Question

please show steps

Let X ∼ N (100, 25) be a normal random value.

Find: (a) P[93 < X < 107] (b) P[93 < X < 100] (c) P[93 < X < 109]

(d) P[X < 109] (e) P[X > 107] (f) P[|X| > 107] (g) P[|X| < 93]

(h) 67-th percentile n(0.67) (i) 23-th percentile n(0.23)

Answer 1

By the law of normal distribution,

i.e.

where and is the cdf of T

(a)

  

[Obtained from the Biometrika table]

(b)

  

[Obtained from the Biometrika table]

(c)

  

[Obtained from the Biometrika table]

(d)

  

[Obtained from the Biometrika table]

(e)

  

[Obtained from the Biometrika table]

(f)

  

[Obtained from the Biometrika table]

(g)

  

[Obtained from the Biometrika table]

(h)

To plot the 67-th percentile, we are to plot the value of from Biometrika table of Standard Normal variables, such that

As the percentile is more than 0.5, the value will be positive.

From the table, it is observed that for

Hence T=0.44

The required value is 102.2

(i)

To plot the 23-th percentile, we are to plot the value of from Biometrika table of Standard Normal variables, such that

As the percentile is less than 0.5, the value will be negative.

From the table, it is observed that for

Hence T=1-0.74=0.26

The required value is 101.3

Hopefully this will help you. In case of any query, do comment. If you are satisfied with the answer, give it a like. Thanks.