Question

11.   The high temperature in Chicago for the month of August is approximately normally distributed with mean µ = 80 F and σ = 8 F. Draw a picture of the distribution, write the z-score, and use the appropriate notation.

a.   What is the middle 99.7% of the data?
b.   What is the probability that a randomly selected day in August has a temperature less than 78 F
c.   What is the probability that the temperature is between 80o and 83o for the same sample?
d.   What is the probability that P(X≥90)?
e.   What temperature divides the lower 80% from the upper 20% of temperatures?

Answer 1

Part a)

P ( a < X < b ) = 0.997
Dividing the area 0.997 in two parts we get 0.997/2 = 0.4985
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.4985
Area above the mean is b = 0.5 + 0.4985
Looking for the probability 0.0015 in standard normal table to calculate critical value Z = -2.97
Looking for the probability 0.9985 in standard normal table to calculate critical value Z = 2.97

-2.97 = ( X - 80 ) / 8
a = 56.24
2.97 = ( X - 80 ) / 8
b = 103.76
P ( 56.24 < X < 103.76 ) = 0.997

Part b)

P ( X < 78 )
Standardizing the value

Z = ( 78 - 80 ) / 8
Z = -0.25

P ( X < 78 ) = P ( Z < -0.25 )
P ( X < 78 ) = 0.4013

part c)

P ( 80 < X < 83 )
Standardizing the value

Z = ( 80 - 80 ) / 8
Z = 0
Z = ( 83 - 80 ) / 8
Z = 0.38
P ( 0 < Z < 0.38 )
P ( 80 < X < 83 ) = P ( Z < 0.38 ) - P ( Z < 0 )
P ( 80 < X < 83 ) = 0.6462 - 0.5
P ( 80 < X < 83 ) = 0.1462

part d)

P ( X >= 90 ) = 1 - P ( X < 90 )
Standardizing the value

Z = ( 90 - 80 ) / 8
Z = 1.25

P ( Z > 1.25 )
P ( X >= 90 ) = 1 - P ( Z < 1.25 )
P ( X >= 90 ) = 1 - 0.8944
P ( X >= 90 ) = 0.1056

Part e)

Lowest 80%

P ( X < ? ) = 80% = 0.8
Looking for the probability 0.8 in standard normal table to calculate critical value Z = 0.84

0.84 = ( X - 80 ) / 8
X = 86.72
P ( X < 86.72 ) = 0.8

Upper 20%

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.2 = 0.8
Looking for the probability 0.8 in standard normal table to calculate critical value Z = 0.84

0.84 = ( X - 80 ) / 8
X = 86.72
P ( X > 86.72 ) = 0.2
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