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Ch. 3 - Describing, Exploring, and Comparing Data
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 3 - Describing, Exploring, and Comparing DataProblem 3.2.11
Chapter 3, Problem 3.2.11

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.


Smart Thermostats Listed below are selling prices (dollars) of smart thermostats tested by Consumer Reports magazine. Are any of the resulting statistics helpful in selecting a smart thermostat for purchase?




250 170 225 100 250 250 130 200 150 250 170 200 180 250

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Step 1: Organize the data. List all the given sample data points: 250, 170, 225, 100, 250, 250, 130, 200, 150, 250, 170, 200, 180, 250. Ensure the data is clearly written and ready for calculations.
Step 2: Calculate the range. The range is the difference between the maximum and minimum values in the dataset. Identify the maximum value (max) and the minimum value (min), then compute the range using the formula: Range=max-min.
Step 3: Compute the variance. First, calculate the mean (average) of the dataset using the formula: Mean=∑xn, where ∑x is the sum of all data points and n is the number of data points. Then, for each data point, calculate the squared difference from the mean, sum these squared differences, and divide by n-1 (since this is a sample). The formula for sample variance is: s2=∑(x-mean)2n-1.
Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance. Use the formula: s=s2, where s2 is the variance calculated in Step 3.
Step 5: Interpret the results. Discuss whether the range, variance, and standard deviation provide meaningful insights for selecting a smart thermostat. Consider whether the variation in prices is significant enough to influence a purchasing decision.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Range

The range is a measure of variation that indicates the difference between the highest and lowest values in a data set. It provides a simple way to understand the spread of the data. For example, in the given prices of smart thermostats, subtracting the lowest price from the highest price will yield the range, helping to assess the price variability.
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Variance

Variance quantifies the degree to which data points differ from the mean of the data set. It is calculated by averaging the squared differences between each data point and the mean. A higher variance indicates greater dispersion among the data points, which can be crucial for understanding the reliability of the prices of smart thermostats.
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Standard Deviation

Standard deviation is the square root of the variance and provides a measure of the average distance of each data point from the mean. It is expressed in the same units as the data, making it more interpretable. In the context of smart thermostat prices, a lower standard deviation would suggest that the prices are closely clustered around the mean, which can inform purchasing decisions.
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Related Practice
Textbook Question

Mean of Roller Coaster Speeds Listed below are maximum speeds (km/h) of randomly selected roller coasters in the United States. Find the mean.

7076978157151194651176545105

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Textbook Question

Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 0.58%, 0.29%, 0.13%, 0.14%, 0.15%, and 0.19%. Identify the single percentage growth rate that is the same as the six consecutive growth rates by computing the geometric mean of 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019.

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Textbook Question

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U.S. Presidents Based on Data Set 22 “Presidents” in Appendix B, at the time of their first inauguration, presidents have a mean age of 55.2 years and a standard deviation of 6.9 years. Is the minimum required 35-year age for a president significantly low?

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Textbook Question

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Textbook Question

The Empirical Rule Based on Data Set 1 “Body Data” in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/) Using the empirical rule, what is the approximate percentage of women with platelet counts


a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

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Textbook Question

Normal Distribution Examine the distribution shown in the histogram from Exercise 6. Does it appear that the sample data are from a population with a normal distribution? Why or why not?

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