Can you please explain to me question # 8 please? Thank you Let X be a geometric
ID: 3126663 • Letter: C
Question
Can you please explain to me question # 8 please?
Thank you
Let X be a geometric(p) random variable for some p (0,1). For n, k N, show that P(X=n+k|X > n)= k). Now, explain this property using the interpretation of X as the first successful trial among independent trials each with success probability p. 8. 355 students are enrolled in a Math 20B class at UC San Diego. At the end of the quarter, the combined scores are approximately normally distributed with mean 72 and standard deviation 15. (a) The top 10% of the class receive an A. What is the grade cutoff for an A? (b) The grade cutoff for a B is 80. In a group of 10 randomly chosen students, what is the probability that at least one got a B? (c) How many students got bonus points that resulted in a combined score 100?Explanation / Answer
a)
First, we get the z score from the given left tailed area. As
Left tailed area = 1 - 0.10 = 0.9
Then, using table or technology,
z = 1.281551566
As x = u + z * s,
where
u = mean = 72
z = the critical z score = 1.281551566
s = standard deviation = 15
Then
x = critical value = 91.22327348 [ANSWER]
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b)
Getting a B means 80 to 91.223, as above.
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 80
x2 = upper bound = 91.223
u = mean = 72
s = standard deviation = 15
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = 0.533333333
z2 = upper z score = (x2 - u) / s = 1.281533333
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.703098571
P(z < z2) = 0.8999968
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.19689822
Note that the probability of x successes out of n trials is
P(n, x) = nCx p^x (1 - p)^(n - x)
where
n = number of trials = 10
p = the probability of a success = 0.19689822
x = the number of successes = 0
Thus, the probability is
P ( 0 ) = 0.111610714
Thus, P(at least one) = 1 - P(0) = 0.888389286 [ANSWER]
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c)
We first get the z score for the critical value. As z = (x - u) / s, then as
x = critical value = 100
u = mean = 72
s = standard deviation = 15
Thus,
z = (x - u) / s = 1.866666667
Thus, using a table/technology, the right tailed area of this is
P(z > 1.866666667 ) = 0.030974076
Hence, 0.030974*355 = 10.995 = 11 students [ANSWER, 11]