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Chapter 7: Problem 11

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 (a) to the left of \(Z=-2\) or to the right of \(Z=2\) (b) to the left of \(Z=-1.56\) or to the right of \(Z=2.56\) (c) to the left of \(Z=-0.24\) or to the right of \(Z=1.20\)

Short Answer

Expert verified
Part (a): 0.0456, Part (b): 0.0646, Part (c): 0.5203.

Step by step solution

01

Understand the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The area under the curve represents probabilities, and the total area under the curve is 1.
02

Use Z-tables or Calculators for Areas

To find the areas under the standard normal curve, the Z-tables or statistical calculators are used. These tables provide the area to the left of a given Z-score.
03

Determine the Area for Part (a)

For part (a), find the area to the left of Z = -2 and to the right of Z = 2. Using Z-tables, the area to the left of Z = -2 is approximately 0.0228. By symmetry, the area to the right of Z = 2 is also 0.0228. The total area is 0.0228 + 0.0228 = 0.0456.
04

Determine the Area for Part (b)

For part (b), find the area to the left of Z = -1.56 and to the right of Z = 2.56. The area to the left of Z = -1.56 is approximately 0.0594. The area to the right of Z = 2.56 is approximately 0.0052. So, the total area is 0.0594 + 0.0052 = 0.0646.
05

Determine the Area for Part (c)

For part (c), find the area to the left of Z = -0.24 and to the right of Z = 1.20. The area to the left of Z = -0.24 is approximately 0.4052. The area to the right of Z = 1.20 is approximately 0.1151. So, the total area is 0.4052 + 0.1151 = 0.5203.
06

Summarize the Results

Parts (a), (b), and (c) have the total areas under the standard normal curve as follows: (a) 0.0456, (b) 0.0646, and (c) 0.5203.

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Key Concepts

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

Z-score
The Z-score, also known as the standard score, is a measure that describes a data point's position in relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. The formula for the Z-score is \(Z = \frac{X - \mu}{\sigma}\), where:\(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. A Z-score of 0 indicates that the data point's score is identical to the mean score. Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below the mean.
Normal Distribution
A normal distribution, often referred to as a bell curve, is a probability distribution that is symmetric about the mean. In a normal distribution, most observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions, forming a bell-shaped curve. The standard normal distribution is a special case with a mean of 0 and a standard deviation of 1. This standard distribution allows us to use Z-scores to find probabilities and areas under the curve.
Area Under the Curve
The area under the normal distribution curve represents probabilities. Since the total area under the curve is equal to 1, any subsection of the curve corresponds to a probability between 0 and 1. For example, if we want to find the probability of a data point falling below a specific Z-score, we look at the area to the left of that Z-score on the standard normal distribution curve. Similarly, to find the probability of a data point falling above a specific Z-score, we look at the area to the right of that Z-score. This concept is crucial for determining probabilities in statistics.
Probability
Probability is a measure of the likelihood that an event will occur. In the context of the normal distribution, probability can be visualized as the area under the curve. For example, finding the probability that a value is less than a certain Z-score involves identifying the area under the curve to the left of that Z-score. The sum of all probabilities for all possible outcomes (the total area under the curve) is 1, representing 100% certainty.
Z-tables
Z-tables, also known as standard normal tables, are used to find the area (or probability) to the left of a given Z-score in the standard normal distribution. Each value in the Z-table represents the cumulative probability from the far left of the distribution (negative infinity) up to that Z-score. For example, if the Z-score is -2, the Z-table tells us that approximately 0.0228 of the data falls to the left of this Z-score. Similarly, for Z = 2, the Z-table shows the same area on the right side because of the symmetric nature of the distribution.

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Most popular questions from this chapter

Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, \(5 \%\) of the patients in the study experienced insomnia as a side effect. Suppose a random sample of 400 Clarinex-D users is obtained. Use the normal approximation to the binomial to (a) approximate the probability that exactly 20 patients experienced insomnia as a side effect. (b) approximate the probability that 15 or fewer patients experienced insomnia as a side effect. (c) approximate the probability that 30 or more patients experienced insomnia as a side effect. (d) approximate the probability that between 10 and 32 patients, inclusive, experienced insomnia as a side effect.

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 0.2

According to the Current Population Survey (Internet release date: September 15,2004 ) \(55 \%\) of males between the ages of 18 and 24 years lived am home in 2003 . (Unmarried college students living in a dorm are counted as living at home.) Suppose that a survey is administered at a community college to 200 randomly select ed male students between the ages of 18 and 24 years and that 130 of them respond that they live at home. (a) Approximate the probability that such a survey will result in at least 130 of the respondents living at home under the assumption that the true percentage is \(55 \%\) (b) Does the result from part (a) contradict the results of the Current Population Survey? Explain.

State the properties of the standard normal curve.

Assume the random variable \(X\) is normally distributed with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. $$P(X>35)$$
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