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Chapter 6: Problem 48

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About \(\quad \%\) of the area is between \(z=-3.5\) and \(z=3.5\) (or within \(3.5\) standard deviations of the mean).

Short Answer

Expert verified
About 99.96% of the area is between \( z=-3.5 \) and \( z=3.5 \).

Step by step solution

01

Understanding the Problem

The task is to find the area under the standard normal distribution curve between the z-scores of -3.5 and 3.5. This will indicate the percentage of data within 3.5 standard deviations of the mean.
02

Use the Standard Normal Distribution Table

Locate the z-scores of -3.5 and 3.5 in the standard normal distribution table. The table provides the cumulative probability from the left tail up to the given z-score.
03

Find the Cumulative Probabilities

The cumulative probability for \( z = -3.5 \) is very close to 0, and the cumulative probability for \( z = 3.5 \) is very close to 1. Precisely, \( P(Z \leq -3.5) \approx 0.0002 \) and \( P(Z \leq 3.5) \approx 0.9998 \).
04

Calculate the Area Between the Z-Scores

To find the area between \( z = -3.5 \) and \( z = 3.5 \), subtract the cumulative probability at \( z = -3.5 \) from the cumulative probability at \( z = 3.5 \): \[ P(-3.5 \leq Z \leq 3.5) = P(Z \leq 3.5) - P(Z \leq -3.5) = 0.9998 - 0.0002 = 0.9996. \]
05

Convert the Area to a Percentage

To convert the area to a percentage, multiply the result by 100: \[ 0.9996 \times 100 = 99.96\text{\%} \].
06

Fill in the Blank

Insert the calculated percentage into the blank to complete the statement.

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Key Concepts

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

z-scores
A z-score represents the number of standard deviations a data point is from the mean. In a standard normal distribution, the mean is zero and the standard deviation is one. Therefore, a z-score indicates how far and in what direction a data point deviates from the mean.
For example, a z-score of 1.5 means the data point is 1.5 standard deviations above the mean. Z-scores can be positive or negative, depending on whether the data point is above or below the mean.
To calculate a z-score, you can use the formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
  • X is the data point
  • \(\mu\) is the mean
  • \(\sigma\) is the standard deviation.
cumulative probability
Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a given point. In the context of the standard normal distribution, it represents the area under the curve to the left of a given z-score.
For instance, a cumulative probability of 0.84 means there is an 84% chance the variable falls below a certain value.
When solving problems involving cumulative probability, standard normal distribution tables are typically used. These tables provide the cumulative probability for different z-scores, helping to determine the likelihood of certain outcomes.
area under the curve
The area under the curve in a probability distribution represents the total probability, which equals 1 (or 100%). In a standard normal distribution, the curve is symmetrical around the mean (zero), and the area under the curve provides insights into the probability corresponding to different ranges of z-scores.
For example, the area between z-scores of -1 and 1 is approximately 68%, indicating that about 68% of the data falls within one standard deviation of the mean.
Calculating areas under the standard normal curve is essential for understanding probabilities and making predictions about data.
range rule of thumb
The range rule of thumb is a simple way to estimate the standard deviation of a dataset based on its range. It's based on the empirical observation that most data in a normal distribution falls within a specific range.
According to the rule:
  • Approximately 68% of data falls within one standard deviation of the mean
  • About 95% within two standard deviations
  • And around 99.7% within three standard deviations.
This rule helps to quickly determine if data points are usual or unusual, based on how many standard deviations they are from the mean.
empirical rule
The empirical rule, also known as the 68-95-99.7 rule, provides a guideline for understanding the spread of data in a normal distribution. It states that:
  • About 68% of data falls within one standard deviation of the mean.
  • Approximately 95% within two standard deviations.
  • Roughly 99.7% within three standard deviations.
This rule is useful because it allows for quick estimations of what percentage of values lie within a certain range. It's especially handy for predicting data behavior and identifying outliers in a dataset.
For instance, in the given problem, finding the proportion of data within 3.5 standard deviations of the mean aligns with applying the empirical rule.

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Most popular questions from this chapter

A common design requirement is that an environment must fit the range of people who fall between the 5 th percentile for women and the 95 th percentile for men. In designing an assembly work table, we must consider sitting knee height, which is the distance from the bottom of the feet to the top of the knee. Males have sitting knee heights that are normally distributed with a mean of \(21.4\) in. and a standard deviation of \(1.2\) in.; females have sitting knee heights that are normally distributed with a mean of \(19.6\) in. and a standard deviation of \(1.1\) in. (based on data from the Department of Transportation). a. What is the minimum table clearance required to satisfy the requirement of fitting \(95 \%\) of men? Why is the 95 th percentile for women ignored in this case? b. The author is writing this exercise at a table with a clearance of \(23.5\) in. above the floor. What percentage of men fit this table, and what percentage of women fit this table? Does the table appear to be made to fit almost everyone?

Because they enable efficient procedures for evaluating answers, multiple choice questions are commonly used on standardized tests, such as the SAT or ACT. Such questions typically have five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of " \(a\)." a. After listing the 25 different possible samples, find the proportion of correct answers in each sample, then construct a table that describes the sampling distribution of the sample proportions of correct responses. b. Find the mean of the sampling distribution of the sample proportion. c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of correct responses? Does the mean of the sampling distribution of proportions always equal the population proportion?

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Instead of using \(0.05\) for identifying significant values, use the criteria that a value \(x\) is significantly high if \(P(x\) or greater \() \leq 0.025\) and a value is significantly low if \(P(x\) or less \() \leq 0.025 .\) Find the female back-to-knee length, separating significant values from those that are not significant. Using these criteria, is a female back-to-knee length of 20 in. significantly low?

According to the website www.torchmate.com, "manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter." Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of \(18.2\) in. and a standard deviation of \(1.0\) in. (based on data from the National Health and Nutrition Examination Survey). a. What percentage of men will fit into the manhole? b. Assume that the Connecticut's Eversource company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder breadth is less than \(18.5\) in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of \(18.5 \mathrm{in} . ?\) Why or why not?

What's wrong with the following statement? "Because the digits 0,1 , \(2, \ldots, 9\) are the normal results from lottery drawings, such randomly selected numbers have a normal distribution."
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