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Chapter 6: Problem 47

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 _______ \(\%\) of the area is between \(z=-3\) and \(z=3\) (or within 3 standard deviations of the mean).

Short Answer

Expert verified
About 99.74% of the area is between z = -3 and z = 3.

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. When calculating areas under this curve, we'll refer to the z-scores, which measure the number of standard deviations away from the mean.
02

Identify the Z-Scores

In this exercise, the z-scores are given as -3 and 3. These z-scores represent the range within 3 standard deviations from the mean.
03

Use Z-Table or Calculator

Consult a standard normal distribution table (z-table) or use a calculator to find the cumulative area to the left of each z-score. For z = -3, this area is approximately 0.0013. For z = 3, this area is approximately 0.9987.
04

Calculate the Area Between Z-Scores

Subtract the cumulative area at z = -3 from the cumulative area at z = 3. This will be the area under the standard normal distribution curve between z = -3 and z = 3: \[ 0.9987 - 0.0013 = 0.9974 \]
05

Convert to Percentage

To convert the area to a percentage, multiply by 100: \[ 0.9974 \times 100 = 99.74\text{\text{%}} \]
06

Final Answer

The area under the curve between z = -3 and z = 3, or within 3 standard deviations of the mean, is about 99.74%.

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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 is a measure that tells you how many standard deviations away a particular value is from the mean of the distribution. When dealing with the standard normal distribution, the mean \( \mu \) is 0 and the standard deviation \( \sigma \) is 1. Calculating z-scores helps in comparing data points from different distributions or understanding where a specific data point lies within a distribution. The formula to calculate the z-score of a value x is: \[ z = \frac{x - \mu}{\sigma} \] By knowing z-scores, we can use standard normal distribution tables (z-tables) to find probabilities and areas under the curve.
Area Under the Curve
The area under the curve of a standard normal distribution represents probabilities or the proportion of data points. The total area under the curve is always equal to 1 or 100%. To find the area between two z-scores, we look at the cumulative area up to each z-score using a z-table. For example, to find the area between z = -3 and z = 3, we find:
  • Area to the left of z = -3 is approximately 0.0013.
  • Area to the left of z = 3 is approximately 0.9987.
To get the area between these z-scores, subtract the smaller cumulative area from the larger one. This gives us: \[ 0.9987 - 0.0013 = 0.9974 \] This area can then be converted to a percentage by multiplying by 100, resulting in 99.74%.
Empirical Rule
The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule for normal distributions. It states that:
  • About 68% of the data falls within 1 standard deviation of the mean.
  • About 95% of the data falls within 2 standard deviations of the mean.
  • About 99.7% of the data falls within 3 standard deviations of the mean.
This rule helps in understanding the spread and distribution of data in a normal distribution. The exercise highlights that about 99.74% of data falls within 3 standard deviations of the mean, which aligns closely with the empirical rule鈥檚 prediction of 99.7%.
Range Rule of Thumb
The range rule of thumb is a simple way to estimate the spread of data in a normal distribution. It provides an intuitive understanding by stating that most data within a normal distribution will lie within 4 standard deviations of the mean. According to this rule:
  • The minimum usual value can be estimated as \( \mu - 2 \sigma \)
  • The maximum usual value can be estimated as \( \mu + 2 \sigma \)
This rule is useful for making quick approximations about the spread of data without detailed calculations. In this exercise, the concept is demonstrated by finding that approximately 99.74% of the area is within 3 standard deviations of the mean, providing a more concrete figure for the distribution range.

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Most popular questions from this chapter

A normal distribution is informally described as a probability distribution that is bell-shaped when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

Common tests such as the SAT, ACT, Law School Admission test (LSAT), and Medical College Admission Test (MCAT) use multiple choice test questions, each with possible answers of \(a, b, c, d, e\), and each question has only one correct answer. We want to find the probability of getting at least 25 correct answers for someone who makes random guesses for answers to a block of 100 questions. If we plan to use the methods of this section with a normal distribution used to approximate a binomial distribution, are the necessary requirements satisfied? Explain.

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.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ For females, find the first quartile \(Q_{1}\), which is the length separating the bottom \(25 \%\) from the top \(75 \%\).

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.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Find the probability that a female has a back-to-knee length greater than \(24.0 \mathrm{in}\).

Assume that females have pulserates that are normally distributed with a mean of 74.0 beats per minute and a standard deviation of 12.5 beats per minute (based on Data Set 1 鈥淏ody Data鈥� in Appendix B). a. If 1 adult female is randomly selected, find the probability that her pulse rate is between 78 beats per minute and 90 beats per minute. b. If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean between 78 beats per minute and 90 beats per minute. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed \(30 ?\)
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