Sampling with Playing Cards
Take a standard deck of 52 playing cards. Shuffle them well. For our purposes, let’s assume the Ace is worth 1 point, 2 through 10 are worth their respective points, Jack is worth 11 points, Queen is worth 12 points, and King is worth 13 points. So there are 4 suits with values of 1 through 13 in them. The average value of all cards is 7.00.
Round 1) Deal the cards into 26 pairs. Compute the averages of those 26 pairs. It is possible to get two Aces, in which case the average is 1.00, and it is possible to get two Kings, in which case the average is 13.00. Most likely the averages will tend to be in the 5 to 9 range. Report (1a) the lowest of your 26 averages, (1b) the highest of your 26 averages, (1c) how many of the 26 had an average between 6 and 8 inclusive (within 1 point of 7), and (1d) how many of the 26 had an average of exactly 7.
Round 2) Deal the cards into 13 piles of 4 cards. Compute the 13 averages. Report (2a) the lowest average, (2b) the highest average, (2c) how many of the averages were between 6 and 8 inclusive (within 1 point of 7), and (2d) how many averages were exactly 7.
Round 3) Deal the cards into 4 piles of 13 cards. Compute the 4 averages. Report (3a) the lowest average, (3b) the highest average, (3c) how many of the averages were between 6 and 8 inclusive (within 1 point of 7), and (3d) how many averages were exactly 7.
After each student has posted their results, the instructor will summarize the findings. What we should see is that with samples of size 2, the averages will be spread out greatly, and only a few averages will be close to 7. And with samples of size 4, the spread of the averages will be greatly reduced, with almost every average being between 4 and 10, and most of them close to 7. Samples of size 13 should result in almost every one being close to 7 (although getting exactly 7 is less likely).
TRUSTING THE RESULTS
If you were to read the results of a study showing that daily use of a certain exercise machine resulted in an average 10-pound weight loss, what more would you want to know about the numbers in addition to the average? (Hint: Do you think everyone who used the machine lost 10 pounds?)
WHAT’S IN A SAMPLE?
Why is it important that a sample be random and representative when conducting hypothesis testing?
The plain M&M’s Milk Chocolates are mass-produced with a distribution of 24% blue, 20% orange, 16% green, 14% yellow, 13% red and 13% brown. The peanut M&M’s are mass-produced with a distribution of 23% blue, 23% orange, 15% green, 15% yellow, 12% red and 12% brown.
Open a packet of your favorite M&Ms, plain or peanut.
Count the number of M&M’s in the packet.
Count number of BLUE, ORANGE, GREEN and RED candies.
Using the Excel file for One Sample Hypothesis Testing, conduct a test of proportions for each of the four colors. Use two-tailed tests and a .05 significance level in each case. Be sure to use the proportions shown above as the hypothesized values.
After each student has posted their results, the instructor will summarize the findings. What we should see is that about 5% of the samples will likely result in a rejection of the null hypothesis
If a random sample of 10 people found that 9 were pro-life (i.e., 90%), while another random sample of 1000 people from the same population found that 550 were pro-life (i.e., 55%), which would you find to be more significant? Why?
PROBING THE POLLSTERS
When pollsters report results of a poll, they often include the margin of error but not much more. If candidate X has 54% support of the voters with a margin of error of 3%, that means the candidate is predicted to have between 51% and 57% support in the election. Before jumping to conclusions, what other information would you like the pollster to provide?