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Chapter 7: Problem 17

Find the indicated \(Z\) -score. Be sure to draw a standard normal curve that depicts the solution. Find the \(Z\) -score such that the area under the standard normal curve to the left is \(0.98 .\)

Short Answer

Expert verified
The Z-score is 2.05.

Step by step solution

01

- Understanding the problem

We need to find the Z-score such that the area under the standard normal curve to the left of this Z-score is 0.98.
02

- Utilize Z-score table

To find this Z-score, we will use a Z-score table (also known as the standard normal table) which shows the area to the left of a given Z-score.
03

- Locate the desired area in the table

Look for the area that is closest to 0.98 in the Z-score table. The closest value to 0.98 in the Z-score table is 0.9798, which corresponds to a Z-score of 2.05.
04

- Verification using symmetry property

Verify using the properties of the standard normal distribution that Z = 2.05 matches the area of 0.98 to the left. This is confirmed since the table value at Z = 2.05 is closest to the desired area.
05

- Draw the standard normal curve

Draw a standard normal curve and shade the area to the left of the Z-score. Label the axis, and clearly indicate the Z-score of 2.05.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Normal Distribution
The standard normal distribution is a special type of normal distribution. It is characterized by a mean of 0 and a standard deviation of 1. Normal distributions are symmetrical and bell-shaped, meaning most of the data lies close to the mean, tapering off equally on both sides.

In a standard normal distribution:
  • The horizontal axis represents Z-scores.
  • The peak at Z = 0 is the mean.
  • The area under the curve totals to 1, representing the entirety of the data.
We often use this distribution to find probabilities and percentiles for data that follows a normal distribution by converting values to Z-scores.
Z-score Table
A Z-score table, also called a standard normal table, helps us find the area to the left of a particular Z-score. This table can be used to determine the likelihood of observing a value less than a given Z-score in a standard normal distribution.

Here's how to use it:
  • The row in the Z-score table indicates the first two digits of the Z-score.
  • The column represents the second decimal place of the Z-score.
  • The intersection of the row and column gives the cumulative probability, i.e., the area to the left of the Z-score.
For example, to find the area to the left of Z = 2.05, locate the row for 2.0 and the column for 0.05. The value at their intersection, 0.9798, tells us that approximately 97.98% of the values lie to the left of Z = 2.05.
Area Under the Curve
The area under the curve in a standard normal distribution represents probabilities or cumulative proportions. Because the entire area under the standard normal curve totals 1, it helps in calculating specific probabilities.

When you look for the area to the left of a Z-score, you're determining the cumulative probability. For instance, finding the area to the left of Z = 2.05 (0.98 or 97.98%) means there's a 98% chance a data point lies below this Z-score.

Here are important concepts related to the area under the curve:
  • The total area under the curve is always 1.
  • The area to the left of the Z-score is called the cumulative area.
  • The area to the right can be found by subtracting the cumulative area from 1.
These principles make the Z-score table extremely useful in various statistical calculations.
Symmetry Property
One of the intriguing features of the normal distribution is its symmetry. The curve is perfectly symmetric around the mean (Z = 0 in a standard normal distribution). This symmetry simplifies many calculations and verifications.

Key points about symmetry in the standard normal distribution:
  • The left side is a mirror image of the right side.
  • The area to the left of a positive Z-score is equal to the area to the right of its negative counterpart. For example, the area to the left of Z = 2.05 is the same as the area to the right of Z = -2.05.
Using the symmetry property can confirm calculations. For instance, if you find the area to the left of Z = 2.05 is around 0.98, the symmetry property lets you know that the area to the right of Z = -2.05 is also about 0.98.

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Most popular questions from this chapter

True or False: Suppose \(X\) is a binomial random variable. To approximate \(P(3 \leq X<7)\) using the normal probability distribution, we compute \(P(3.5 \leq X<7.5)\).

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