After the running of the Toronto International Marathon last year, the following

Question

After the running of the Toronto International Marathon last year, the following data was compiled for all persons who completed it:

All runners N= 13,280

Mean time = 250 minutes

Standard deviation = 68 minutes

All men

N = 7,150

Mean time = 240 minutes

Standard deviation = 70

All women

N = 6,130

Mean time = 260 minutes

Standard deviation 66 minutes

Let us assume, for estimation purposes, that the distribution of runners by time is a perfect normal distribution for all three distributions.

1a. What proportion of all runners ran between the mean time and 245 minutes? (5 marks)

1b. About how many men likely ran a sub-2.5 hour marathon (i.e. 150 minutes or less)? (5 marks) Hint: if you have the proportion, you can calculate the corresponding number

1c. What percentage of all runners likely ran a marathon of between 175 minutes and 185 minutes)? (5 marks)

Explanation / Answer

1a)

proportion of all runners ran between the mean time and 245 minutes :

1b)

proportion of men  ran a sub-2.5 hour marathon :

hence number of men  ran a sub-2.5 hour marathon =np=7150*0.0993 =~710

1c)

percentage of all runners likely ran a marathon of between 175 minutes and 185 minutes:

for normal distribution z score =(X-)/ here mean=       = 250.000 std deviation   == 68

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