Standard Normal Probabilities Extremes

Question

Hello! I need help understanding how to find the answer using the table. I am currently in college level Elementary statistics.

Determine the value of z* such that satisfies the conditions below. (Assume that the requested value of z* is positive. Round your answers to two decimal places.)



(a) −z* and z* separate the middle 94% of all z values from the most extreme 6%

z* = 

(b) −z* and z* separate the middle 90% of all z values from the most extreme 10%

z* = 

(c) −z* and z* separate the middle 97% of all z values from the most extreme 3%

z* = 

(d) −z* and z* separate the middle 92% of all z values from the most extreme 8%

z* = 

Answer 1

Hi Cara,

For these questions, think (1-alpha)/2; alpha=your extreme level--i.e. 6% for question a

a. alpha= 0.06

alpha/2= 0.06

1-(alpha/2)= 0.97

Look at the interior of the z-table for the value closest to 0.97. Once you have that, look to the row and column for the z-score. On my z-table, the closest value to 0.97 is 0.9699, which corresponds to

z*= 1.88

b. This is worth memorizing. It's z*= 1.645. This will become essential when you compute 90% z-confidence intervals, which you likely will later in the course. You can still use the 1-(alpha/2) method here, but this is a key critical value so I suggest just committing to memory.

c. Again, think 1 - (alpha/2); alpha = 0.03

0.03/2= 0.015

1 - 0.015 = 0.985

Again, look in interior of z-table for closest value to 0.985. In my z-table, I have 0.9850 right on at:

z*= 2.17

d. Same procedure; alpha = 0.08, compute 1-(alpha/2)

alpha/2= 0.08/2 =0.04

1 - 0.04 = 0.96

Again, check interior of z-table for 0.96; closest I have is 0.9599 at:

z*= 1.75

I hope this helps.

Answer 2

The Standard Normal Curve is the bell shaped curve. It is completely symmetrical. This is important because you are trying to find the middle portion that contains the given percentages (94,90,97, and 92). With each of these, the process is identical. I'll explain with the 94% so that you can copy the process for the rest. I am operating under the assumption that you have a z-score table. I am using the z-score table here: https://www.z-table.com/

Each row in the leftmost column (that has a z above it) represents the ones digit and the tenths digit of the z-score. Each column to the right of the leftmost column represents a different hundredths digit of the z-score. If you combine the ones, tenths and hundredths digits by tracking the corresponding row and column, you will see a decimal. This decimal represents the amount of area there is under the curve to the left of that z-score. Now, the area of the entire Standard Normal curve is exactly 1. So each decimal is the percentage of the area under the curve to the left of the z-score.

(a) Since the Standard Normal curve is symmetric and you are trying to find the middle 94%, we know that 6% is not covered in the area. Of that 6%, half (3%) is in the left tail, and the other half is in the right tail. You need to find the z-score corresponding to 3% (this will be a negative z-score since the percentage is less than 50). You do this by looking at the decimal portion (where the area is located) of the z-score table for the decimal form of 3% (.03). You will find the decimal .0301 corresponds to the z-score z = -1.88. So there is roughly 3% area to the left of the z-score -1.88, and by symmetry, there is roughly 3% area to the right of z = 1.88. This means the numbers you are looking for are z = -1.88 and z = 1.88. So z* = 1.88.

I feel like this answer might be a little over explained, but hopefully it makes sense. You can follow this procedure to find parts (b), (c) and (d).