle the case of an empty vector by printing a message.
#     (Hint: use if/else and length(x) == 0)
calculate_average <- function(v){
  if(length(v) != 0) return(mean(as.numeric(v)))
  else return("vector is empty")}
# Q3. Write a function called check_even_odd that takes an integer and prints whether
#     it is "Even" or "Odd".
#     Test it on 14 and 27.
#     (Hint: use the %% modulus operator)
check_even_odd <- function(i){
  if(i %% 2 == 0) return("Even")
  else return("Odd")}

check_even_odd(14)
## [1] "Even"
check_even_odd(27)
## [1] "Odd"

Part 2 — Hypothesis Testing

Problem 1 — Two-Proportion z-test

In 2017, of the 144,790 students who took the AP Biology exam, 84,200 were female. That same year, of the 211,693 students who took the AP Calculus AB exam, 102,598 were female.

Is there enough evidence to show that the proportion of female students taking the Biology exam is HIGHER than the proportion taking the Calculus AB exam? Test at the 5% level.

State your hypotheses:

  • H₀: p₁ = p₂
  • H₁: p₁ > p₂
# Q4. Run the appropriate two-proportion test.
#     (Hint: prop.test(c(84200, 102598), c(144790, 211693), alternative = "greater"))
prop.test(c(84200, 102598), c(144790, 211694), alternative = "greater")
## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  c(84200, 102598) out of c(144790, 211694)
## X-squared = 3235, df = 1, p-value < 2.2e-16
## alternative hypothesis: greater
## 95 percent confidence interval:
##  0.09409171 1.00000000
## sample estimates:
##    prop 1    prop 2 
## 0.5815319 0.4846524
# Q5. What is the p-value? At α = 0.05, do you reject H₀?
# p-value: <2.20e-16. Reject null hypothesis.

Q6. Write your conclusion in plain English (one or two sentences):

It can be said with 95% confidence that the proportion of female students taking the Biology exam is higher than the proportion of female students taking the Calculus AB exam in 2017.

Problem 2 — Paired t-test

A vitamin K shot is given to infants soon after birth. Researchers want to see if how the infants are handled can reduce the pain. They measured how long (in seconds) the infant cried after the shot. One group received the shot the conventional way; the other group received it while the mother held the infant.

Is there enough evidence to show that infants cried LESS on average when held by their mothers vs. the conventional method? Test at the 5% level.

Old <- 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)

New <- 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)

State your hypotheses:

  • H₀: μ_d = 0
  • H₁: μ_d != 0
# Q7. Run a paired t-test.
#     (Hint: t.test(Old, New, paired = TRUE))
t.test(Old, New, paired = TRUE)
## 
##  Paired t-test
## 
## data:  Old and New
## 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
# Q8. What is the p-value? At α = 0.05, do you reject H₀?
# p-value: 0.977. Cannot reject null hypothesis.

Q9. Write your conclusion in plain English. Does the data support the claim that the new method reduces crying time?

There is not enough evidence to reject the claim that infants held by their mothers during the shot cried less than those that received the shot in the conventional way.