We are told that the running times in a race are normally distributed with a mean of 63 minutes and a standard deviation of 4 minutes. To qualify, a runner has to be in the fastest 16 % of all applicants. We are asked to find the qualifying time.
Qualifying time=?
To do that we will draw the probability distribution. We can begin by finding the times that are one, two, and three standard deviations away from the mean. For convenience, the mean will be represented by the letter m and the standard deviation will be represented by s.
m-3s
m-2s
m-s
m
m+s
m+2s
m+3s
Substitute
63-3( 4)
63-2( 4)
63- 4
63
63+ 4
63+2( 4)
63+3( 4)
Simplify
51
55
59
63
67
71
75
Now, let's draw vertical lines with the calculated values.
Finally, we can sketch the normal curve. The highest point of the curve should be at the mean, 63
The normal curve is divided into sections of standard deviation widths. Let's label the percentages of each section.
Now, let's use this information to find the section with the best 16 % of the times. Because the graph is symmetric we know that the right half of the distribution consists of 50 % of the data.
Therefore, 50 % +34 %=84 % of the times are greater than or equal to 59 minutes.
This means that the fastest 100 % -84 % =16 % of the times are lesser than or equal to 59 minutes!
To be in the fastest 16 % of the runners, an applicant has to finish the race under 59 minutes. Therefore, the qualifying time is also 59 minutes.
