Question

X ~ N(70, 11). Suppose that you form random samples of 25 from this distribution. Let X be the random variable of averages. Let ΣX be the random variable of sums.

1. Find the 30th percentile. (Round your answer to two decimal places.)

2. Sketch the graph, shade the region, label and scale the horizontal axis for X, and find the probability. (Round your answer to four decimal places.)

P(66 < X < 72) =

3. Sketch the graph, shade the region, label and scale the horizontal axis for X,  and find the probability. (Round your answer to four decimal places.)

P(32 < X < 66) =

Answer 1

Solution:-

Given that,

mean = = 70

standard deviation = = 11

Using standard normal table,

1 ) P(Z < z) = 30%

= P(Z < z) = 0.30

= P(Z < -0.524 ) = 0.25

z = -0.52

Using z-score formula,

x = z * +

x = -0.52 * 11+70

x = 64.28

The 30th percentile = 64.28

2) P (66 < x < 72 )

P ( 66- 70 / 27) < ( x -  / ) < ( 72 - 70 / 11)

P ( - 4 / 11 < z < 2 / 11 )

P (-0.36 < z < 0.18 )

P ( z < 0.18 ) - P ( z < -0.36)

Using z table

= 0.5714 - 0.3594

= 0.2120

Probability =0.2120

3 ) P (32 < x < 66 )

P ( 32 - 70 / 27) < ( x -  / ) < ( 66 - 70 / 11)

P ( - 38 / 11 < z < - 4 / 11 )

P (-3.45 < z < -0.36 )

P ( z < -0.36) - P ( z < -3.45)

Using z table

= 0.3594 -0.0003

= 0.3591

Probability =0.3591