Question

Assume a normal distribution and find the following probabilities. (Round the values of z to 2 decimal places. Round your answers to 4 decimal places.) (a) P(x < 23 | μ = 26 and σ = 4) enter the probability of fewer than 23 outcomes if the mean is 26 and the standard deviation is 4 (b) P(x ≥ 58 | μ = 40 and σ = 9) enter the probability of 58 or more outcomes if the mean is 40 and the standard deviation is 9 (c) P(x > 35 | μ = 40 and σ = 6) enter the probability of more than 35 outcomes if the mean is 40 and the standard deviation is 6 (d) P(18 < x < 26 | μ = 24 and σ = 4) enter the probability of more than 18 and fewer than 26 outcomes if the mean is 24 and the standard deviation is 4 (e) P(x ≥ 76 | μ = 60 and σ = 2.82) enter the probability of 76 or more outcomes if the mean is 60 and the standard deviation is 2.82.

Answer 1

a) µ = 26, σ = 4

P(X < 23) =

= P( (X-µ)/σ < (23-26)/4 )

= P(z < -0.75)

Using excel function:

= NORM.S.DIST(-0.75, 1)

= 0.2266

b) µ = 40, σ = 9

P(X >= 58) =

= P( (X-µ)/σ >= (58-40)/9)

= P(z >= 2)

= 1 - P(z < 2)

Using excel function:

= 1 - NORM.S.DIST(2, 1)

= 0.0228

c) µ = 40, σ = 6

P(X > 35) =

= P( (X-µ)/σ > (35-40)/6)

= P(z > -0.83)

= 1 - P(z < -0.83)

Using excel function:

= 1 - NORM.S.DIST(-0.83, 1)

= 0.7967

d) µ = 24, σ = 4

P(18 < X < 26) =

= P( (18-24)/4 < (X-µ)/σ < (26-24)/4 )

= P(-1.5 < z < 0.5)

= P(z < 0.5) - P(z < -1.5)

Using excel function:

= NORM.S.DIST(0.5, 1) - NORM.S.DIST(-1.5, 1)

= 0.6247

e) P(X >= 76) =

= P( (X-µ)/σ >= (76-60)/2.82)

= P(z >= 5.67)

= 1 - P(z < 5.67)

Using excel function:

= 1 - NORM.S.DIST(5.67, 1)

= 0