Math Problem Statement
In each case, determine the value of the constant c that makes the probability statement correct. (Round your answers to two decimal places.)
(a) Φ(c) = 0.9850
(b) P(0 ≤ Z ≤ c) = 0.2939
(c) P(c ≤ Z) = 0.1292
(d) P(−c ≤ Z ≤ c) = 0.6528
(e) P(c ≤ |Z|) = 0.0128
Solution
We need to determine the value of the constant that makes each probability statement correct. For these types of problems, we are dealing with the standard normal distribution , which has a mean of 0 and a standard deviation of 1. We can use the cumulative distribution function (CDF) , the properties of the standard normal distribution, and tables or technology to find the values of .
(a)
represents the cumulative distribution function for a standard normal distribution, meaning gives the probability that is less than or equal to .
We need to find such that: Using a standard normal table or calculator, we find that:
(b)
This represents the probability that lies between 0 and , so we need to find such that: Since , we have: Using the standard normal table:
(c)
This represents the probability that is greater than or equal to . This is the complementary probability of the cumulative distribution: So, we have: Using the standard normal table:
(d)
This represents the probability that lies between and , which is: Solving for : Using the standard normal table:
(e)
This represents the probability that is greater than or equal to in absolute value, meaning: Thus: Using the standard normal table:
Final answers:
- (a)
- (b)
- (c)
- (d)
- (e)
Would you like more details on any of these solutions? Here are 5 related questions to expand the topic:
- How can you calculate probabilities using the standard normal distribution without a table?
- What is the relationship between the CDF and the probability for a normal distribution?
- How can we interpret the value of in real-world applications?
- How does the symmetry of the normal distribution affect these types of problems?
- What is the meaning of and when is it useful?
Tip: Always remember that the cumulative distribution function gives the area under the curve from to , representing the probability that a standard normal variable is less than or equal to .
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Math Problem Analysis
Mathematical Concepts
Probability
Standard Normal Distribution
Cumulative Distribution Function (CDF)
Formulas
Φ(c): cumulative distribution function of the standard normal distribution
P(a ≤ Z ≤ b): probability that Z lies between a and b
P(Z ≥ c) = 1 - Φ(c)
P(-c ≤ Z ≤ c) = 2Φ(c) - 1
Theorems
Properties of the Standard Normal Distribution
Cumulative Distribution Function (CDF)
Suitable Grade Level
College-level Probability or Advanced High School Statistics
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