Question

In this game, there is only one deck of cards. You play with a friend and the deck of cards belongs to him/her. Numbered cards are worth their face value, jacks are worth 11, queen 12, kings 13 and aces 14. You have a suspicion that in this deck of cards, your friend has replaced some high cards in the deck with low cards.

You take 10 cards and quickly calculate the average value: 4.5. You do the math: In a regular deck of cards the average should be mean =7.5 and standard deviation = 4.18

Now assume that you do not know the standard deviation of the “population of cards” and that you use the standard deviation in your sample and that this is equal to s = 2.4

1. Calculate a test statistic to test whether you can reject your null hypothesis based on your estimated standard deviation. What are the degrees of freedom for your test?
2. Calculate the p-value.
3. Do you reject the null hypothesis at the 1% significance level?

Answer 1

(a)

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:μ = 7.5

Ha: μ ≠ 7.5

(b)test statistic:

t=​​~t_n-1

t= ​=−2.27 ~t_9,0.1   (calculated)

t_9,0.1=1.833 (tabulated)

since,   t_cal > t_tab we reject null hypothesis or 7.5 is accepted.

(c) p-value is p=0.0494, and since p=0.0494<0.1, it is concluded that the null hypothesis is rejected or 7.5 is accepted.

(d) Since, we know that the average value in a regular cards of deck =7.5

but since, in this case we rejected our null hypothesis or 7.5 is accepted.

hence, yes we can blame our friend for cheating.

please rate my answer and comment for doubts.