The probability that the top card is spades in more than 30% of the sample in a nontraditional deck of cards is 0.224.
<h3>How to get the z scores?</h3>
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
![Z = \dfrac{X - \mu}{\sigma}, \\](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D%2C%20%5C%5C)
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
![P(Z \leq z) = P(Z < z) )](https://tex.z-dn.net/?f=P%28Z%20%5Cleq%20z%29%20%3D%20P%28Z%20%3C%20z%29%20%29)
Also, know that if we look for Z = z in z tables, the p value we get is
![P(Z \leq z) = \rm p \: value](https://tex.z-dn.net/?f=P%28Z%20%5Cleq%20z%29%20%3D%20%5Crm%20p%20%5C%3A%20value)
A nontraditional deck of cards has 30 total cards: 5 hearts, 10 clubs, 8 spades, and 7 diamonds. the cards are shuffled, and the top card is noted. this process is repeated 100 times.
Here, the sample size n is 100. IN the 30 cards 8 cards are spades. Thus, the probability of a card to be spade is,
![P=\dfrac{8}{30}\\P=0.2667](https://tex.z-dn.net/?f=P%3D%5Cdfrac%7B8%7D%7B30%7D%5C%5CP%3D0.2667)
Thus, the mean of it is,
![\mu=100\times0.2667\\\mu=26.67](https://tex.z-dn.net/?f=%5Cmu%3D100%5Ctimes0.2667%5C%5C%5Cmu%3D26.67)
The value of standard deviation is,
![\sigma=\sqrt{\mu(1-p)}\\\sigma=\sqrt{26.67(1-0.2667)}\\\sigma=4.42](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Cmu%281-p%29%7D%5C%5C%5Csigma%3D%5Csqrt%7B26.67%281-0.2667%29%7D%5C%5C%5Csigma%3D4.42)
Now for the P(X>30), z-score is,
![Z=\dfrac{x-\mu}{\sigma}\\Z=\dfrac{30-26.67}{4.42}\\Z=0.754\\P(Z > 0.754)=0.224](https://tex.z-dn.net/?f=Z%3D%5Cdfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5CZ%3D%5Cdfrac%7B30-26.67%7D%7B4.42%7D%5C%5CZ%3D0.754%5C%5CP%28Z%20%3E%200.754%29%3D0.224)
Thus, the probability that the top card is spades in more than 30% of the sample in a nontraditional deck of cards is 0.224.
Learn more about the z score here;
brainly.com/question/13299273
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