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Chapter 7 – Normal Distribution
Recall the Standard Normal:• Since for every value of and , there is a different
normal distribution, we transform a normal random variable to a standard normal distribution with = 0 and = 1 using the formula:
z = x –
• Denoted N(0,1)
• Appendix C-2 allows you to find all of the area under the curve left of z. (Hand-out)
• Shift the point of symmetry to zero by subtracting from x. • Divide by to scale the distribution to a normal with = 1.
Chapter 7 – Normal DistributionExample: Using the Std. Normal transformation• Daily sales at a bicycle shop are normally distributed with mean
$15,000 and standard deviation $4000.• Find the probability that sales will exceed $20,000 on a randomly-
selected day.
• Find the probability that sales will be less than $12,000 on a randomly selected day.
• Find the probability that sales will be between $12,000 and $20,000 on a randomly selected day.
1056.8944.1)25.1(1)(1 40001500020000 ZPZP
P(X > 20000) = 1 – P(X < 20000)
P(X < 12000) = 2266.)75.0()( 40001500012000 ZPZP
P(12000 < X < 20000) = P (X < 20000) – P(X < 12000)
6678.2266.8944.)75.0()25.1( ZPZP
Clickers
If the starting salary for students majoring in Business is normally distributed with a mean of $45,000 and a standard deviation of $5,000, find the probability that the starting salary of a randomly selected student will be less than $50,000.
A = 0.1056
B = 0.3085
C = 0.6915
D = 0.8413
Clickers
If the starting salary for students majoring in Business is normally distributed with a mean of $45,000 and a standard deviation of $5,000, find the probability that the starting salary of a randomly selected student will be at least $50,000.
A = 0.1056
B = 0.1587
C = 0.8413
D = 0.8944
Clickers
If the starting salary for students majoring in Business is normally distributed with a mean of $45,000 and a standard deviation of $5,000, find the probability that the starting salary of a randomly selected student will be between $45,000 and $50,000.
A = 0.3413
B = 0.3944
C = 0.8413
D = 0.8944
Chapter 7 – Normal Distribution
Basis for the Empirical Rule:
• Approximately 68% of the area under the curve is between + 1
• Approximately 95% of the area under the curve is between + 2
• Approximately 99.7% of the area under the curve is between + 3
Chapter 7 – Normal Distribution
Finding z for a Given Area:• Appendices C-1 and C-2 be used to find the
z-value corresponding to a given probability.• For example, what z-value defines the top 1% of a
normal distribution?• This implies that 99% of the area lies to the left of z.• Or that 1% of the area lies to the left of –z.• Or that 49% of the area lies between 0 and z.
Chapter 7 – Normal Distribution
Finding z for a Given Area:
• Look for an area of .4900 in Appendix C-1 (or for an area of 0.9900 in the Appendix C-2 – the handout):
• Without interpolation, the closest we can get is z = 2.33
Chapter 7 – Normal Distribution
, xz 282.17
75x
974.83 x
Finding Areas by Standardizing:• Suppose John took an economics exam on which the
class mean was 75 with a standard deviation of 7. What score would place John in the upper 10th percentile?
• Find the value of x such that P(X > x) = .10 or P(X < x) = 0.90.
• From the previous slide, we know P(Z > 1.282) = .10.
• Since 974.875x
• A score of 84 or better would place John in the top 10% of his class.
Clickers
Suppose the starting salary for students majoring in Business is normally distributed with a mean of $45,000 and a standard deviation of $5,000. If Jane Wants a starting salary in the top 25%, approximately what salary should she negotiate for?
A = $56,630
B = $54,800
C = $53,225
D = $51,410
E = $48,375