Early on the morning of April 15 1912 the RMS Titanic collid

Early on the morning of April 15, 1912, the RMS Titanic collided with an iceberg as she made her maiden voyage from Southampton UK to New York City USA. Although this British passenger liner was designed with advanced safety features including watertight compartments with remotely activated watertight doors, she only carried enough lifeboats for 1,178 people due to outdated maritime safety regulations. The ship\'s hull plates buckled inwards along her starboard side due to the collision, flooding five of her sixteen watertight compartments. 1,514 of the 2,224 aboard perished, making this sinking one of the deadliest commercial peacetime maritime disasters.

1.Suppose that 30% of those onboard came onboard at Cherbourg. What is the probability (correct to three decimal places) that in a random sample of 30 people onboard between 30% and 38% came onboard at Cherbourg?

2.

Suppose that ticket prices for first class parlor suites were normally distributed with a mean of 890 pounds and a standard deviation of 25 pounds. What is the probability (correct to three decimal places) that a random sample of 30 first class parlor suite passengers would have paid an average of more than 900 pounds for their tickets?

3.

Suppose that 30% of those onboard came onboard at Cherbourg. What is the probability (correct to three decimal places) that at most 4 of 10 randomly selected people onboard came onboard at Cherbourg?

Solution

1.

Note that

u = n p = 9
s = sqrt(n p (1 - p)) = 2.50998008

and the upper and lower bounds of the actual numbers are

x1 = 0.30*30 = 9
x2 = 0.38*30 = 11.4

Thus, we first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    9      
x2 = upper bound =    11.4      
u = mean = 9
n = sample size =    30      
s = standard deviation =    2.50998008      
          Thus, the two z scores are          
          
z1 = lower z score =    0      
z2 = upper z score =    5.237229366      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.5      
P(z < z2) =    0.999999918      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.5000

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