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Chapter 8: Problem 6

The number of engines owned per fire department was obtained from a random sample taken from the profiles of fire departments from across the United States (Firehouse/June 2003). $$\begin{array}{lllllll} \hline 29 & 8 & 7 & 33 & 21 & 26 & 0 & 11 & 4 & 54 & 7 & 4 \\ \hline \end{array}$$ Use the data to find a point estimate for each of the following parameters. a. Mean b. Variance c. Standard deviation

Short Answer

Expert verified
The Mean is the average value of the data, the Variance is a measure of how spread out the data is, and the Standard Deviation is a measure of the dispersion of the data set from the mean. The exact values depend on the actual computation using the given data.

Step by step solution

01

Calculating the Mean

Add all the numbers together and divide by the count of the numbers. The formula for mean is \[Mean = \frac{\sum_{i=1}^{n} x_i}{n}\] where \(x_i\) are the individual values and \(n\) is the total number of values.
02

Calculating the Variance

Subtract the mean from each value, square the result, add those together, and then divide by the count of numbers. The formula for variance is \[Variance = \frac{\sum_{i=1}^{n} (x_i - mean)^2}{n}.\] where \(x_i\) are the individual values, \(mean\) is the mean of the set, and \(n\) is the total number of values.
03

Calculating the Standard Deviation

Standard deviation is the square root of the variance. The formula for standard deviation is \[StdDev = \sqrt{Variance}\] where \(Variance\) is the value calculated in the previous step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
To find the mean, we begin by summing up all the given data points. In this case, these are the numbers of engines owned by various fire departments: 29, 8, 7, 33, 21, 26, 0, 11, 4, 54, 7, and 4. Start by adding them:
  • 29 + 8 + 7 + 33 + 21 + 26 + 0 + 11 + 4 + 54 + 7 + 4 = 204
Once you've obtained the sum, which is 204, you divide it by the total number of data points. Here, we have 12 values as our data set:
  • Mean = \( \frac{204}{12} \)
  • Mean = 17
The mean is a measure of central tendency that provides us with an average value. It tells us, on average, how many engines each fire department owns based on our sample data.
Variance Calculation
The variance helps us understand how spread out the data points are around the mean. To calculate variance, first, subtract the mean from each individual data point and square the result:
  • (29 - 17)^2 = 144
  • (8 - 17)^2 = 81
  • (7 - 17)^2 = 100
  • (33 - 17)^2 = 256
  • (21 - 17)^2 = 16
  • (26 - 17)^2 = 81
  • (0 - 17)^2 = 289
  • (11 - 17)^2 = 36
  • (4 - 17)^2 = 169
  • (54 - 17)^2 = 1369
  • (7 - 17)^2 = 100
  • (4 - 17)^2 = 169
Sum these squared differences:
  • 144 + 81 + 100 + 256 + 16 + 81 + 289 + 36 + 169 + 1369 + 100 + 169 = 2810
Then, divide this sum by the total number of data points, which is 12:
  • Variance = \( \frac{2810}{12} \)
  • Variance \approx 234.17
This variance value tells us about the variability of the number of engines each fire department owns.
Standard Deviation Calculation
The standard deviation is the square root of the variance and provides an average distance of each data point from the mean. It is a crucial measure that helps in understanding the distribution of data. Let's find it using the variance calculated previously:
  • Standard Deviation = \( \sqrt{234.17} \)
  • Standard Deviation \approx 15.31
The standard deviation is particularly useful as it is in the same unit as the data points themselves, which in this case is the number of engines. This provides a more intuitive understanding of the data spread and how much individual department ownership numbers stray from the average.

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Most popular questions from this chapter

Suppose you wanted to test the hypothesis that the mean minimum home service call charge for plumbers is at most \(\$ 95\) in your area. Explain the conditions that would exist if you made an error in decision by committing a a. type I error. b. type II error.

With a nationwide average drive time of about 24.3 minutes, Americans now spend more than 100 hours a year commuting to work, according to the U.S. Census Bureau's American Community Survey. Yes, that's more than the average 2 weeks of vacation time (80 hours) taken by many workers during a year. A random sample of 150 workers at a large nearby industry was polled about their commute time. If the standard deviation is known to be 10.7 minutes, is the resulting sample mean of 21.7 minutes significantly lower than the nationwide average? Use \(\alpha=0.01\) a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

A normally distributed population is known to have a standard deviation of \(5,\) but its mean is in question. It has been argued to be either \(\mu=80\) or \(\mu=90,\) and the following hypothesis test has been devised to settle the argument. The null hypothesis, \(H_{o}: \mu=80,\) will be tested by using one randomly selected data value and comparing it with the critical value of \(86 .\) If the data value is greater than or equal to \(86,\) the null hypothesis will be rejected. a. Find \(\alpha,\) the probability of the type I error. b. Find \(\beta,\) the probability of the type II error.

a. What decision is reached when the test statistic falls in the critical region? b. What decision is reached when the test statis- tic falls in the noncritical region?

The weights of full boxes of a certain kind of cereal are normally distributed with a standard deviation of 0.27 oz. A sample of 18 randomly selected boxes produced a mean weight of 9.87 oz. a. Find the \(95 \%\) confidence interval for the true mean weight of a box of this cereal. b. Find the \(99 \%\) confidence interval for the true mean weight of a box of this cereal. c. What effect did the increase in the level of confidence have on the width of the confidence interval?
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