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Chapter 3: Problem 10

Find the population variance and standard deviation or the sample variance and standard deviation as indicated. $$ \text { Population: } 4,10,12,12,13,21 $$

Short Answer

Expert verified
Population Variance = 25, Population Standard Deviation = 5.

Step by step solution

01

- Find the Mean

First, find the mean (average) of the population values. Add all the values together and divide by the number of values. Mean, \(\bar{x}\) = (4 + 10 + 12 + 12 + 13 + 21) / 6 = 72 / 6 = 12.
02

- Subtract the Mean

Next, subtract the mean from each population value to find the deviation for each value. \(4 - 12 = -8\), \(10 - 12 = -2\), \(12 - 12 = 0\), \(12 - 12 = 0\), \(13 - 12 = 1\), \(21 - 12 = 9\).
03

- Square the Deviations

Square each of the deviations obtained in Step 2. \((-8)^2 = 64\), \((-2)^2 = 4\), \((0)^2 = 0\), \((0)^2 = 0\), \((1)^2 = 1\), \((9)^2 = 81\).
04

- Find the Sum of Squared Deviations

Add all the squared deviations together. \(64 + 4 + 0 + 0 + 1 + 81 = 150\).
05

- Calculate the Population Variance

To find the population variance, divide the sum of squared deviations by the number of values in the population. Population Variance, \(\text{蟽}^2\) = \(150 / 6 = 25\).
06

- Calculate the Standard Deviation

Finally, take the square root of the population variance to find the population standard deviation. Population Standard Deviation, \(\text{蟽}\) = \(\text{鈭�25} = 5\).

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Key Concepts

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

Mean Calculation
The mean is the first step in calculating population variance and standard deviation. To find the mean, simply add up all the values and then divide by the number of values. For instance, in our exercise, we have the population values 4, 10, 12, 12, 13, and 21. We add these up to get 72. We then divide by the total number of values, which is 6. So, the mean \(\bar{x}\) is calculated as follows: \[ \bar{x} = \frac{4 + 10 + 12 + 12 + 13 + 21}{6} = \frac{72}{6} = 12 \] This mean provides a central value in the dataset that will be used in subsequent steps.
Deviation from the Mean
The next step is to calculate the deviation of each value from the mean. Deviation measures how far each value is from the mean. To do this, subtract the mean from each value in the dataset. This helps in understanding how spread out the values are from the central point (mean). For our data:
  • \( 4 - 12 = -8 \)
  • \( 10 - 12 = -2 \)
  • \( 12 - 12 = 0 \)
  • \( 12 - 12 = 0 \)
  • \( 13 - 12 = 1 \)
  • \( 21 - 12 = 9 \)
These deviations show how each value compares to the mean.
Variance Calculation
Variance measures the average degree to which each value differs from the mean. First, we need to square each deviation to eliminate negative values and emphasize larger deviations. In our example:
  • \( (-8)^2 = 64 \)
  • \( (-2)^2 = 4 \)
  • \( (0)^2 = 0 \)
  • \( (0)^2 = 0 \)
  • \( (1)^2 = 1 \)
  • \( (9)^2 = 81 \)
Then, sum these squared deviations: \[ 64 + 4 + 0 + 0 + 1 + 81 = 150 \] Finally, divide by the number of values to get the population variance: \[\sigma^2 = \frac{150}{6} = 25 \] Variance helps in understanding how data points are spread out within the population.
Standard Deviation Calculation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It's the square root of the variance, bringing the units back to the original scale. For our data, the population variance was found to be 25. Taking the square root of this value, we get: \[\sigma = \sqrt{25} = 5 \] The standard deviation tells us, on average, how much the values deviate from the mean. A lower standard deviation indicates that the values are closer to the mean, whereas a higher standard deviation indicates more spread.

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Most popular questions from this chapter

Babies born after a gestation period of 32-35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34 -week gestation period baby weighs 2400 grams and a 40 -week gestation period baby weighs 3300 grams. What is the \(z\) -score for the 34 -week gestation period baby? What is the \(z\) -score for the 40 -week gestation period baby? Which baby weighs less relative to the gestation period?

A sample of 30 registered voters was surveyed in which the respondents were asked, "Do you consider your political views to be conservative, moderate, or liberal?" The results of the survey are shown in the table. $$ \begin{array}{lll} \hline \text { Liberal } & \text { Conservative } & \text { Moderate } \\ \hline \text { Moderate } & \text { Liberal } & \text { Moderate } \\ \hline \text { Liberal } & \text { Moderate } & \text { Conservative } \\ \hline \text { Moderate } & \text { Conservative } & \text { Moderate } \\ \hline \text { Moderate } & \text { Moderate } & \text { Liberal } \\ \hline \text { Liberal } & \text { Moderate } & \text { Liberal } \\ \hline \text { Conservative } & \text { Moderate } & \text { Moderate } \\ \hline \text { Liberal } & \text { Conservative } & \text { Liberal } \\ \hline \text { Liberal } & \text { Conservative } & \text { Liberal } \\ \hline \text { Conservative } & \text { Moderate } & \text { Conservative } \\\ \hline \end{array} $$ (a) Determine the mode political view. (b) Do you think it would be a good idea to rotate the choices conservative, moderate, or liberal in the question? Why?

According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: (a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? (c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

In December \(2014,\) the average price of regular unleaded gasoline excluding taxes in the United States was \(\$ 3.06\) per gallon, according to the Energy Information Administration. Assume that the standard deviation price per gallon is \(\$ 0.06\) per gallon to answer the following. (a) What minimum percentage of gasoline stations had prices within 3 standard deviations of the mean? (b) What minimum percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean? (c) What is the minimum percentage of gasoline stations that had prices between \(\$ 2.94\) and \(\$ 3.18 ?\)

The following data represent the percentage of workers who carpool to work for the 50 states plus Washington, D.C. Note: The minimum observation of \(7.2 \%\) corresponds to Maine and the maximum observation of \(16.4 \%\) corresponds to Hawaii. $$ \begin{array}{rrrrrrrrr} \hline 7.2 & 8.5 & 9.0 & 9.4 & 10.0 & 10.3 & 11.2 & 11.5 & 13.8 \\ \hline 7.8 & 8.6 & 9.1 & 9.6 & 10.0 & 10.3 & 11.2 & 11.5 & 14.4 \\ \hline 7.8 & 8.6 & 9.2 & 9.7 & 10.0 & 10.3 & 11.2 & 11.7 & 16.4 \\ \hline 7.9 & 8.6 & 9.2 & 9.7 & 10.1 & 10.7 & 11.3 & 12.4 & \\ \hline 8.1 & 8.7 & 9.2 & 9.9 & 10.2 & 10.7 & 11.3 & 12.5 & \\ \hline 8.3 & 8.8 & 9.4 & 9.9 & 10.3 & 10.9 & 11.3 & 13.6 & \\ \hline \end{array} $$ (a) Find the five-number summary. (b) Construct a boxplot. (c) Comment on the shape of the distribution.
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