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Chapter 3: Problem 76

A large population has a mean of 310 and a standard deviation of 37 . Using the empirical rule, find what percentage of the observations fall in the intervals \(\mu \pm 1 \sigma, \mu \pm 2 \sigma\), and \(\mu \pm 3 \sigma\).

Short Answer

Expert verified
Approximately 68% of the observations fall within the range \(\mu \pm 1 \sigma\), 95% fall within \(\mu \pm 2 \sigma\), and 99.7% of observations fall within \(\mu \pm 3 \sigma\).

Step by step solution

01

Apply the Empirical Rule for \(\mu \pm 1 \sigma\)

The first step is to calculate the range for the first interval which is \(\mu \pm 1 \sigma\). But in this case, it's not required because we will be using the Empirical Rule, which simply states that approximately 68% of the values lie within one standard deviation from the mean.
02

Apply the Empirical Rule for \(\mu \pm 2 \sigma\)

The next step is to calculate percentage of the values that lie within two standard deviations from the mean. Again, we directly apply the Empirical Rule, which states that about 95% of the values fall within this range.
03

Apply the Empirical Rule for \(\mu \pm 3 \sigma\)

The final step is to calculate the percentage of the values that fall within three standard deviations from the mean. According to the empirical rule, almost all the values (99.7%) fall within this range.

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Key Concepts

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

Standard Deviation
Standard deviation is a key concept in statistics. It's a measure of how spread out the numbers in a data set are.
Imagine you have a group of numbers, like quiz scores. If everyone scored around the same, the scores would be close to the mean, resulting in a low standard deviation.
  • Low standard deviation: Data points are close to the mean.
  • High standard deviation: Data points are spread out widely.
Mathematically, the standard deviation is calculated using the formula: \[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}\]where:
  • \(\sigma\) represents the standard deviation,
  • \(N\) is the number of data points,
  • \(x_i\) are the individual data points, and
  • \(\mu\) is the mean of the data set.
Understanding standard deviation helps in assessing the variability or consistency within a set of data.
Mean
The mean is what most people refer to as the "average." It provides a central value of a data set and is calculated by adding all the data points together and then dividing by the number of points.
For example, if you have the test scores 75, 85, 90, and 95, the mean would be calculated as: \[\mu = \frac{75 + 85 + 90 + 95}{4} = 86.25\]
  • The mean helps summarize data with a single number.
  • It is affected by extremely high or low values, also known as outliers.
The mean is essential for understanding data trends, but must be used with other statistical measures for comprehensive insights.
Normal Distribution
A normal distribution looks like a bell curve when you graph it. It is symmetric about the mean, where the mean, median, and mode are all equal.
In a normal distribution:
  • Approximately 68% of the data falls within one standard deviation of the mean.
  • About 95% falls within two standard deviations.
  • About 99.7% falls within three standard deviations.
The normal distribution is important because many natural phenomena and test results follow this pattern.
It helps in predicting outcomes and making decisions in real-world scenarios. The shape of the distribution often defines the precision and reliability of the derived statistics.
Statistics
Statistics is the science of collecting, analyzing, and interpreting large amounts of numerical data. It helps us understand what data is telling us and make informed decisions.
Key areas of statistics include:
  • Descriptive statistics: Summarizing and explaining data using numbers like mean, median, mode, and standard deviation.
  • Inferential statistics: Drawing conclusions from data, often involving some form of sampling and estimation.
Statistics are everywhere, from daily weather forecasts to deciding whether a new drug is effective.
It provides the tools for determining trends, testing theories, and communicating data-driven findings objectively. In practice, good statistical understanding can lead to more meaningful conclusions from data.

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Most popular questions from this chapter

One disadvantage of the standard deviation as a measure of dispersion is that it is a measure of absolute variability and not of relative variability. Sometimes we may need to compare the variability of two different data sets that have different units of measurement. The coefficient of variation is one such measure. The coefficient of variation, denoted by CV, expresses standard deviation as a percentage of the mean and is computed as follows: For population data: \(\mathrm{CV}=\frac{\sigma}{\mu} \times 100 \%\) For sample data: \(\quad \mathrm{CV}=\frac{s}{\bar{x}} \times 100 \%\) The yearly salaries of all employees who work for a company have a mean of \(\$ 62,350\) and a standard deviation of \(\$ 6820\). The years of experience for the same employees have a mean of 15 years and a standard deviation of 2 years. Is the relative variation in the salaries larger or smaller than that in years of experience for these employees?

The following data give the prices (in thousands of dollars) of 20 houses sold recently in a city. \(\begin{array}{llllllllll}184 & 297 & 365 & 309 & 245 & 387 & 369 & 438 & 195 & 390 \\ 323 & 578 & 410 & 679 & 307 & 271 & 457 & 795 & 259 & 590\end{array}\) Find the \(20 \%\) trimmed mean for this data set.

Although the standard workweek is 40 hours a week, many people work a lot more than 40 hours a week. The following data give the numbers of hours worked last week by 50 people. \(\begin{array}{llllllllll}40.5 & 41.3 & 41.4 & 41.5 & 42.0 & 42.2 & 42.4 & 42.4 & 42.6 & 43.3 \\ 43.7 & 43.9 & 45.0 & 45.0 & 45.2 & 45.8 & 45.9 & 46.2 & 47.2 & 47.5 \\ 47.8 & 48.2 & 48.3 & 48.8 & 49.0 & 49.2 & 49.9 & 50.1 & 50.6 & 50.6 \\ 50.8 & 51.5 & 51.5 & 52.3 & 52.3 & 52.6 & 52.7 & 52.7 & 53.4 & 53.9 \\\ 54.4 & 54.8 & 55.0 & 55.4 & 55.4 & 55.4 & 56.2 & 56.3 & 57.8 & 58.7\end{array}\) a. The sample mean and sample standard deviation for this data set are \(49.012\) and \(5.080\), respectively. Using Chebyshev's theorem, calculate the intervals that contain at least \(75 \%, 88.89 \%\), and \(93.75 \%\) of the data. b. Determine the actual percentages of the given data values that fall in each of the intervals that you calculated in part a. Also calculate the percentage of the data values that fall within one standard deviation of the mean. c. Do you think the lower endpoints provided by Chebyshev's theorem in part a are useful for this problem? Explain your answer. d. Suppose that the individual with the first number (54.4) in the fifth row of the data is a workaholic who actually worked \(84.4\) hours last week and not \(54.4\) hours. With this change now \(\bar{x}=49.61\) and \(s=7.10\). Recalculate the intervals for part a and the actual percentages for part b. Did your percentages change a lot or a little? e. How many standard deviations above the mean would you have to go to capture all 50 data values? What is the lower bound for the percentage of the data that should fall in the interval, according to the Chebyshev theorem.

Refer to Exercise \(3.115\). Suppose the times taken to learn the basics of this word processor by all students have a bell-shaped distribution with a mean of 200 minutes and a standard deviation of 20 minutes. a. Using the empirical rule, find the percentage of students who will learn the basics of this word processor in i. 180 to 220 minutes ii. 160 to 240 minutes "b. Using the empirical rule, find the interval that contains the time taken by \(99.7 \%\) of all students to learn this word processor.

Prepare a box-and-whisker plot for the following data: \(\begin{array}{llllllll}36 & 43 & 28 & 52 & 41 & 59 & 47 & 61 \\ 24 & 55 & 63 & 73 & 32 & 25 & 35 & 49 \\ 31 & 22 & 61 & 42 & 58 & 65 & 98 & 34\end{array}\)
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