Concept explainers
Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used.
- a. P(−1.22 < Z < 2.15)
- b. P(−3.00 < Z < 2.00)
- c. P(Z < 2.00)
- d. P(Z = 0)
a.
Find the standard normal area for
Answer to Problem 15SE
The standard normal area for
Explanation of Solution
Calculation:
Normal distribution:
A continuous random variable X is said to follow normal distribution if the probability density function of X is,
Standard normal distribution:
A continuous random variable Z is said to follow standard normal distribution if the probability density function of Z is,
For a standard normal variable it is known that,
The probability
Calculate the value of
From Appendix C-1: Table “STANDARD NORMAL AREAS”,
- Locate the z value 2.1 in column of z.
- Locate the probability value corresponding to z-value 2.1 in the column 0.05.
Thus,
Calculate the value of
From Appendix C-2: Table “CUMULATIVE STANDARD NORMAL DISTRIBTION”,
- Locate the z value –1.2 in column of z.
- Locate the probability value corresponding to z-value –1.2 in the column 0.02.
Thus,
Hence,
Hence, the standard normal area for
b.
Find the standard normal area for
Answer to Problem 15SE
The standard normal area for
Explanation of Solution
Calculation:
For a standard normal variable it is known that,
The probability
Calculate the value of
From Appendix C-1: Table “STANDARD NORMAL AREAS”,
- Locate the z value 2.0 in column of z.
- Locate the probability value corresponding to z-value 2.0 in the column 0.00.
Thus,
Calculate the value of
From Appendix C-2: Table “CUMULATIVE STANDARD NORMAL DISTRIBTION”,
- Locate the z value –3.0 in column of z.
- Locate the probability value corresponding to z-value –3.0 in the column 0.00.
Thus,
Hence,
Hence, the standard normal area for
c.
Find the standard normal area for
Answer to Problem 15SE
The standard normal area for
Explanation of Solution
Calculation:
For a standard normal variable it is known that,
The probability
From part (b), it is found that
Hence,
Hence, the standard normal area for
d.
Find the standard normal area for
Answer to Problem 15SE
The standard normal area for
Explanation of Solution
Calculation:
Calculate the value of
From Appendix C-1: Table “STANDARD NORMAL AREAS”,
- Locate the z value 0.00 in column of z.
- Locate the probability value corresponding to z-value 0.00 in the column 0.00.
Thus,
Moreover, it can be said that in case of continuous distribution the probability value for a particular point is 0.
Therefore, the standard normal area for
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