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Chapter 6: Problem 19

Use the table or technology to answer each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. a. Find the area in a standard Normal curve to the left of \(1.02\) by using the Normal table. (See the excerpt provided.) Note the shaded curye. b. Find the area in a standard Normal curve to the right of \(1.02\). Remember that the total area under the curve is \(1 .\)

Short Answer

Expert verified
The area in the standard Normal curve to the left of \(1.02\) is \(0.8461\) or \(84.61\%\), and the area to the right of \(1.02\) is \(0.1539\) or \(15.39\%\).

Step by step solution

01

Find area to the left of \(1.02\)

To find the area to the left of \(1.02\), look up \(1.02\) in the Normal distribution table. The value corresponds to the area to the left of the \(z\)-value, which is \(0.8461\). This means that \(84.61\%\) of the data falls to the left of \(1.02\) under the curve.
02

Sketch the curve for part a

Sketch a Normal curve and shade the area to the left of \(1.02\). Note that the shaded area represents \(84.61\%\) of the total area under the curve.
03

Find area to the right of \(1.02\)

To find the area to the right of \(1.02\), subtract the area to the left of \(1.02\) obtained from the Normal table from the total area under the curve, which is \(1\). So, \(1 - 0.8461 = 0.1539\). This means \(15.39\%\) of the data falls to the right of \(1.02\) under the curve.
04

Sketch the curve for part b

Sketch another Normal curve and shade the area to the right of \(1.02\). Note that the shaded area represents \(15.39\%\) of the total area under the curve.

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Key Concepts

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

Standard Normal Curve
The Standard Normal Curve is a crucial element in statistics. Imagine it as a smooth, bell-shaped curve that mirrors how certain sets of data are distributed.
It's a type of Normal distribution where the mean is zero and the standard deviation is one. Key properties of the Standard Normal Curve include:
  • A mean (\( \mu \)) of 0.
  • A standard deviation (\( \sigma \)) of 1.
  • Symmetrical about the mean (which is zero).
  • The total area under the curve equals 1, which represents 100% of the distribution.
Understanding the Standard Normal Curve helps in determining probabilities and allows us to convert various Normal distributions into a standardized form, thereby simplifying data analysis significantly.
Z-value
The Z-value, also called a Z-score, is a way to quantify how far away a particular data point is from the mean, in terms of standard deviations. Calculating the Z-score of an individual data point involves:
  • Subtracting the mean from the data point.
  • Dividing the result by the standard deviation.
This provides a Z-value, which can be positive or negative:
  • A positive Z-value indicates the data point is above the mean.
  • A negative Z-value indicates it's below the mean.
In the context of the Standard Normal Curve, a Z-value tells us about the location of a particular score within the distribution. For example, in our exercise, a Z-value of 1.02 indicates that the data point is 1.02 standard deviations above the mean.
Area under the curve
The area under the curve of the Standard Normal Curve offers insightful information about the probability and distribution of data. In probability, any specific region beneath the curve corresponds to the likelihood of selecting a data point within that particular region. Two key concepts of area under the curve include:
  • Area to the left of a Z-value: Represents the probability that a data point is less than the Z-value. In our exercise, the area to the left of 1.02 is 0.8461, or 84.61%.
  • Area to the right of a Z-value: Represents the probability that a data point is greater than the Z-value. Here, the area to the right of 1.02 is 0.1539, or 15.39%.
Together, these areas sum up to 1 or 100%, showing the entirety of possible outcomes. Understanding these areas gives a comprehensive overview of where data points lie within a Standard Normal distribution.

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Most popular questions from this chapter

You may have heard that drunk driving is dangerous, but what about drunk walking? According to federal information (reported in the Ventura County Star on August \(6 .\) 2013 ), \(50 \%\) of the pedestrians killed in the United States had a blood-alcohol level of \(0.08 \%\) or higher. Assume that two randomly selected pedestrians who were killed are studied. a. If the pedestrian was drunk (had a blood-alcohol level of \(0.08 \%\) or higher), we will record a D, and if the pedestrian was not drunk, we will record an \(\mathrm{N}\). List all possible sequences of \(\mathrm{D}\) and \(\mathrm{N}\). b. For each sequence, find the probability that it will occur, by assuming independence. c. What is the probability that neither of the two pedestrians was drunk? d. What is the probability that exactly one out of two independent pedestrians was drunk? c. What is the probability that both were drunk?

For each situation, identify the sample size \(n\), the probability of success \(p\), and the number of successes \(x .\) When asked for the probability, state the answer in the form \(b(n, p, x) .\) There is no need to give the numerical value of the probability. Assume the conditions for a binomial experiment are satisfied. a. According to the Federal Highway Research Institute in Germany, 2 out of 3 persons in an accident get killed. In a random sample of 27 persons meeting with an accident, what is the probability that exactly 12 persons would have died? b. Twenty-five percent of the persons killed in accidents are pedestrians. If we randomly select 27 persons who have died in an accident, what is the probability that 12 persons are pedestrians?

Stanford-Binet IQ scores for children are approximately Normally distributed and have \(\mu=100\) and \(\sigma=15 .\) What is the probability that a randomly selected child will have an IQ below 115 ?

The distribution of the math portion of SAT scores has a mean of 500 and a standard deviation of 100 , and the scores are approximately Normally distributed. a. What is the probability that one randomly selected person will have an SAT score of 550 or more? b. What is the probability that four randomly selected people will all have SAT scores of 550 or more? c. For 800 randomly selected people, what is the probability that 250 or more will have scores of 550 or more? d. For 800 randomly selected people, on average how many should have scores of 550 or more? Round to the nearest whole number. e. Find the standard deviation for part d. Round to the nearest whole number. f. Report the range of people out of 800 who should have scores of 550 or more from two standard deviations below the mean to two standard deviations above the mean. Use your rounded answers to part \(\mathrm{d}\) and \(\mathrm{e}\). g. If 400 out of 800 randomly selected people had scores of 550 or more, would you be surprised? Explain.

ACT scores are approximately Normally distributed with a mean of 21 and a standard deviation of 5, as shown in the figure. (ACT scores are test scores that some colleges use for determining admission.) What is the probability that a randomly selected person scores 24 or more?
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