Concept explainers
Find the standard normal area for each of the following. Sketch the normal curve and shade in the area represented below.
- a. P(Z < −1.28)
- b. P(Z > 1.28)
- c. P(−1.96 < Z < 1.96)
- d. P(−1.65 < Z < 1.65)
a.
Find the standard normal area for
Answer to Problem 17SE
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-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 0.08.
Thus,
Hence, the standard normal area for
b.
Find the standard normal area for
Answer to Problem 17SE
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 1.2 in column of z.
- Locate the probability value corresponding to z-value 1.2 in the column 0.08.
Thus,
Hence,
Hence, the standard normal area for
c.
Find the standard normal area for
Answer to Problem 17SE
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 1.9 in column of z.
- Locate the probability value corresponding to z-value 1.9 in the column 0.06.
Thus,
Calculate the value of
From Appendix C-2: Table “CUMULATIVE STANDARD NORMAL DISTRIBTION”,
- Locate the z value –1.9 in column of z.
- Locate the probability value corresponding to z-value –1.9 in the column 0.06.
Thus,
Hence,
Hence, the standard normal area for
c.
Find the standard normal area for
Answer to Problem 17SE
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 1.6 in column of z.
- Locate the probability value corresponding to z-value 1.6 in the column 0.05.
Thus,
Calculate the value of
From Appendix C-2: Table “CUMULATIVE STANDARD NORMAL DISTRIBTION”,
- Locate the z value –1.6 in column of z.
- Locate the probability value corresponding to z-value –1.6 in the column 0.05.
Thus,
Hence,
Hence, the standard normal area for
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