The length of time that it takes for a ferry to reach a summer resort from the mainland is approximately normally distributed with a mean of 2 hours and a standard deviation of 12 minutes. Over many past trips, what proportion (area) of times has the ferry reaches the island in A) less than 1 hour, 45 minutes? B) more than 2 hours, 5 minutes? C) between 1 hour, 50 minutes and 2 hours, 20 minutes?

A) 0.10565

B) 0.33846

C)

Step-by-step explanation:

The length of time that it takes for a ferry to reach a summer resort from the mainland is approximately normally distributed with a mean of 2 hours and a standard deviation of 12 minutes.

Note that, all units must be in minutes

We solve this question using z score formula

= z = (x-μ)/σ

where

x is the raw score

μ is the population mean = 2 hours

Converting to minutes

1 hour = 60 minutes

2 hour = x

Cross Multiply

x = 2 × 60 minutes

x = 120 minutes

Mean = 120 minutes

σ is the population standard deviation = 12 minutes

Over many past trips, what proportion (area) of times has the ferry reaches the island in

A) less than 1 hour, 45 minutes?

Converting 1 hour 45 minutes to minutes

60 minutes = 1 hour

Hence:

60 + 45 minutes = 105 minutes

For x < 105 minutes

= 105 - 120/12

= -1.25

Probability value from Z-Table:

P(x<105) = 0.10565

B) more than 2 hours, 5 minutes?

Converting 2 hours 5 minutes to minutes

Hence:

1 hour = 60 minutes

2 hour = x

Cross Multiply

x = 2 × 60 minutes

x = 120 minutes

Hence:

120 + 5 minutes = 125 minutes

For x < 125 minutes

z = 125 - 120/12

z = 0.41667

Probability value from Z-Table:

P(x<125) = 0.66154

P(x>125) = 1 - P(x<125) = 0.33846

C) between 1 hour, 50 minutes and 2 hours, 20 minutes?

Converting 1 hour, 50 minutes to minutes

= 60 minutes = 1 hour

Hence:

60 + 50 minutes = 110 minutes

For x = 110 minutes

= 110 - 120/12

= -0.83333

Probability value from Z-Table:

P(x = 110) = 0.20233

Converting 2 hours, 20 minutes to minutes

= 1 hour = 60 minutes

2 hour = x

Cross Multiply

x = 2 × 60 minutes

x = 120 minutes

Hence:

120 + 20 minutes = 140 minutes

For x = 140minutes

z = 140 - 120/12

=1.66667

Probability value from Z-Table:

P(x = 140) = 0.95221