Find the following probabilities for the standard normal random variable z: a. P(−1.36 ≤ z ≤ 1.38)= ________ b. P(−2.27 ≤ z ≤ 1.
2 answers:
Answer:
a) 0.8293
b) 0.9379
c) 0.9082
d) 0.7517
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
a. P(−1.36 ≤ z ≤ 1.38)=
This is the pvalue of Z = 1.38 subtracted by the pvalue of Z = -1.36
Z = 1.38 has a pvalue of 0.9162
Z = -1.36 has a pvalue of 0.0869
0.9162 - 0.0869 = 0.8293
b. P(−2.27 ≤ z ≤ 1.64)
Same logic as a.
Z = 1.64 has a pvalue of 0.9495
Z = -2.27 has a pvalue of 0.0116
0.9495 - 0.0116 = 0.9379
c. P(z ≤ 1.33)=
This is the pvalue of Z = 1.33.
Z = 1.33 has a pvalue of 0.9082
d. P(z > −0.68)=
This is 1 subtracted by the pvalue of Z = -0.68. So
Z = -0.68 has a pvalue of 0.2483
1 - 0.2483 = 0.7517
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