Caleb is selecting cards from a set of 12 cards numbered 1-2-4-4-5-5-5-5-6-9-10-13.
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Using the normal distribution, the percentages are given as follows:
a) 9.18%.
b) 97.72%.
c) 50%.
d) 4.27%.
e) 0.13%.
f) 59.29%.
g) 2.46%.
h) 50%.
i) 50%.
<h3>Normal Probability Distribution</h3>The z-score of a measure X of a normally distributed variable with mean and standard deviation
is given by:
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
For this problem, the mean and the standard deviation are given as follows:
For item a, the proportion is the <u>p-value of Z when Z = 167</u>, hence:
Z = (167 - 247)/60
Z = -1.33.
Z = -1.33 has a p-value of 0.0918.
Hence the percentage is of 9.18%.
For item b, the proportion is the <u>p-value of Z when Z = 367</u>, hence:
Z = (367 - 247)/60
Z = 2.
Z = 2 has a p-value of 0.9772.
Hence the percentage is of 97.72%.
For item c, the proportion is <u>one subtracted by the p-value of Z when X = 247</u>, hence:
Z = (247 - 247)/60
Z = 0
Z = 0 has a p-value of 0.5.
Hence the percentage is of 50%.
For item d, the proportion is <u>one subtracted by the p-value of Z when X = 350</u>, hence:
Z = (350 - 247)/60
Z = 1.72
Z = 1.72 has a p-value of 0.9573.
1 - 0.9573 = 0.0427.
Hence the percentage is of 4.27%.
For item e, the proportion is the <u>p-value of Z when Z = 67</u>, hence:
Z = (67 - 247)/60
Z = -3.
Z = -3 has a p-value of 0.0013.
Hence the percentage is of 0.13%.
For item f, the proportion is the <u>p-value of Z when X = 300 subtracted by the p-value of Z when X = 200</u>, hence:
X = 300:
Z = (300 - 247)/60
Z = 0.88.
Z = 0.88 has a p-value of 0.8106.
X = 200:
Z = (200 - 247)/60
Z = -0.78.
Z = -0.78 has a p-value of 0.2177.
0.8106 - 0.2177 = 0.5929.
Hence the percentage is 59.29%.
For item g, the proportion is the <u>p-value of Z when X = 400 subtracted by the p-value of Z when X = 360</u>, hence:
X = 400:
Z = (400 - 247)/60
Z = 2.55.
Z = 2.55 has a p-value of 0.9946.
X = 360:
Z = (360 - 247)/60
Z = 1.88.
Z = 1.88 has a p-value of 0.97.
0.9946 - 0.97 = 0.0246
Hence the percentage is 2.46%.
For items h and i, the distribution is symmetric, hence median = mean and the percentages are of 50%.
More can be learned about the normal distribution at brainly.com/question/24808124
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