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Chapter 3: Problem 53

All values of a data set must be within \(s \sqrt{n-1}\) of the mean. If a person collected 25 data values that had a mean of 50 and a standard deviation of 3 and you saw that one data value was \(67,\) what would you conclude?

Short Answer

Expert verified
The value 67 is an outlier as it falls outside the valid range.

Step by step solution

01

Understand the Condition

The exercise specifies that all data points must be within \(s \sqrt{n-1}\) of the mean, where \(s\) is the standard deviation and \(n\) is the number of data points in the set. This means each data point \(x\) must satisfy \(|x - \text{mean}| < s \sqrt{n-1}\).
02

Identify Given Values

We are given that the mean \(\mu = 50\), the standard deviation \(s = 3\), and the number of data points \(n = 25\). Also, we have a specific data point \(x = 67\) that we need to check.
03

Calculate the Limit

Calculate \(s \sqrt{n-1}\) to determine the allowed range around the mean. First, calculate \(n-1\):\(n - 1 = 25 - 1 = 24\).Now, find \(\sqrt{24}\):\(\sqrt{24} \approx 4.899\).Finally, multiply by the standard deviation \(s\):\(3 \times 4.899 \approx 14.697\).
04

Determine the Allowed Range

The allowed range around the mean is:\([\mu - s \sqrt{n-1}, \mu + s \sqrt{n-1}]= [50 - 14.697, 50 + 14.697]= [35.303, 64.697]\).
05

Check the Data Value

The data value \(67\) must be within the range \([35.303, 64.697]\).Since \(67 > 64.697\), \(67\) does not fall within the valid range.
06

Conclusion

Since the data point \(67\) is outside the allowed range, it does not conform to the requirement that all values be within \(s \sqrt{n-1}\) of the mean. Therefore, \(67\) should be considered an outlier or an error in data collection.

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Key Concepts

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

Understanding the Mean
The mean, often referred to as the "average," is a measure of central tendency in a data set. To calculate the mean of a data set, you sum up all the values and then divide by the total number of values. It provides a single value that represents a central point of the data distribution.

For example, if you have a set of numbers like 3, 7, and 14, the mean would be \[ \frac{3 + 7 + 14}{3} = \frac{24}{3} = 8.\]
  • The mean is useful in understanding the general trend of the data.
  • It is sensitive to extremely high or low values, known as outliers.
Defining a Data Set
A data set is a collection of data points that are gathered and organized for analysis. Each point in a data set is a single observation that, when combined, forms a larger picture of the subject being studied. Data sets can vary greatly in size, from just a few entries to thousands, or even millions, of points.

In our given example, a data set consists of 25 values. The range and distribution of these values influence the analysis, like calculating the mean or identifying outliers.
  • Larger data sets can provide more reliable statistical conclusions.
  • The structure of a data set affects how certain calculations, like mean and standard deviation, are performed.
Identifying an Outlier
An outlier is a value in a data set that is significantly higher or lower than the other values. Outliers can greatly affect statistical calculations, such as the mean and standard deviation. They may indicate a variability in measurement, a unique factor, or require further investigation altogether.

In the context of our exercise, the data point 67 was identified as an outlier because it fell outside the acceptable range: \([35.303, 64.697]\).
  • Outliers can skew analysis and lead to misleading results if not addressed.
  • Identifying outliers is crucial for accurate interpretation of data.
Exploring Range in Data Sets
The range of a data set is the difference between the highest and lowest values. It provides insight into the spread or dispersion of the data. Calculating the range can quickly highlight the variability in a set and signal the potential presence of outliers.

To find the range: \[\text{Range} = \text{Maximum Value} - \text{Minimum Value}\]In our exercise, the acceptable range of values was calculated to determine if each data point fits within a certain standard deviation from the mean: \([35.303, 64.697]\).
  • Range is simple to calculate and gives an immediate sense of data variability.
  • A limited range suggests less variability, while a larger range indicates more.

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Most popular questions from this chapter

Listed below are the enrollments for selected independent religiously controlled 4-year colleges that offer bachelor鈥檚 degrees only. Construct a grouped frequency distribution with six classes and find the mean and modal class. \(\begin{array}{llllllllll}{1013} & {1867} & {1268} & {1666} & {2309} & {1231} & {3005} & {2895} & {2166} & {1136} \\ {1532} & {1461} & {1750} & {1069} & {1723} & {1827} & {1155} & {1714} & {2391} & {2155} \\ {1412} & {1688} & {2471} & {1759} & {3008} & {2511} & {2577} & {1082} & {1067} & {1062} \\\ {1319} & {1037} & {2400} & {} & {} & {}\end{array}\)

Team batting averages for major league baseball in 2015 are represented below. Find the variance and standard deviation for each league. Compare the results. \(\begin{array}{lcccc}{} & {\mathrm{NL}} & {} & {\mathbf{A} \mathbf{L}} & {} & {} \\ \hline 0.242-0.246 & {3} & {0.244-0.249} & {3} \\ {0.247-0.251} & {6} & {0.250-0.255} & {6} \\ {0.252-0.256} & {1} & {0.256-0.261} & {2} \\\ {0.257-0.261} & {11} & {0.262-0.267} & {1} \\ {0.262-0.266} & {11} & {0.268-0.273} & {3} \\ {0.267-0.271} & {1} & {0.274-0.279} & {0}\end{array}\)

The geometric mean (GM) is defined as the nth root of the product of n values. The formula is $$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$ The geometric mean of 4 and 16 is $$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$ The geometric mean of \(1,3,\) and 9 is $$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$ The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a 20% raise after 1 year of service and a 10% raise after the second year of service, the average percentage raise per year is not 15 but 14.89%, as shown. $$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$ or $$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$ His salary is \(120 \%\) at the end of the first year and \(110 \%\) at the end of the second year. This is equivalent to an average of \(14.89 \%,\) since \(114.89 \%-100 \%=\) \(14.89 \% .\) $$\begin{array}{l}{\text { This answer can also be shown by assuming that }} \\\ {\text { the person makes } \$ 10,000 \text { to start and receives two }} \\\ {\text { raises of } 20 \% \text { and } 10 \% .}\end{array}$$ $$\begin{array}{l}{\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000} \\ {\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200}\end{array}$$ $$\text { His total salary raise is } \$ 3200 . \text { This total is equivalent to }$$ $$\begin{array}{l}{\$ 10,000 \cdot 14.89 \%=\$ 1489.00} \\ {\$ 11,489 \cdot 14.89 \%=\$ 1710.71}\end{array}$$ $${\$ 3199.71} \approx \$ 3200$$ Find the geometric mean of each of these. a. The growth rates of the Living Life Insurance Corporation for the past 3 years were 35, 24, and 18%. b. A person received these percentage raises in salary over a 4-year period: 8, 6, 4, and 5%. c. A stock increased each year for 5 years at these percentages: 10, 8, 12, 9, and 3%. d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were 1, 3, and 5.5%.

In a distribution of 160 values with a mean of 72, at least 120 fall within the interval 67鈥�77. Approximately what percentage of values should fall in the interval 62鈥�82? Use Chebyshev鈥檚 theorem.

The average age of Senators in the 114th congress was 61.7 years. If the standard deviation was 10.6, find the z scores of a senator who is 48 years old and one who is 66 years old.
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