Textbook Question
A student’s test grade of 75 represents the 65th percentile of the grades. What percent of students scored higher than 75?
Ch. 2 - Descriptive Statistics
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- 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: 9780137493470Not the one you use?Change textbook
Chapter 2, Problem 2.R.28
In Exercises 27 and 28, find the range, mean, variance, and standard deviation of the sample data set.
Salaries (in dollars) of a random sample of teachers
62,222 56,719 50,259 45,120 47,692 45,985 53,489 71,534
Verified step by step guidance1
Step 1: Calculate the range of the data set. The range is the difference between the maximum and minimum values in the data set. Identify the maximum value (71,534) and the minimum value (45,120), then compute the range as Range = Max - Min.
Step 2: Calculate the mean (average) of the data set. Add all the salary values together and divide by the total number of data points. Use the formula: Mean = (Σx) / n, where Σx is the sum of all data points and n is the number of data points.
Step 3: Compute the variance of the sample. First, subtract the mean from each data point to find the deviation of each value. Then square each deviation, sum them up, and divide by (n - 1), where n is the number of data points. Use the formula: Variance = (Σ(x - Mean)²) / (n - 1).
Step 4: Calculate the standard deviation of the sample. The standard deviation is the square root of the variance. Use the formula: Standard Deviation = √Variance.
Step 5: Summarize the results. Report the range, mean, variance, and standard deviation of the sample data set, ensuring all calculations are accurate and clearly presented.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Descriptive Statistics
Descriptive statistics summarize and describe the main features of a data set. This includes measures such as the mean (average), variance (measure of data spread), and standard deviation (average distance from the mean). These statistics provide a quick overview of the data's central tendency and variability, which are essential for understanding the overall distribution of the sample.
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Range
The range is a simple measure of variability that indicates the difference between the highest and lowest values in a data set. It provides a quick sense of the spread of the data but does not account for how the values are distributed within that range. In the context of the given salaries, calculating the range helps to understand the extent of salary variation among the sampled teachers.
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Variance and Standard Deviation
Variance measures how far each number in the data set is from the mean and thus from every other number. It is calculated as the average of the squared differences from the mean. Standard deviation, the square root of variance, provides a more interpretable measure of spread in the same units as the data. Both are crucial for assessing the consistency and reliability of the data set.
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Related Practice
Textbook Question
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.
Textbook Question
The towing capacities (in pounds) of all the pickup trucks at a dealership have a bell-shaped distribution, with a mean of 11,830 pounds and a standard deviation of 2370 pounds. In Exercises 45– 48, use the corresponding z-score to determine whether the towing capacity is unusual. Explain your reasoning.
5,500 pounds
Textbook Question
In Exercises 21 and 22, determine whether the approximate shape of the distribution in the histogram is symmetric, uniform, skewed left, skewed right, or none of these.
Textbook Question
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.
Textbook Question
Describe the shape of the distribution for the histogram you made in Exercise 3 as symmetric, uniform, skewed left, skewed right, or none of these.
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