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Magazine Chapter 7: Problem 8
Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the right of (a) \(Z=-3.49\) (b) \(Z=-0.55\) (c) \(Z=2.23\) (d) \(Z=3.45\)
Short Answer
Expert verified
a) 0.9998 b) 0.7088 c) 0.0128 d) 0.0003
Step by step solution
01
Understanding the Standard Normal Curve
The standard normal curve is a bell-shaped curve that is symmetrical about the mean (which is 0) and has a standard deviation of 1. The curve extends infinitely in both directions, but most of the area under the curve is within 3 standard deviations of the mean.
02
Using Z-Table for Area Calculation
To find the area under the standard normal curve, use the Z-Table (Standard Normal Table), which provides the cumulative probability from the mean to a given Z value.
03
Area to the Right of Z=-3.49
Look up the Z value of -3.49 in the Z-table. The Z-table gives the area to the left of Z. Area to the left of Z=-3.49 is approximately 0.0002. The area to the right is obtained by subtracting this value from 1.
04
Calculate the Area to the Right of Z=-3.49
The area to the right is calculated as follows: \[ 1 - 0.0002 = 0.9998 \] Therefore, the area to the right of Z=-3.49 is 0.9998.
05
Area to the Right of Z=-0.55
Look up the Z value of -0.55 in the Z-table. The Z-table gives the area to the left of Z. Area to the left of Z=-0.55 is approximately 0.2912. The area to the right is obtained by subtracting this value from 1.
06
Calculate the Area to the Right of Z=-0.55
The area to the right is calculated as follows: \[ 1 - 0.2912 = 0.7088 \] Therefore, the area to the right of Z=-0.55 is 0.7088.
07
Area to the Right of Z=2.23
Look up the Z value of 2.23 in the Z-table. The Z-table gives the area to the left of Z. Area to the left of Z=2.23 is approximately 0.9872. The area to the right is obtained by subtracting this value from 1.
08
Calculate the Area to the Right of Z=2.23
The area to the right is calculated as follows: \[ 1 - 0.9872 = 0.0128 \] Therefore, the area to the right of Z=2.23 is 0.0128.
09
Area to the Right of Z=3.45
Look up the Z value of 3.45 in the Z-table. The Z-table gives the area to the left of Z. Area to the left of Z=3.45 is approximately 0.9997. The area to the right is obtained by subtracting this value from 1.
10
Calculate the Area to the Right of Z=3.45
The area to the right is calculated as follows: \[ 1 - 0.9997 = 0.0003 \] Therefore, the area to the right of Z=3.45 is 0.0003.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Z-table
The Z-table, or Standard Normal Table, is a fundamental tool in statistics used to find the cumulative probability associated with a Z value on the standard normal curve. Each entry in the table represents the area under the curve to the left of a given Z score. The Z score is a measure of how many standard deviations an element is from the mean. For example, if you have a Z value of -3.49, you can look it up in the Z-table to find the related cumulative probability. But remember, the Z-table only gives you the area to the left of the Z score.
Cumulative Probability
Cumulative probability refers to the probability that a random variable will take a value less than or equal to a specific value. When using the Z-table, you are finding the cumulative probability from the mean (0 on the standard normal curve) to the Z value you're interested in.
For instance, if the Z value is -0.55, the cumulative probability or the area to the left might be 0.2912. This means that 29.12% of the data falls below the Z value of -0.55. For problems involving the area to the right of a Z value, you would subtract this cumulative probability from 1.
For instance, if the Z value is -0.55, the cumulative probability or the area to the left might be 0.2912. This means that 29.12% of the data falls below the Z value of -0.55. For problems involving the area to the right of a Z value, you would subtract this cumulative probability from 1.
Area Under the Curve
In the context of the standard normal curve, 'area under the curve' refers to the total probability represented by a specific portion of the curve. The entire area under the curve sums up to 1 (or 100% probability). This area corresponds to the likelihood of an event occurring.
To determine the area under the curve to the right of a Z value, you subtract the cumulative probability found in the Z-table from 1. For example, to find the area to the right of Z=2.23, if the Z-table gives an area of 0.9872 to the left, the area to the right would be 1 - 0.9872 = 0.0128 or 1.28%.
To determine the area under the curve to the right of a Z value, you subtract the cumulative probability found in the Z-table from 1. For example, to find the area to the right of Z=2.23, if the Z-table gives an area of 0.9872 to the left, the area to the right would be 1 - 0.9872 = 0.0128 or 1.28%.
Z Value Calculations
Calculating Z values and interpreting them is crucial for understanding areas under the standard normal curve. Z values help determine how far a data point is from the population mean, measured in terms of standard deviations.
For example, if you have a Z value of -0.55, the Z-table gives an area of 0.2912 to the left of this Z value. To find the area to the right, you subtract this from 1: 1 - 0.2912 = 0.7088. So, 70.88% of the data lies to the right of a Z score of -0.55. This process remains consistent whether the Z value is positive or negative.
For example, if you have a Z value of -0.55, the Z-table gives an area of 0.2912 to the left of this Z value. To find the area to the right, you subtract this from 1: 1 - 0.2912 = 0.7088. So, 70.88% of the data lies to the right of a Z score of -0.55. This process remains consistent whether the Z value is positive or negative.
