Question 421445: Please Help with this question urgently....:(
The speed of motorists on a highway is normally distributed with a mean of 120 miles/hour and variance of 240. If the metro-police wants to stop the 5% fastest motorists, what is the cut-off speed so that all cars which travel at a speed faster than the cut-off speed, will be stopped?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean is 120 mph
variance is 240.
standard deviation would be square root of the variance = sqrt(240) = 15.49193338
simplest way is to use the calculator from the following website.
http://davidmlane.com/hyperstat/z_table.html
use the bottom calculator from that website.
enter a mean of 120.
enter a standard deviation of 15.49 (rounded to nearest hundred)
enter shaded area of .05
click on above
answer will be above 145.4788 mph which you would round to be simply 145 mph.
if you do not have this calculator, then you need to use your z-tables.
first you have to go into the z-table and find out the z-score of the value that is equivalent to .95 of the area under the curve.
.95 means that 95% of your scores will be below that number.
this means that 5% of your scores will be above that number.
from your z-table, you should be able to find this value to be a z-score of 1.645.
that means that you would need a raw score that is 1.645 z-scores above the mean.
since your standard deviation is the equivalent of 1 z-score, then 1.645 times your standard deviation would need to be added to your mean to give you the raw score.
you would get 120 + 1.645 * (15.49) = 145.48105 mph which you would round down to 145 mph.
the police would be looking to ticket motorist that exceed 145 mph.
are you sure it's mph and not kmph (kilometers per hour).
kmph would make more sense, since 120 kmph is equivalent to approximately 75 miles per hour (a more reasonable speed) and 245 kmph is equivalent to approximately 90 mph.
one z-score table is attached here for your reference.
http://www.sjsu.edu/faculty/gerstman/EpiInfo/z-table.htm
in this table (they don't all work the exact same way), you would look for a value of .95 in the table and you would then reference the first column number and add to it the top row number.
your actual value is somewhere in between, so some interpolation would need to be performed.
in this table, i found a value of .9495 and .9505 that were bracketing the .95 that i was looking for.
.9495 correlated to 1.64
.9505 correlated to 1.65
i interpolated that to be equal to 1.645 since the number i was looking for (.95) was exactly in between those other 2 numbers.
with a z-table, the mean is always 0 and the standard deviation is always 1.
to convert from a raw score to a z-score, you use the following formula:
z-score = (raw score minus mean) / standard deviation.
example:
raw score = 100 and mean = 100 and standard deviation = 10
you would get a z-score of 0 because (100-100)/10 = 0
raw score = 110 and mean = 100 and standard deviation = 10
you would get a z-score of 1 because (110-100)/10 = 1.
the z-score tells you how many standard deviations you are above or below the mean.
raw score = 90 and mean = 100 and standard deviation = 10
you would get a z-score of -1 because (90-100)/10 = -1.
similarly, if you have the z-score, you can then correlate to the raw score by simply reversing the process as we did up above.
example:
z-score = 2 and mean = 100 and standard deviation = 10
raw score would be equal to 2 * 10 + 100 = 120.
reverse this and you get a z-score of (120-100) / 10 = 20/10 = 2.
a z-score of 2 means you are 2 standard deviations above the mean.
use of the z-table calculator is definitely easier but you should know how to use the z-tables as well.
|
|
|
