Question

The average daily rate of a hotel in Canada as of August 2018 was $182.65. Assume the average daily rate follows a normal distribution with a standard deviation of $25.70.

Standard Normal Distribution Table

a. What is the probability that the average daily rate of a Canadian hotel will be:

(i) less than $175
P(X < 175)=P(X < 175)=

(ii) more than $200
P(X > 200)=P(X > 200)=

(iii) Between $150 and $190
P(150 < X < 190)=P(150 < X < 190)=

b. Determine the average daily rates that separate the:

(i) top 4% of average daily rates from the rest of the daily rates or from the bottom 96% of average daily rates
x=x=

(ii) bottom 15% of average daily rates from the rest of the daily rates
x=x=

(iii) middle 65% of average daily rates from the rest of the daily rates

  < x <  < x <  

Please provide 6 decimal places.

Please provide correct answers. Thanks.

Answer 1

Solution :

Given that ,

mean = = 182.65

standard deviation = = 25.70

P(x < 175 ) = P[(x - ) / < ( 175 - 182.65 ) / 25.70 ]

= P(z < -0.30 )

Using z table,

= 0.3821

Probability = 0.3821

2 .

P(x > 200 ) = 1 - P( x < 200)

=1- P[(x - ) / < ( 200 - 182.65 ) / 25.70]

=1- P(z < 0.68 )

Using z table,

= 1 - 0.7517

= 0.2483

Probability = 0.2483

3 .

P( 150 < x < 190 )

= P[( 150 - 182.65) / 25.70) < (x - ) /  < ( 190 - 182.65 ) / 25.70 ) ]

= P( -1.27 < z < 0.29)

= P(z < 0.29 ) - P(z < -1.27 )

Using z table,

= 0.6141 - 0.1020

= 0.5121

Probability = 0.5121

( b )

1 )

The z - distribution of the 4% is

P(Z > z) = 4%

= 1 - P(Z < z ) = 0.04

= P(Z < z ) = 1 - 0.04  

= P(Z < z ) = 0.96

= P(Z < 1.751) = 0.96

z = 1.751

Using z-score formula,

x = z * +

x = 1.751 * 25.70 + 182.65

x = 227.65

2.)

The z - distribution of the 15%

P(Z < z) = 15%

= P(Z < z ) = 0.15.

= P(Z < -1.036 ) = 0.15.  

z = -1.036

Using z-score formula,

x = z * +

x = -1.036 * 25.70 + 182.65

x = 156.02

3 )

middle 65%

= 1 - 65%  

= 1 - 0.65 = 0.35

/2 = 0.175

1 - /2 = 1 - 0.157 = 0.825

Z/2 = Z0.175 = -0.935

Z1 - /2 = Z 0.825 = +0.935

Using z-score formula,

x = z * +

x = -0.935 * 25.70 + 182.65

x = 158.620500

Using z-score formula,

x = z * +

x = 0.935 * 25.70 + 182.65

x = 206.679500

middle 65% of average daily rates from the rest of the daily rates

158.620500 < x <  206.679500