Question

The average daily rate of a hotel in Canada as of August 2018 was $172.75. Assume the average daily rate follows a normal distribution with a standard deviation of $28.60.

Standard Normal Distribution Table ( note you need access to this table in order to do this question)

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 $205
P(X > 205)=P(X > 205)=

(iii) Between $145 and $185
P(145 < X < 185)=P(145 < X < 185)=

b. Determine the average daily rates that separate the:

(i) top 7% of average daily rates from the rest of the daily rates or from the bottom 93% of average daily rates
x=x=

Round to 2 decimal places.

(ii) bottom 25% of average daily rates from the rest of the daily rates
x=x=

Round to 2 decimal places.

(iii) middle 70% of average daily rates from the rest of the daily rates

  < x <  < x <  

Round to 2 decimal places.

h(i,x)=

{10if  program i halts on input x,otherwise.{1if  program i halts on input x,0otherwise.

Answer 1

Solution :

Given that ,

mean = = 172.75

standard deviation = = 28.60

(i)  P(x < 175 ) = P[(x - ) / < ( 175 - 172.75 ) / 28.60 ]

= P(z < 0.08)

Using z table,

= 0.5319

Probability = 0.5319

(ii)

P(x > 205 ) = 1 - P( x < 205)

=1- P[(x - ) / < ( 205 - 172.75 ) / 28.60 ]

=1- P(z < 1.13 )

Using z table,

= 1 - 0.8708

= 0.1292

Probability = 0.1292

(iii)

P( 145 < x < 185)

= P[( 145 - 172.75 ) / 28.60 ) < (x - ) /  < ( 185 - 172.75 ) / 28.60 ) ]

= P( -0.97 < z < 0.43 )

= P(z < 0.43 ) - P(z < -0.97)

Using z table,

= 0.6664 - 0.1660

= 0.5004

Probability = 0.5004

(i)

The z - distribution of the 7% is

P(Z > z) = 7%

= P(Z < z ) = 0.07

= 1 - P(Z < z) = 1 - 0.07

= P(Z < z ) = 0.93

= P(Z < 1.476 ) = 0.93

z = 1.476

Using z-score formula,

x = z * +

x = 1.476 * 28.60 + 172.75

x = 214.9636

x = 214.96

(ii)

The z - distribution of the 25% is

P(Z < z) = 25%

= P(Z < z) = 0.25  

= P(Z < -0.6745 ) = 0.25

z = -0.6745

Using z-score formula,

x = z * +

x = -0.6745 * 28.60 + 172.75

x = 153.4593

x = 153.46

(iii)

Middle 70%

= 1 -70%  

= 1 - 0.70 = 0.30

/2 = 0.15

1 - /2 = 1 - 0.15 = 0.85

Z/2 = Z0.15 = -1.036

Z1 - /2 = Z0.85 = 1.036

Using z-score formula,

x = z * +

x = -1.036 * 28.60 + 172.75

x = 143.12

Using z-score formula,

x = z * +

x = 1.036 * 28.60 + 172.75

x = 202.38

Answer = 143.12 < x < 202 .38