In: Statistics and Probability
Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. (Round your answers to four decimal places.)
(a) P(0 ≤ Z ≤ 2.63)
(b) P(0 ≤ Z ≤ 2)
(c) P(−2.60 ≤ Z ≤ 0)
(d) P(−2.60 ≤ Z ≤ 2.60)
(e) P(Z ≤ 1.63)
(f) P(−1.15 ≤ Z)
(g) P(−1.60 ≤ Z ≤ 2.00)
(h) P(1.63 ≤ Z ≤ 2.50)
(i) P(1.60 ≤ Z)
(j) P(|Z| ≤ 2.50)
You may need to use the appropriate table in the Appendix of Tables to answer this question.
a) P(0 < Z < 2.63) =
= P(z < 2.63) - P(z < 0)
Using excel function:
= NORM.S.DIST(2.63, 1) - NORM.S.DIST(0, 1)
= 0.4957
b) P(0 < Z < 2) =
= P(z < 2) - P(z < 0)
Using excel function:
= NORM.S.DIST(2, 1) - NORM.S.DIST(0, 1)
= 0.4772
c) P(−2.60 ≤ Z ≤ 0) =
= P(z < 0) - P(z < -2.6)
Using excel function:
= NORM.S.DIST(0, 1) - NORM.S.DIST(-2.6, 1)
= 0.4953
(d) P(−2.60 ≤ Z ≤ 2.60) =
= P(z < 2.6) - P(z < -2.6)
Using excel function:
= NORM.S.DIST(2.6, 1) - NORM.S.DIST(-2.6, 1)
= 0.9907
(e) P(Z ≤ 1.63)
= P(z < 1.63)
Using excel function:
= NORM.S.DIST(1.63, 1)
= 0.9484
(f) P(−1.15 ≤ z) = P(z > -1.15)
= 1 - P(z < -1.15)
Using excel function:
= 1 - NORM.S.DIST(-1.15, 1)
= 0.8749
(g) P(−1.60 ≤ Z ≤ 2.00)
= P(z < 2) - P(z < -1.6)
Using excel function:
= NORM.S.DIST(2, 1) - NORM.S.DIST(-1.6, 1)
= 0.9225
(h) P(1.63 ≤ Z ≤ 2.50)
= P(z < 2.5) - P(z < 1.63)
Using excel function:
= NORM.S.DIST(2.5, 1) - NORM.S.DIST(1.63, 1)
= 0.0453
(i) P(1.60 ≤ Z) = P(z > 1.6)
= 1 - P(z < 1.6)
Using excel function:
= 1 - NORM.S.DIST(1.6, 1)
= 0.0548
(j) P(|Z| ≤ 2.50)
= P(-2.5 < z < 2.5)
= P(z < 2.5) - P(z < -2.5)
Using excel function:
= NORM.S.DIST(2.5, 1) - NORM.S.DIST(-2.5, 1)
= 0.9876
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