Show all work in order to receive credits. The scores on an undergraduate statis
ID: 3269428 • Letter: S
Question
Show all work in order to receive credits. The scores on an undergraduate statistics test are normally distributed with a mean of 73 and a standard deviation of 7.
a) what is the probability that a randomly selected student received a score of 88 or above?
b) what is the probability that a randomly selected student receives a score between 70 and 80 on this exam?
c) what score on the statistics test is the 75th percentile?
d) suppose that your professor added six points to everyone's test score. What is the new mean test score? What is the new standard deviation of test scores?
Explanation / Answer
Here the mean of test scores = 73
Standard deviation = 7
(a) Probabillity that a students received a score of 88 or above
Pr (X >= 88; 73; 7) = 1 - Pr(X <88 ; 73; 7)
Z = (88 - 73)/ 7 = 2.143
so Pr (X >= 88; 73; 7) = 1- (2.143)
where is the cumulative normal probability distribution.
Pr (X >= 88; 73; 7) = 1- 0.9838 = 0.0162
(b) Here we have to calculate
Pr ( 70 <= X < = 80; 73; 7) = Pr(X <80 ; 73; 7) - Pr(X <70 ; 73; 7)
calculation Z score : Z1 = (80 - 70)/ 7 = 1
Z2 = (70 - 73)/ 7 = -0.4285
Pr ( 70 <= X < = 80; 73; 7) = (1) - ( -0.43)
where is the cumulative normal probability distribution.
Pr ( 70 <= X < = 80; 73; 7) = 0.8413 - 0.3336 = 0.5077
(c) 75% percentile means
Pr (x <= X ; 73; 7) = 0.75
as per Z - table the relative Z -value = 0.675
(x - 73)/7 = 0.675
X = 73 + 7 * 0.675 = 77.725
(d) New mean test score = 73 + 6 = 79
adding six points will not change the standard deviation so new standard deviation = 7