Question 933350: the mean is 50, and standard deviation is 6. what can you say about the fraction of numbers that lie between 32-68?
find the least possible percentage of numbers in a data set lying within 3/2 standard deviation of the mean?
I need to fingure these our for a final This is not the fianl but if i can figure some of these out I can plug the numbers in my instruct said. I have many more im struggling so bad
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first question:
mean is 50
standard deviation is 6.
fraction that lies between 32 and 68 is calculated as follows:
first you have to find the z-score, then you find the percent using the z-score table.
the formuls for z-score is z = (x-m) / s, where z = the z-score and x is the raw score and m is the mean and s is the stnadard deviation.
you want the z-score for 32 and 68.
formula for raw score of 32 is z = (32 - 50) / 6 which gets you z = -3
formula for raw score of 68 is z = (68 - 50) / 6 which gets you z = 3
you are looking for the area under the normal distribution curve that is between a z-score of -3 and a z-score of 3.
look in the z-score table to find the percent of the area to the left of the z-score.
from the table, you will find that the ratio to the left of -3 is equal to .0013 and the ratio0 to the left of 3 is equal to .9987.
subtract the smaller ratio from the larger ratio to get the ratio that is between -3 and 3 is equal to .9974.
a ratio of .9974 means that 99.74% of the area under the normal distribution curve is between a z-score of -3 and 3.
this means that the fraction of numbers between 32 and 68 is equal to .9987 which means that 99.87 of all the numbers in the data set that gave you a mean of 50 with a standard deviation of 6 can be found between 32 and 68.
if you grabbed a number at random from the data set, you would find that the number was between 32 and 68 99.74% of the number of times that you grabbed, and would be less than 32 and greater than 68 only .26% of the number of times that you grabbed.
put another way, if you grabbed a number at random 10,000 times, you would get a number between 32 and 68 for 9974 of those times and get a number less than 32 or greater than 68 only 26 of those times.
second question:
if the number lies with 3/2 standard deviations from the mean, then you would calculate the percent as follows.
a z-score tells you the number of standard deviations you are from the mean.
a z-score of 3 tells you that you are 3 standard deviations above the mean.
a z-score of -3 tells you that you are 3 standard deviations below the mean.
we solved for that in the first problem.
in this problem, you are looking for a z-swcore of between -3/2 and 3/2 because:
a z-score of 3/2 means you are 3/2 standard deviations above the mean.
a z-score of -3/2 means you are 3/2 standard deviations below the mean.
same drill.
look for a z-score of 3/2 in the z-score table and find the ratio to the left of that z-score.
look for a z-score of -3/2 in the z-score table and find the ratio to the left of that z-score.
subtract the smaler ratio from the larger ratio and you have the ratio between those z-scores.
multiply by 100 and you have the percent of the area under the normal distribution curve that is between the z-scores of -3/2 and 3/2.
a z-score of 3/2 is equal to a z-score of 1.5.
a z-score of -3/2 is equal to a z-score of -1.5.
area to the left of a z-score of 1.5 is equal to .9332
area to the left of a z-score of -1.5 is equal to .0668
.9332 - .0668 = .8664
this is the ratio or proportion of the area under the normal distribution curve that is between a z-score of -1.5 and 1.5.
the percent is equal to 100 * .8664 which is equal to 86.64
the z-score table i am using can be found here:\
http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf
the area shown is the area that is to the left of the z-score.
that area is the ratio of the area to the left of the z-score to the total area under the normal distribution curve.
the total area under the normal distribution curve is always equal to 1.
the ratio multiplied by 100 is equal to the percent.
|
|
|
