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Chapter 2: Problem 130

Rough Rule of Thumb for the Standard Deviation According to the \(95 \%\) rule, the largest value in a sample from a distribution that is approximately symmetric and bell-shaped should be between 2 and 3 standard deviations above the mean, while the smallest value should be between 2 and 3 standard

Short Answer

Expert verified
For a distribution that is approximately symmetric and bell-shaped, according to the \(95\%\) rule, the largest value should be between \(碌 + 2蟽\) and \(碌 + 3蟽\), while the smallest value should be between \(碌 - 2蟽\) and \(碌 - 3蟽\), where '碌' is the mean and '\(蟽\)' is the standard deviation.

Step by step solution

01

Understanding the 95% Rule for Standard Deviation

According to the \(95\%\) rule, which is also known as the Empirical Rule or the Normal Distribution Rule, approximately \(95\%\) of the data values in a symmetric and bell-shaped distribution (also known as a normal distribution) will fall within 2 standard deviations of the mean.
02

Apply the 95% Rule to Determine the Range for the Largest Value

Since the largest value in the sample should be between 2 and 3 standard deviations above the mean, we find this range by adding 2 and 3 times the standard deviation to the mean. If '碌' is the mean and '\(蟽\)' is the standard deviation, then the largest value should fall in the range \(碌 + 2蟽\) to \(碌 + 3蟽\).
03

Apply the 95% Rule to Determine the Range for the Smallest Value

Similarly, since the smallest value in the sample should be between 2 and 3 standard deviations below the mean, we find this range by subtracting 2 and 3 times the standard deviation from the mean. Again, if '碌' is the mean and '\(蟽\)' is the standard deviation, then the smallest value should fall in the range \(碌 - 2蟽\) to \(碌 - 3蟽\).

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Key Concepts

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

Normal Distribution
The concept of a normal distribution is a fundamental idea in statistics, often used to describe a data set. Imagine drawing a bell-shaped curve. This curve represents how data points are distributed along a number line. In many natural processes, data tends to cluster around a central point, creating this characteristic shape.

In a normal distribution:
  • The mean (average), median, and mode of the data set are all located at the peak of the curve.
  • The distribution is symmetric 鈥� the left side is a mirror image of the right side.
  • The spread or width of the curve is determined by the standard deviation.
  • Approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is known as the Empirical Rule.
By understanding normal distribution, you can make predictions about where the majority of data points will fall relative to the mean.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. It gives you an idea of how much variance exists from the average (mean) value.

The steps to calculate standard deviation are:
  • Find the mean of the data set.
  • Subtract the mean from each data point and square the result to eliminate negative numbers.
  • Calculate the average of these squared differences.
  • Take the square root of this average to obtain the standard deviation.
A low standard deviation indicates that data points are close to the mean, suggesting less variability. Conversely, a high standard deviation indicates a wide spread around the mean. Knowing the standard deviation helps in understanding the distribution of data, as well as in identifying outliers. It is vital in applying the Empirical Rule to predict where a particular data point may fall.
Symmetric Distribution
In a symmetric distribution, the left and right sides of the distribution are mirror images of each other. This symmetry is a core characteristic of normal distribution, but not all symmetric distributions must be normal.

Some features of a symmetric distribution include:
  • The center of the distribution coincides with the peak, and this is where the mean, median, and mode are equal.
  • Outliers, if present, are evenly distributed on both sides, maintaining the symmetry.
  • These distributions are easier to analyze and interpret because of their predictable nature.
Understanding symmetric distribution allows us to apply specific statistical rules, like the Empirical Rule, effectively. Recognizing symmetry in data can help in making accurate predictions about data behavior around the mean.

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Most popular questions from this chapter

US Obesity Levels by State over Many Years Exercise 2.237 deals with some graphs showing information about the distribution of obesity rates in states over three different years. The website http://stateofobesity.org/adult- obesity/ shows similar graphs for a wider selection of years. Use the graphs at the website to answer the questions below. (a) What is the first year recorded in which the 15 \(19 \%\) category was needed, and how many states are in that category in that year? What is the first year the \(20-24 \%\) category was needed? The \(25-29 \%\) category? The \(30-34 \%\) category? The \(35 \%+\) category? (b) If you are in the US right now as you read this, what state are you in? In what obesity rate category did that state fall in \(1990 ?\) In what category is it in now? If you are not in the US right now as you read this, find out the current percent obese of the country you are in. Name a state (and year, if needed) which roughly matches that value.

For the datasets. Use technology to find the following values: (a) The mean and the standard deviation. (b) The five number summary. 25, 72, 77, 31, 80, 80, 64, 39, 75, 58, 43, 67, 54, 71, 60

Sketch a curve showing a distribution that is symmetric and bell-shaped and has approximately the given mean and standard deviation. In each case, draw the curve on a horizontal axis with scale 0 to 10. Mean 3 and standard deviation 1.

2.240 What Do You Call a Sweetened Carbonated Beverage? If you reach for a sweetened carbonated beverage, do you refer to it as soda, pop, coke, or a soft drink? Different regions of the United States use different terms, as shown in this heat map: discovermagazine.com/galleries/2013/june/regionalus-language-dialect. \(^{88}\) If you live in the United States, specify where you live and which term is predominantly used there. If you do not live in the United States, choose a location in the US and specify the location and which term is predominantly used there.

Sketch a curve showing a distribution that is symmetric and bell-shaped and has approximately the given mean and standard deviation. In each case, draw the curve on a horizontal axis with scale 0 to 10. Mean 5 and standard deviation 2.
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