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

1. State a reasonable null and alternative hypothesis and use mathematical notation.
2. Calculate a test statistic to test whether you can reject your null hypothesis.
3. Calculate the p-value.
4. You consider a significance level of 10% so that you can be 90 percent confident before blaming your friend of cheating. Would you do so or not?
5. If you would like to be even more confident in your statement and use a significance level of 1%, which corresponds to being 99 percent confident, would you need a higher or lower p-value to reject the Null hypothesis?

Answer 1

Ho :   µ =   7.5                  
Ha :   µ <   7.5       (Left tail test)          
                          
Level of Significance ,    α =    0.10                  
population std dev ,    σ =    4.1800                  
Sample Size ,   n =    10                  
Sample Mean,    x̅ =   4.5000                  
                          
Standard Error , SE = σ/√n =   4.1800   / √    10   =   1.3218      
Z-test statistic= (x̅ - µ )/SE = (   4.500   -   7.5   ) /    1.3218   =   -2.27
                          
      
p-Value   =   0.0116   [ Excel formula =NORMSDIST(z) ]             
Decision:   p-value<α, Reject null hypothesis                       
Conclusion: There is enough evidence that card have been replaced.

If you would like to be even more confident in your statement and use a significance level of 1%, which corresponds to being 99 percent confident,

We would you need a lower p-value to reject the Null hypothesis

For 99% confidence , Level of Significance ,    α =    0.01  

p-Value   =   0.0116   [ Excel formula =NORMSDIST(z) ]             
Decision:   p-value>α, Do not Reject null hypothesis

Thanks in advance!

revert back for doubt

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