Question

if I took a full deck of cards, and pulled 1 card out of the deck, then noted the value of the card as a sample (ignoring suits), then took 500 samples and measured the frequency of each card.. How unusual or normal would it be to see a distribution of
A = 7.96%
K = 6.09%
Q = 9.13%
4 = 8.20%
2 = 9.60%
How would I go about finding a z score for each result, and an expected standard deviation and variance for the data?

Answer 1

is 500 samples enough? how could I tell?

Answer 2

would you need more data? the full data set for the samples is..
A: 7.96%
K: 6.09%
Q 9.13%
J 7.96%
T 6.79%
9 9.84%
8 6.32%
7 7.49%
6 7.49%
5 5.62%
4 8.20%
3 7.49%
2 9.60%

Answer 3

\[\langle x\rangle = \sum xP(x)\]\[\langle x^2\rangle = \sum x^2P(x)\]\[\sigma^2=\langle x^2\rangle-\langle x\rangle^2\]

Answer 4

you have a table of P(x) values

you will need to assign numbers for the x values of J,Q,K, & A, say: 11,12,13, & 14

Answer 5

\[\langle x\rangle = 14\times6.09\%+13\times7.96\%+\dots \]

\[\langle x^2\rangle = 14^2\times6.09\%+13^2\times7.96\%+\dots \]

Answer 6

What do you get for ⟨x⟩? the average value of the cards?

What do you get for ⟨x^2⟩? the average of the square of the value of the cards?

Answer 7

Thank you Uncle Rhaukaus for your help.. my responses might be a little slow..
Here is what I have so far ...

P(x) x P(x) x^2 P(x)
14 7.96% 1.114 15.6016
13 6.09% 0.7917 10.2921
12 9.13% 1.0956 13.1472
11 7.96% 0.8756 9.68316
10 6.79% 0.6790 6.7900
9 9.84% 0.8856 7.9704
8 6.32% 0.5056 4.0448
7 7.49% 0.5243 3.6701
6 7.49% 0.4494 2.6964
5 5.62% 0.2810 1.4050
4 8.20% 0.3280 1.3120
3 7.49% 0.2257 0.6741
2 9.60% 0.1920 0.3840

Sample Mean = 7.9469
x^2 = 77.6193
σ^2 = 77.6193 - 63.1532 = 14.4661
σ = 3.8034

Answer 8

P(x) x P(x) x^2 P(x)
14 0.07692 1.0769 15.0769
13 0.07692 1.0000 13.0000
12 0.07692 0.9231 11.0769
11 0.07692 0.8462 9.3077
10 0.07692 0.7692 7.6923
9 0.07692 0.6923 6.2308
8 0.07692 0.6154 4.9231
7 0.07692 0.5385 3.7692
6 0.07692 0.4615 2.7692
5 0.07692 0.3846 1.9231
4 0.07692 0.3077 1.2308
3 0.07692 0.2308 0.6923
2 0.07692 0.1538 0.3077

The same logic applied to draws of 1 card out of 13

Sample Mean (x) = 8
sum (x^2) P(x) = 78
σ^2 = 78 - (8^2) = 78 - 64 = 14
σ = 3.7417

Answer 9

the z-score for each data point is given by

z = (x - mean) / σ

Answer 10

Most of the data should have a z-score less than 1.

The first point, A: 7.96%, is near the mean, so it's z score should be very small.