Chi-Squared Goodness of Fit

I. Initialization Block

Initializing RStudio

If you execute the code block below, RStudio will prompt you to install Mosaic if needed.

library(mosaic)

II. Exercises

1. A baseball card company claims that 25% of its cards are rookies, 65% are veterans but not All-Stars, and 10% are veteran All-Stars. Suppose a random sample of 200 cards has 70 rookies, 120 veterans, and 10 All-Stars. Is the company’s claimed distribution credible? Test using \(\chi^2\) GOF with a 0.05 level of significance.

2. A supposedly fair die is rolled 50 times with the following results: 9 ones, 15 twos, 9 threes, 8 fours, 6 fives and 13 sixes. Test at the 0.05 level whether the die is actually fair using \(\chi^2\) GOF.

3. Helena buys custom M&M’s for a bridal shower she’s throwing for her sister. She orders 10% yellow, 20% pale blue, 30% red and 40% pink. When she receives her 10 lb. package, she sees almost not yellow and far too many pale blue. She randomly selects 200 of the M&M’s and finds the following observed counts: \[\begin{array}{cccc}\text{Yellow}&\text{Blue}&\text{Red}&\text{Pink}\\ \hline 8&54&64&74\end{array}\]

III. Code Blocks

observed = c(11, 11, 6)
expected = c(0.5, 0.25, 0.25)

xchisq.test(x = observed,
           p = expected)

pups = sample(c("black","blue","fawn"), 28, replace=TRUE, prob = c(0.5, 0.25, 0.25))
tally(pups)

N = 0
# Do the randomization 1,000 time
for (j in 1:1000) {
# Create random sample of 28 pups
pups = sample(c("black","blue","fawn"), 28, replace=TRUE, prob = c(0.5, 0.25, 0.25))
black = 0
blue = 0
fawn = 0
# Count number of pups for each color
for (i in 1:28) {
  if (pups[i] == "black") { black = black + 1}
  if (pups[i] == "blue") { blue = blue + 1}
  if (pups[i] == "fawn") { fawn = fawn + 1}
}
# Calculate Chi-Squared statistic
chiObserved = (black - 14)^2/14 +(blue - 7)^2/7+(fawn - 7)^2/7
if ( chiObserved > 43/14) {N = N + 1}
} 
N

observed = c(161,39)
expected = c(.75,.25)

xchisq.test(x = observed,
           p = expected)

observed = c(11, 29, 16, 14, 17, 23, 9)
expected = c(1/7,1/7,1/7,1/7,1/7,1/7,1/7)

xchisq.test(x = observed,
           p = expected)