Question

Between what two z-scores, to the nearest three decimal places, would we find (each question has two answers as the question is asking for two z-score):

a) 85.62% of the distribution's area?  

b) 99.995% of the distribution's area?  

c) 53.75% of the distribution's area?  

d) 13.96% of the distribution's area?

Please show your work! Any help is appreciated :)

Answer 1

Ans:

a) 85.62% of distribution area lies between -1.462 and 1.462

b) 99.995% of distribution area lies between -4.056 and 4.056

c) 53.75% of distribution area lies between -0.735

and 0.735

d) 13.96% of distribution area lies between -0.176 and 0.176

###########Explanation#########

Here we want to find the z-score for which the following % of distribution area lies between -z and z.

a) 85.62% of the distribution's area

So find z such that

Solving

From Normal Distribution table

Hence from the table using interpolation;

z = 1.4617

Rounding to three decimal places

z= 1.462

85.62% of distribution area lies between -1.462 and 1.462

b)

Now 99.995% of the distribution's area is given as

So find z such that

Solving

Obtaining the value of z-score since 0.000025 is very small we will use technology

Hence

z = 4.0556

Rounding to three decimal places

z= 4.056

99.995% of distribution area lies between -4.056 and 4.056

c)  53.75% of the distribution's area

So find z such that

Solving

Obtaining z-value from normal distribution table

Hence from the table using interpolation;

z = 0.7347

Rounding to three decimal places

z= 0.735

53.75% of distribution area lies between -0.735

and 0.735

d) 13.96% of the distribution's area

So find z such that

Solving

Obtaining the z-value form the normal distribution table

Hence from the table using interpolation;

z = 0.1758

Rounding to three decimal places

z= 0.176

13.96% of distribution area lies between -0.176 and 0.176