Domanda del libro di testo
From a random sample of airplanes, the number of defects found in their fuselages are listed. Find the sample mean and the sample standard deviation of the data.
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Ch. 2 - Descriptive Statistics
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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TOC.tableOfContents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 2.R.31
The mean sale per customer for 40 customers at a gas station is \$32.00, with a standard deviation of \$4.00. Using Chebychev’s Theorem, determine at least how many of the customers spent between \$24.00 and \$40.00.
Guida verificata passo dopo passo1
Step 1: Recall Chebychev's Theorem, which states that for any dataset (regardless of distribution), at least \(1 - \frac{1}{k^2}\) of the data values lie within \(k\) standard deviations of the mean, where \(k > 1\).
Step 2: Calculate the number of standard deviations \(k\) that the interval \([24, 40]\) represents. Use the formula \(k = \frac{|X - \mu|}{\sigma}\), where \(X\) is the boundary value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. For the lower boundary \(24\), calculate \(k = \frac{32 - 24}{4}\). For the upper boundary \(40\), calculate \(k = \frac{40 - 32}{4}\).
Step 3: Verify that the \(k\) values for both boundaries are the same, as the interval is symmetric around the mean. Use this \(k\) value in Chebychev's formula \(1 - \frac{1}{k^2}\) to determine the proportion of data within this range.
Step 4: Multiply the proportion obtained from Chebychev's formula by the total number of customers (40) to find the minimum number of customers who spent between \(24\) and \(40\).
Step 5: Interpret the result in the context of the problem, ensuring that the answer represents the minimum number of customers as guaranteed by Chebychev's Theorem.
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Mean
The mean is the average value of a set of numbers, calculated by summing all values and dividing by the count of values. In this context, the mean sale per customer is $32.00, indicating the typical amount spent by customers at the gas station.
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Percorso guidato
04:52
Calculating the Mean
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. Here, the standard deviation of $4.00 shows how much individual customer spending varies from the average.
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Percorso guidato
08:45Calculating Standard Deviation
Chebyshev's Theorem
Chebyshev's Theorem states that for any dataset, regardless of its distribution, at least (1 - 1/k²) of the data values will fall within k standard deviations of the mean. This theorem is useful for determining the minimum proportion of data within a specified range, allowing us to calculate how many customers spent between $24.00 and $40.00 based on the given mean and standard deviation.
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