Question

1. The high temperature in Chicago for the month of August have a normal distribution with a mean μ= 80 F and a standard deviation of  σ= 8 F.

A. Using this distribution, what percent of days in August would you expect to have a high temperature over 95 F?

B. What percentage of days would you expert the high temperature to be between 85 and 95 F.?

C. What would be the high temperature required for a day to be in the bottom 10% of the distribution?

D. Determine the temperature range that marks the middle 60% of high temperatures in August?

Answer 1

Solution :

Given that,

mean = = 80

standard deviation = = 8

A ) P ( x > 95 )

= 1 - P (x < 95 )

= 1 - P ( x -  / ) < ( 95 - 80 / 8 )

= 1 - P ( z < 15 / 8 )

= 1 - P ( z < 1.87)

Using z table

= 1 - 0.9693

= 0.0337

Probability = 0.0337

B ) P (85 < x < 95 )

P ( 85 - 80 / 8 ) <( x -  / ) < ( 95 - 80 / 8 )

P ( 5 / 8 < z < 15 / 8 )

P ( 0. 62 < z < 1.87)

P ( z < 1.87) - P ( z < 0.62 )

Using z table

= 0.9693 - 0.7324

= 0.2369

Probability = 0.2369

C ) P(Z <  z) = 10%

P(Z < z) = 0.10

P(Z < -1.282 ) = 0.10

Using standard normal table,

z = -1.28

Using z-score formula,

x = z * +

x = -1.28 * 8 + 80

= 69.76

The high temperature required for a day is 69.76 F

D ) P(-z < Z < z) = 60%
P(Z < z) - P(Z < z) = 0.60
2P(Z < z) - 1 = 0.60
2P(Z < z ) = 1 + 0.60
2P(Z < z) = 1.60
P(Z < z) = 1.60 / 2
P(Z < z) = 0.80

Using standard normal table,
z = 0.84 znd z = - 0.84

Using z-score formula,

x = z * +

x = 0.84 * 8 + 80

= 86.72

The Maximum temperature required for a day is 86.72 F in August

Using z-score formula,

x = z * +

x = - 0.84 * 8 + 80

= 73.28

The Minimum temperature required for a day is 73.28 F in August