Question

1. for a normal distribution, find the z - score that separates the distribution as follows:

a) separate the highest 30% from the rest of the distribution.
b) separate the lowest 40% from the rest of the distribution.
c) separate the highest 75% from the rest of the distribution.

Explanation:

Part (a)

Step1: Determine the area to the left

To find the z - score that separates the highest 30% from the rest, the area to the left of the z - score is \(1 - 0.3=0.7\).

Step2: Use the z - table or invNorm function

Using a standard normal distribution table (z - table) or a calculator with an inverse normal function (invNorm), we look for the z - value corresponding to an area of 0.7. Using a calculator or z - table, we find that \(z\approx0.5244\) (the value can be approximated from the z - table or calculated more precisely using a calculator. For example, in a TI - 84 Plus, invNorm(0.7,0,1) gives approximately 0.5244).

Part (b)

Step1: Determine the area to the left

To find the z - score that separates the lowest 40% from the rest, the area to the left of the z - score is 0.4.

Step2: Use the z - table or invNorm function

Using a standard normal distribution table or a calculator with an inverse normal function, we look for the z - value corresponding to an area of 0.4. Using a calculator (invNorm(0.4,0,1)) or z - table, we find that \(z\approx - 0.2533\).

Part (c)

Step1: Determine the area to the left

To find the z - score that separates the highest 75% from the rest, the area to the left of the z - score is \(1 - 0.75 = 0.25\).

Step2: Use the z - table or invNorm function

Using a standard normal distribution table or a calculator with an inverse normal function, we look for the z - value corresponding to an area of 0.25. Using a calculator (invNorm(0.25,0,1)) or z - table, we find that \(z\approx- 0.6745\).

Answer:

s:
a) \(\boldsymbol{z\approx0.52}\) (or more precisely \(z\approx0.5244\))
b) \(\boldsymbol{z\approx - 0.25}\) (or more precisely \(z\approx- 0.2533\))
c) \(\boldsymbol{z\approx - 0.67}\) (or more precisely \(z\approx - 0.6745\))