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Ch. 8 - Hypothesis Testing
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 8 - Hypothesis TestingProblem 8.c.1d
Chapter 8, Problem 8.c.1d

Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.
46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20
d. Variance

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Step 1: Understand the concept of variance. Variance measures the spread of a data set by calculating the average squared deviation of each data point from the mean. The formula for variance is: \( \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \), where \( x_i \) represents each data point, \( \bar{x} \) is the mean, and \( n \) is the number of data points.
Step 2: Calculate the mean (\( \bar{x} \)) of the data set. Add all the data points together and divide by the total number of data points. For this data set: \( 46, 51, 44, 51, 43, 32, 38, 48, 45, 27, 34, 29, 26, 28, 23, 26, 28, 40, 16, 20 \). Use the formula \( \bar{x} = \frac{\sum x_i}{n} \).
Step 3: Compute the squared deviations for each data point. Subtract the mean (\( \bar{x} \)) from each data point (\( x_i \)) and square the result. For example, if \( x_1 = 46 \) and \( \bar{x} \) is the mean, calculate \( (46 - \bar{x})^2 \). Repeat this for all data points.
Step 4: Sum all the squared deviations. Add together all the squared deviations calculated in Step 3. This gives \( \sum (x_i - \bar{x})^2 \).
Step 5: Divide the sum of squared deviations by the number of data points (\( n \)) to find the variance. Use the formula \( \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \). This final step provides the variance of the data set.

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Key Concepts

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

Variance

Variance is a statistical measure that represents the degree of spread or dispersion of a set of values. It quantifies how much the individual data points differ from the mean of the dataset. A higher variance indicates that the data points are more spread out, while a lower variance suggests they are closer to the mean. Variance is calculated by taking the average of the squared differences from the mean.
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Mean

The mean, often referred to as the average, is a measure of central tendency that summarizes a set of values by dividing the sum of all values by the number of values. It provides a single value that represents the center of the data distribution. In the context of the lightning deaths data, calculating the mean is essential for determining how far each individual death count deviates from this central value when computing variance.
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Standard Deviation

Standard deviation is a statistic that measures the amount of variation or dispersion in a set of values. It is the square root of the variance and provides a more interpretable measure of spread in the same units as the original data. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates a wider spread. Understanding standard deviation is crucial for interpreting the variance in the context of the lightning deaths data.
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Related Practice
Textbook Question

RESAMPLING

c. When testing a claim about a proportion or mean or standard deviation, what is an important advantage of using a resampling method instead of the parametric method described in the preceding sections of this chapter?

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Textbook Question

Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.

46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20

e. Range

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Textbook Question

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.


c. Use the sample data to construct a 95% confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1?

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Textbook Question

Statistical Literacy and Critical Thinking

In Exercises 1–4, use the results from a Hankook Tire Gauge Index survey of a simple random sample of 1020 adults. Among the 1020 respondents, 86% rated themselves as above average drivers. We want to test the claim that more than 3/4 of adults rate themselves as above average drivers.


Number and Proportions


c. For the hypothesis test, identify the value used for the population proportion and use the symbol that represents it.

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Textbook Question

Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of recent and consecutive years. Find the values of the indicated statistics.

46 51 44 51 43 32 38 48 45 27 34 29 26 28 23 26 28 40 16 20

c. standard deviation

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Textbook Question

Lightning Deaths Based on the results given in Cumulative Review Exercise 6, assume that for a randomly selected lightning death, there is a 0.8 probability that the victim is a male.

a. Find the probability that three random people killed by lightning strikes are all males.

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