Description
1. Hypothesis Testing for a single population (Total 10 points)
Consider the 10 samples: {1.68, 1.34, 1.98, 0.97, 1.09, 2.65, 1.23, 1.78, 0.6, 0.85}. Use the K‐S test to
check whether these samples are from the Uniform(0, 3) distribution. First, set up the hypotheses. Then,
create a 10 X 6 table with entries: ሾ𝑥, 𝐹ሺ𝑥ሻ, 𝐹
ିሺ𝑥ሻ, 𝐹
ାሺ𝑥ሻ, ห𝐹
ିሺ𝑥ሻ െ 𝐹ሺ𝑥ሻห, |𝐹
ାሺ𝑥ሻ െ 𝐹ሺ𝑥ሻ|ሿ, where
𝐹
ିሺ𝑥ሻ and 𝐹
ାሺ𝑥ሻ are the values of the eCDF to the left and right of x, and 𝐹ሺ𝑥ሻ is the CDF of
Uniform(0, 3) at x; this is the same notation as in class. Finally, compare the max difference with the
threshold of 0.37 to Reject/Accept. Show all rows and columns.
2. Toy Example for Permutation Test (Total 10 points)
Let X = {2, 9} and Y = {4}. The null hypothesis is that X and Y are from the same distribution. Use the
permutation test to decide this using a p‐value threshold of 0.05. Please show all steps for each
permutation clearly.
3. Independence Teststo Save Your Casino (Total 15 points)
Being the owner of Casino 544, you are concerned that you are losing a lot of money because of the
dealers at the blackjack tables. The Null hypothesis is that the outcome of the tables should be
independent of the dealer, but you aren’t sure.
(a) Validate your claim based on the dealer observations for a day, using the χ2 test. Use α=0.05. You can
use tools/online resources to find the CDF of χ2
; one such tool is
https://www.danielsoper.com/statcalc/calculator.aspx?id=62. (10 points)
Dealer A Dealer B Dealer C
Win 48 54 19
Draw 7 5 4
Loose 55 50 25
(b) You want to be more certain about the loyalty of your dealers, so you collect more data: number of
wins from each dealer for 10 days. Find the Pearson correlation coefficient for each pair of
dealers. What can you conclude? (5 points)
Day‐1 Day‐2 Day‐3 Day‐4 Day‐5 Day‐6 Day‐7 Day‐8 Day‐9 Day‐10
Dealer A 48 40 58 53 65 25 52 34 30 45
Dealer B 54 48 51 47 62 35 70 20 25 40
Dealer C 19 40 35 41 38 32 32 37 37 15
4. Real‐World Example Based on NBA (Total 20 points)
NBA veterans often complain about how easy it is to score points in today’s games as opposed to prior
decades. This question attempts to verify this claim. We will compare the distribution of points per
game for all teams between 1999 and 2009, and between 2009 and 2019. The Null hypothesis is that the
scoring distribution remains the same. Refer to the “Team Per Game Stats” table found on
“https://www.basketball‐reference.com/leagues/NBA_2019.html#all_team‐stats‐per_game” (this is
2019 data); you can obtain data for different years by changing the year in the URL. We only care about
the last column i.e. Points (PTS); this column is the points per game. Use the “Share & more” option to
format data. You can use any programming tool for this problem. Submit code for this question as q4.py.
(a) Using Permutation test, check if the Null hypothesis can be accepted or rejected for the two cases
(1999 vs. 2009, and 2009 vs. 2019). Choose n=200 random permutations and use a p‐value threshold
of 0.05. Clearly show the p‐value you obtain. (7 points)
(b) Repeat part (a) with n=2000. (7 points)
(c) Repeat part (a) but using two‐sample K‐S test with a max difference threshold of 0.05. You do not
need to show the full table but do create the two eCDF graphs on the same figure and identify the
max difference in the figure along with its value. Print and attach this figure. (6 points)
5. Type‐1 and Type‐2 error for one‐sided unpaired T‐test (Total 15 points)
Let {𝑋ଵ, 𝑋ଶ,…,𝑋ሽ be i.i.d. from Normal(1, 1
2
) and {𝑌ଵ, 𝑌ଶ,…,𝑌ሽ be i.i.d. from Normal(2, 2
2
). Also
suppose X’s and Y’s are independent, and 1, 1
2
, 2, 2
2
are unknown. Let Sx and SY be the sample
standard deviations of the two populations. Assume that n and m are large. Let H0: 1 > 2 be the null
hypothesis and H1: 1 <= 2 be the alternate hypothesis. Consider the T statistic for the unpaired T test,
as in class, with δ > 0 being the critical value.
(a) For the above test, show that the probability of Type‐1 and Type‐2 errors are given by
Φሺെ𝛿 െ భିమ
ටೄೣమ
ାೄమ
ሻ and 1 െ Φሺെ𝛿 െ భିమ
ටೄೣమ
ାೄమ
ሻ, respectively. (10 points)
(b) Show that the p‐value is given by Φ ቌ തିതିሺభିమሻ
ටೄೣమ
ାೄమ
ቍ. (5 points)
