Probability Calculations and Z-Score Analysis in Normal Distributions
An assignment on probability theory, covering Z-score calculations and normal distributions.
Aiden Campbell
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Probability Calculations and Z-Score Analysis in Normal Distributions
1. Find the following probabilities based on a standard normal variable Z. Use Table 1. (Round your
answers to 4 decimal places.)
a. P(Z > 0.74) 0.2296
b. P(Z ≤ −1.92) 0.0274
c. P(0 ≤ Z ≤ 1.62) 0.4474
d. P(−0.90 ≤ Z ≤ 2.94) 0.8143
2. The cumulative probabilities for a continuous random variable X are P(X ≤ 10) = 0.42 and P(X ≤
20) = 0.66. Calculate the following probabilities. (Round your answers to 2 decimal places.)
a. P(X > 10) 0.58
b. P(X > 20) 0.34
c. P(10 < X < 20) 0.24
3. Let X be normally distributed with mean μ = 120 and standard deviation σ = 20. Use Table 1.
a. Find P(X ≤ 86). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)
P(X ≤ 86) 0.0446
b. Find P(80 ≤ X ≤ 100). (Round "z" value to 2 decimal places and final answer to 4 decimal
places.)
P(80 ≤ X ≤ 100) 0.1359
c. Find x such that P(X ≤ x) = 0.40. (Round "z" value to 2 decimal places and final answer to
nearest whole number.)
x 115
d. Find x such that P(X > x) = 0.90. (Round "z" value to 2 decimal places and final answer to 1
decimal place.)
x 94.4
1. Find the following probabilities based on a standard normal variable Z. Use Table 1. (Round your
answers to 4 decimal places.)
a. P(Z > 0.74) 0.2296
b. P(Z ≤ −1.92) 0.0274
c. P(0 ≤ Z ≤ 1.62) 0.4474
d. P(−0.90 ≤ Z ≤ 2.94) 0.8143
2. The cumulative probabilities for a continuous random variable X are P(X ≤ 10) = 0.42 and P(X ≤
20) = 0.66. Calculate the following probabilities. (Round your answers to 2 decimal places.)
a. P(X > 10) 0.58
b. P(X > 20) 0.34
c. P(10 < X < 20) 0.24
3. Let X be normally distributed with mean μ = 120 and standard deviation σ = 20. Use Table 1.
a. Find P(X ≤ 86). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)
P(X ≤ 86) 0.0446
b. Find P(80 ≤ X ≤ 100). (Round "z" value to 2 decimal places and final answer to 4 decimal
places.)
P(80 ≤ X ≤ 100) 0.1359
c. Find x such that P(X ≤ x) = 0.40. (Round "z" value to 2 decimal places and final answer to
nearest whole number.)
x 115
d. Find x such that P(X > x) = 0.90. (Round "z" value to 2 decimal places and final answer to 1
decimal place.)
x 94.4
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Subject
Statistics