Question

a. Find the area under the standard normal curve that lies between z=-1.11 and 3.21. Round to 4 decimal places. b. Find the z-score for which the area to its left is 0.54. Round to two decimal places.

Answer 1

The answers of part a. and b. are as

a. Using Excel or z table

Answer :- The area under the standard normal curve that lies between z = -1.11 and 3.21 is 0.8658

b. Using Excel or z table

Answer :- The z-score for which the area to its left is 0.54 is 0.10

The complete solution of above questions are as below

a. We find the area under the standard normal curve that lies between z = -1.11 and 3.21 using following Excel function

=NORMSDIST(z)

P(-1.11 < z < 3.21)

Using above z-values in the Excel function as below and then press Enter

=NORMSDIST(3.21) - NORMSDIST(-1.11)

=0.865836812 $\approx$ 0.8658 (Round answer to 4 decimal places)

P(-1.11 < z < 3.21) = 0.8658

The area under the standard normal curve that lies between z = -1.11 and 3.21 is 0.8658

Or

We also find the above area under the standard normal curve that lies between z = -1.11 and 3.21 using z table

P(-1.11 < z < 3.21) = P(z = 3.21) - P(z = -1.11) ---------(1)

Using z table

P(z = 3.21) = 0.9993 and  P(z = -1.11) = 0.1335

Using above values in equation (1) we get

P(-1.11 < z < 3.21) = 0.9993 - 0.1335

P(-1.11 < z < 3.21) = 0.8658

The area under the standard normal curve that lies between z = -1.11 and 3.21 is 0.8658

b. We find the z-score for which the area to its left is 0.54 using following Excel function

=NORMSINV(probability)

Here, probability = 0.54

Using above values in Excel function and then press Enter

=NORMSINV(0.54)

=0.100433721 $\approx$ 0.10 (Round answer to two decimal places)

z = 0.10

The z-score for which the area to its left is 0.54 is 0.10

We also find the z-score for which the area to its left is 0.54 using z table

Here, probability = 0.54

So, Using z table we find z value whose corresponding probability = 0.54 = 0.5400 (Using 4 decimal because z table based on values which are 4 decimal places) or approximately closest value $\approx$ 0.5400 Here, 0.5398 is closest to 0.5400, so we find it's corresponding z as below

=0.10

z = 0.10

The z-score for which the area to its left is 0.54 is 0.10

Summary :-

a. Using Excel or z table

The area under the standard normal curve that lies between z = -1.11 and 3.21 is 0.8658

b. Using Excel or z table

The z-score for which the area to its left is 0.54 is 0.10