Answer the following questions about the normal distribution: (a) What percentag
ID: 3049289 • Letter: A
Question
Answer the following questions about the normal distribution:
(a) What percentage of the area under the curvr is between the mean and the right most end of the curve?
(b) What percentge of the area under the curve is within one standard deviation of the mean (on either side of the mean)?
2. Using the Unit Normal Table, determine the following:
(a) Find the probabilities that correspond to the following z scores: 2,0, 0.5, -0,75, -2.0
(b) Find the z-score that corrspond to the following probabilities: 0.5000, 0.8413, 0.3050
Assume that the following is true: The scale for the SAT is set so that the distribution of scores is approximately normal with mean = 500 and standard deviation = 100.
(a) What is the probability of having an SAT score of 130 or above?
(b) what is the probability of having an SAT score of 120 or above?
(c) What is the probability of having an SAT score of 91 or less?
(d) You think you might need a tutor. You know of a tutoring service for students who score between 350 and 650 on the SAT. You think that you probbably fit within their range.
What is the probability that you will get an SAT score between 350 and 650?
The National Collegiate Athletic Association (NCAA) requires Division I athletes to score at least 820 on the combined mathematics and verbal parts of the SAT exam in order to compare in their first college year. In 1999, the scores of the millions of students taking the SATs where approximatly normal with a mean =1017 and a standard deviation =209. What is the probability of scoring an 820 or less in this distribution?
Explanation / Answer
1 (a) The mean divides the total area into 2 equal halves. So the percentage area between the mean and the right most end of the curve is 50%
(b) The cumulative area under the normal curve upto 1 standard deviation below mean =0.1587
The cumulative area under the normal curve upto 1 standard deviation above mean =0.8423
So the area under the normal curve within 1 standard deviation around the mean = 0.8413 - 0.1587 = 0.6826
The percentge of the area under the curve is within one standard deviation of the mean (on either side of the mean) = 0.6826 x 100 = 68.26%
2. We make use of a standard normal z-table. One can be found here: http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf
(a) The cumulative probabilities for corresponding to z-scores:
2: 0.9772
0.5: 0.6915
-0.75: 0.2266
-2.0: 0.0228
(b) The z-scores corrspond to the probabilities:
0.5000: 0.0
0.8413: 1.0
0.3050: -0.51
Assume that the scale for the SAT is set so that the distribution of scores is approximately normal with mean = 500 and standard deviation = 100.
(a) The probability of having an SAT score of 130 or above = P(X>130) = P(Z> (130-500)/100 ) = P(Z>-3.7) = 1.
(b) The probability of having an SAT score of 120 or above is slightly more than the probability of having an SAT score of 130 or above, but since we have approximated the score of more than 130 to 1.0, the probability of a score of more than 120 is also 1
(c) The probability of having an SAT score of 91 or less is almost 0, because the z-value corresponding to this score of 91 is (91-500)/100 = -4.09 and the probability of having a z-score lesser than -18.29 is almost 0 as can be seen from the standard normal table since there are no values corresponding to it (the probability is almost 0 below -3.5
(d) The probability of getting an SAT score between 350 and 650 = P(350 < X < 650) = P( (350-500)/100) < Z < (650-500)/sqty100) = P( -1.5 < Z < 1.5) = P (Z<1.5) - P(Z<-1.5) = 0.9332 - 0.0668 = 0.8664
3. P(X<820) = P(Z< (820-1017)/209 ) = P(Z< -0.943) = 0.1736 (as read from the standard normal table)