a) What is the 68-95-99.7 Rule and WHEN does it apply? 2. Based on long term dat

Question

a) What is the 68-95-99.7 Rule and WHEN does it apply? 2. Based on long term data, the time to travel a particular bus route is 1 hour with a standard deviation of 10 minutes. Sketch the density curve of the travel time distribution assuming a normal distribution. Use the rule in a) to put the correct scale on the x-axis. 3. a) Define the z-transformation for a Normal distribution within a population, defining all the terms used. b) An individual run of this bus took 65 minutes. What is the z-score of this trip? What proportion of trips is likely to take longer than this?

Explanation / Answer

a)

As per the 68-95-99.7 rule, for a given data set,  

68% observations lie between x-s and x+s

95% observations lie between x-2s and x+2s

99.7% observations lie between x-3s and x+3s

where

x= Mean

s = Standard Deviation

2)

M = 60

S = 10

On X-axis plot the values of X-3S to X+3S

i.e. 30 to 90 where the mean/peak value lies at 60

3)

a)

Z-score for X = (X-M)/S

Where X is the data point

M = Population Mean

S= Population Standard Deviation

It is essentially transforming a X value to the standard normal value

b)

X = 65

M = 60

S = 10

Z-score = (65-60)/10 = 0.5

P(Z>0.5) = 1 -P(X<0.5) = 1- 0.6915 = 0.3085

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