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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.1c
Chapter 8, Problem 8.c.1c

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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Step 1: Understand the formula for standard deviation. The standard deviation measures the spread of data around the mean. The formula is: \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \), where \( \sigma \) is the standard deviation, \( x_i \) are the individual data points, \( \mu \) is the mean of the data, and \( N \) is the number of data points.
Step 2: Calculate the mean (\( \mu \)) 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 \( \mu = \frac{\sum x_i}{N} \).
Step 3: Compute the squared differences from the mean for each data point. For each \( x_i \), subtract the mean \( \mu \), then square the result: \( (x_i - \mu)^2 \). Perform this calculation for all data points in the set.
Step 4: Find the average of the squared differences. Sum all the squared differences calculated in Step 3, then divide by the total number of data points \( N \). This gives the variance: \( \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \).
Step 5: Take the square root of the variance to find the standard deviation. Use the formula \( \sigma = \sqrt{\text{Variance}} \). This final step provides the standard deviation 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.

Standard Deviation

Standard deviation is a statistical measure that quantifies 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 that the values are spread out over a wider range. It is calculated as the square root of the variance, which is the average of the squared differences from the mean.
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Mean

The mean, or average, is a measure of central tendency that is calculated by summing all the values in a dataset and dividing by the number of values. It provides a single value that represents the center of the data distribution. Understanding the mean is essential for calculating the standard deviation, as it serves as the reference point from which deviations are measured.
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Variance

Variance is a statistical measure that represents the degree of spread in a set of data points. It is calculated by taking the average of the squared differences between each data point and the mean. Variance is a key component in determining standard deviation, as the standard deviation is simply the square root of the variance, providing insight into the data's variability.
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Related Practice
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

d. Variance

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.


Null and Alternative Hypotheses and Test Statistic


b. Find the value of the test statistic.

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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?

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.


b. Use the P-value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

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.

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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