If X is N(650, 625), find
(a) P(600 ≤ X < 660).
(b) A constant c > 0 such that P(|X −650| ≤ c) = 0.9544.
Solution 7E
Step1 of 3:
We have We have random variable X which follows normal distribution with mean 650 and variance 625.
That is
= 25.
We need to find,
(a) P(600 ≤ X < 660).
(b) A constant c > 0 such that P(|X −650| ≤ c) = 0.9544.
Step2 of 3:
Let “X” be random variable which follows normal distribution with parameters .
That is X N(
)
X N[
625]
The probability mass function of normal distribution is given by
,
Where,
= mean of X
= variance
= standard deviation.
a).
P(600 ≤ X < 660) =
=
=
=
=
Here, is obtained by using standard normal table(Area under normal curve).
In Area under normal curve we have to see row 0.4 under column 0.00
= 0.0228 In Area under normal curve we have to see row -2.0 under column 0.00
= 0.6554 - 0.0228
= 0.6326.
Hence, P(600 ≤ X < 660) = 0.6326.
Step3 of 3:
b).
P(|X −650| ≤ c) = 0.9544
= 0.9544
= 0.9544
= 0.9772
We have to see where 0.9772 falls in standard normal table(area under normal table).
In area under normal table 0.9772 falls in row 2.0 under column 0.00
Hence,
= 50
Therefore, c = 50.
Conclusion:
Therefore,
a).P(600 ≤ X < 660) = 0.6326.
b). c = 50.
