Download Quiz 7

Quiz 7
Stat 190
1. You are given that the mean, the standard deviation and the z-score of a binomial experiment
are
p
x−µ
µ = np,
σ = np(1 − p),
z=
.
σ
Assume that p = 45% of adult Americans believe that the overall state of moral values in
U.S. is poor. We select n = 500 random adult Americans and look at the number X among
them who feel the state of moral values is poor.
(a) Find the mean and standard deviation of this binomial variable X.
The mean and the standard deviation are
µ =
σ
500(0.45) = 225
p
√
500(0.45)(0.55) = 123.75 = 11.12.
=
(b) Does X have a bell-shape normal distribution? Why?
X has a normal distribution since np(1 − p) = 123.75 ≥ 10.
(c) Find the probability that at most 220 of them feel the state of morals is poor. Show
your work and do not forget continuity correction.
The z-score is
220.5 − 225
z=
= −0.40.
11.12
We look at the normal table and have P (X ≤ 220) = 0.3446.
(d) Find the probability that more than 250 of them feel the state of morals is poor. Show
your work and do not forget continuity correction.
The z-score is
250.5 − 225
z=
= 2.29.
11.12
We look at the normal table and have P (X > 250) = 1 − 0.9890 = 0.011.
(e) Find the probability that between 230 and 255 of them, inclusively, feel the state of
morals is poor. Show your work and do not forget continuity correction.
The two z-scores are
229.5 − 225
11.12
255.5 − 225
11.12
=
0.40
=
2.74.
We look at the normal table and hve P (230 ≤ X ≤ 255) = 0.9969 − 0.6554 = 0.3415.
1