Question

Suppose a population of scores x is normally distributed with μ = 16 and σ = 5. Use the standard normal distribution to find the probability indicated. (Round your answer to four decimal places.)

Pr(16 ≤ x ≤ 18.3)

You may need to use the table of areas under the standard normal curve from the appendix.

Also,

Use the table of areas under the standard normal curve to find the probability that a z-score from the standard normal distribution will lie within the interval. (Round your answer to four decimal places.)

−1.8 ≤ z ≤ 2.9

And,

Suppose a population of scores x is normally distributed with

μ = 30

and

σ = 10.

Use the standard normal distribution to find the probability indicated. (Round your answer to four decimal places.)

Pr(25 ≤ x ≤ 45)

Answer 1

1. Suppose a population of scores x is normally distributed with μ = 16 and σ = 5.

We will solve this problem by standardising.

P(16 ≤ X ≤ 18.3) = P((16 - 16)/5 ≤ (x - µ)/ σ ≤ (18.3 - 16)/5)

                              = P(0 ≤ Z ≤ 0.46 )

                              = P(Z ≤ 0.46) - P(Z ≤ 0)

                              = 0.6772 - 0.5 …… (Using statistical table)

                              = 0.1772

P(16 ≤ X ≤ 18.3) = 0.1772

2. The probabilty that the z-score from a standard normal distribution will lie within interval.

P(-1.8 ≤ z ≤ 2.9) = = P(Z ≤ 2.9) - P(Z ≤ -1.8) = 0.9981 – 0.0359 = 0.9622

3. Suppose a population of scores x is normally distributed with μ = 30 and σ = 10.

We will solve this problem by standardising.

P(25 ≤ X ≤ 45) = P((25 - 30)/(10) ≤ X - µ)/ σ ≤ (45 - 30)/(10)

                              = P(- 0.5 ≤ Z ≤ 1.5)

                              = P(Z ≤ 1.5) - P(Z ≤ -0.5)

                              = 0.9332 – 0.3085…………… (using statistical table)

                             = 0.6247

P(25 ≤ X ≤ 45) = 0.6247