(1) Carry out a Pearson's chi-squared test, report the test-statistic and its p-value. You should have with you a text book with the needed chi-squared table. (2) Some statistician suggests to use the average, X, as a statistic. Un- der the null distribution, E(X) = 3.06, SD(X) = 0.116. Calculate the observed X, and the z-value. (3) Do you think this gadget outputs uniformly-distributed results?
(1) Carry out a Pearson's chi-squared test, report the test-statistic and its p-value. You should have with you a text book with the needed chi-squared table. (2) Some statistician suggests to use the average, X, as a statistic. Un- der the null distribution, E(X) = 3.06, SD(X) = 0.116. Calculate the observed X, and the z-value. (3) Do you think this gadget outputs uniformly-distributed results?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:(1) Carry out a Pearson's chi-squared test, report the test-statistic and
its p-value. You should have with you a text book with the needed
chi-squared table.
(2) Some statistician suggests to use the average, X, as a statistic. Un-
der the null distribution, E(X) = 3.06, SD(X) = 0.116. Calculate
the observed X, and the z-value.
(3) Do you think this gadget outputs uniformly-distributed results?
Transcribed Image Text:Problem 1. The State of California has a state lottery game called Lotto.
To play, you select a 6 set from {1,2, ..., 49}. There are various gadgets sold
to generate a six-element subset of {1, 2, ..., ,49}; one such gadget is pictured
below. There are 49 numbered holes and six balls enclosed by a plastic
cover. One shakes the balls around and uses the six set determined by their
final resting place.
1
1018,26o4o42。
2o1o19o7o3504B。
12。20028o36o4o
13021029。27。45。
O
O
O
O
PICK 6 LOTTO & WIN
3 4 5
O
O
O
O
O
O
O
O
19 20 21
26 27 28 29
O
O
O
O
O
34 35 36 37
O
O
O
O
42 43 44 45
O
O
O
TO HONO
7。15o23o1o39oFo
9o7o25o3oHo49。
NO
6
O
14 15 16
O
O
O
O
8
22 23 24 25
O
O
O
O
38 39 40
30 31 32 33
O
O
46 47 48
O
O
17
O
41
O
O
We wish to test if such gadget indeed lead to uniformly-distributed re-
sults. To test this, 100 trials were performed. The gadget was vigorously
shaken and set down on a flat surface. Following each six set is X - the
number of balls falling on the outer perimeter in that 6-set. For exam-
ple, the first 6-set {10, 11, 13, 25, 36, 42} had 3 outside numbers, 10, 25, 42,
so X 3. There are 25 outside numbers out of 49. If the six sets were
chosen uniformly at random, X would have a hypergeometric distribution
24
H(X = j) = (²5) (²4₁) / (4). These numbers and also the empirical frequen-
cies are given in the table below.
Hypergeometric and Empirical Probabilties for X.
1 2
4 5 6
3
0
H{X=j} .013 .091 .250 .333 .228 .016 .010
Empirical .01 .04 .14 .35 .30 .13 .03
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