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Chapter 6: Problem 3

Calculate the area under the standard normal curve to the left of these values: a. \(z=1.6\) b. \(z=1.83\) c. \(z=.90\) d. \(z=4.18\)

Short Answer

Expert verified
Answer: a) The area to the left of z=1.6 is 94.52%. b) The area to the left of z=1.83 is 96.64%. c) The area to the left of z=0.90 is 81.59%. d) The area to the left of z=4.18 is approximately 100%.

Step by step solution

01

Understand the standard normal curve and z-table

A standard normal curve is a bell-shaped, symmetric curve which represents a normal distribution with a mean of 0 and a standard deviation of 1. The z-table (also known as the standard normal table) helps us find the probabilities (areas) to the left of a given z-score.
02

Find the area to the left of z=1.6

Locate z=1.6 using the z-table. The row represents the first 2 digits (1.6) and the column represents the second decimal digit (0). Therefore, the entry in the table is 0.9452. This means that the area to the left of z=1.6 is 0.9452 or 94.52%.
03

Find the area to the left of z=1.83

Locate z=1.83 in the z-table. The row represents the first 2 digits (1.8) and the column represents the second decimal digit (3). In the table, the entry is 0.9664. Therefore, the area to the left of z=1.83 is 0.9664 or 96.64%.
04

Find the area to the left of z=0.90

Locate z=0.90 in the z-table. The row represents the first 2 digits (0.9) and the column represents the second decimal digit (0). The entry in the table is 0.8159. Therefore, the area to the left of z=0.90 is 0.8159 or 81.59%.
05

Find the area to the left of z=4.18

Since the z-table typically lists probabilities up to a z-score of approximately 3.5, we can use the following property for large positive z-scores: the area to the left of a large positive z-score is approximately equal to 1 (or 100%). This is because almost all the area under the curve falls to the left of a large z-score. In this case, z=4.18 is a large positive z-score, so the area to the left of z=4.18 is approximately equal to 1 or 100%.
06

Summary

We found the areas under the standard normal curve to the left of the given z-scores using the z-table: a. The area to the left of z=1.6 is 0.9452 or 94.52%. b. The area to the left of z=1.83 is 0.9664 or 96.64%. c. The area to the left of z=0.90 is 0.8159 or 81.59%. d. The area to the left of z=4.18 is approximately 1 or 100%.

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