Question

Given a normal distribution with (mean) μ= 50 and (standard deviation) σ = 4, what is the probability that

NOTE: I'd like to learn how to do this in the shortest way possible on ti 84 plus calculator.

a) x>43

b) x<42

c) x>57.5

d) 42 <x<48

e) x<40 or x>55

f) 5% of the values are less than what X value?

g) 60% of the values are between what two X values (symmetrically distributed around the mean)?

h) 85% of the values will be above what X value?

Answer 1

We have given mean = 50 and standard deviation = 4

a) probability of X > 43 = P ( X > 43) = 0.9599

b) Probability of X < 42 = P ( X < 42 ) = 0.0227

c) Probability of X > 57.5 = P ( X > 57.5) = 0.03039

d ) Probability of 42 < X < 48 = P ( 42 < X < 48) = P ( X < 48 ) - P ( X < 42) = 0.3085-0.02275 = 0.2857

e ) Probability of X < 40 or X > 55 = P( X< 40) + P (X>55) = 0.006206 + 0. 1056 = 0.11185

f ) To obtain value of x for which 5% of the values are less than X

it can be obtained such that P ( X < x) = 0.05. therefore x = 43.42

g) To obtain two values of X in which 60 % of the values lies.

i. e. to obtain X₁ and X₂ such that P ( X₁ < X < X₂ ) = 0.60

therefore X₁ = 46.63352 and X₂ = 53.36648

h ) To find x above which 85% of the values lies.

i.e. P ( X > x ) = 0.85, therefore x = 45.8527

All above calculation can be obtained in all statistical softwares, We have used R - software in which the command used for obtaining the values of x are qnorm() and the area for specified x is pnorm().