2.10 x and s by Hand. Radon is a naturally occurring gas and is the second leading cause of lung cancer in the United States." It comes from the natural breakdown of uranium in the soil and enters buildings through cracks and other holes in the foundations. Found throughout the United States, levels vary considerably from state to state. Several methods can reduce the levels of radon in your home, and the Environmental Protection Agency recommends using one of these if the measured level in your home is above 4 picocuries per liter. Four readings from Franklin County, Ohio, where the county average is 8.4 picocuries per liter, were 7.1, 11.6, 8.1, and 13.4. (a) Find the mean step-by-step. That is, find the sum of the four observations and divide by 4. (b) Find the standard deviation step-by-step. That is, find the deviation of each observation from the mean, square the deviations, then obtain the variance- and the standard deviation. Example 2.7 (page 58) shows the method. (c) Now enter the data into your calculator and use the mean and standard deviation buttons to obtain X and s. Do the results agree with your hand calculations?
2.10 x and s by Hand. Radon is a naturally occurring gas and is the second leading cause of lung cancer in the United States." It comes from the natural breakdown of uranium in the soil and enters buildings through cracks and other holes in the foundations. Found throughout the United States, levels vary considerably from state to state. Several methods can reduce the levels of radon in your home, and the Environmental Protection Agency recommends using one of these if the measured level in your home is above 4 picocuries per liter. Four readings from Franklin County, Ohio, where the county average is 8.4 picocuries per liter, were 7.1, 11.6, 8.1, and 13.4. (a) Find the mean step-by-step. That is, find the sum of the four observations and divide by 4. (b) Find the standard deviation step-by-step. That is, find the deviation of each observation from the mean, square the deviations, then obtain the variance- and the standard deviation. Example 2.7 (page 58) shows the method. (c) Now enter the data into your calculator and use the mean and standard deviation buttons to obtain X and s. Do the results agree with your hand calculations?
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Can you please help me with answer 2.10 x and s by hand- a, b and c? Thank you
Transcribed Image Text:**APPLY YOUR KNOWLEDGE**
**2.10 \(\bar{x}\) and \(s\) by Hand.** Radon is a naturally occurring gas and is the second leading cause of lung cancer in the United States. It comes from the natural breakdown of uranium in the soil and enters buildings through cracks and other holes in the foundations. Found throughout the United States, levels vary considerably from state to state. Several methods can reduce the levels of radon in your home, and the Environmental Protection Agency recommends using one of these if the measured level in your home is above 4 picocuries per liter. Four readings from Franklin County, Ohio, where the county average is 8.4 picocuries per liter, were 7.1, 11.6, 8.1, and 13.4.
(a) Find the mean step-by-step. That is, find the sum of the four observations and divide by 4.
(b) Find the standard deviation step-by-step. That is, find the deviation of each observation from the mean, square the deviations, then obtain the variance and the standard deviation. Example 2.7 (page 58) shows the method.
(c) Now enter the data into your calculator and use the mean and standard deviation buttons to obtain \(\bar{x}\) and \(s\). Do the results agree with your hand calculations?
**2.11 \(\bar{x}\) and \(s\) Are Not Enough.** The mean \(\bar{x}\) and standard deviation \(s\) measure center and variability but are not a complete description of a distribution. Data sets with different shapes can have the same mean and standard deviation. To demonstrate this fact, use your calculator to find \(\bar{x}\) and \(s\) for these two small data sets. Then make a stemplot of each and comment on the shape of each distribution.
**DATASET2**
| | Data A | Data B |
|-------|------------|------------|
| Values| 9.14 | 6.58 |
| | 8.14 | 5.76 |
| | 8.74 | 7.71 |
| | 8.77 | 8.84 |
| | 9.26 | 8.47 |
| | 8.10 |
![### Example 2: Calculating the Standard Deviation
#### Overview
Georgia Southern University had 2786 students with regular admission in its freshman class of 2015. For each student, data are available on their SAT and ACT scores (if taken), high school GPA, and the college within the university to which they were admitted. In Exercise 3.49 (page 98), the full data set for the SAT Mathematics scores will be examined. Here are the first five observations from that data set:
\[ 490, 580, 450, 570, 650 \]
We will compute \( \bar{x} \) and \( s \) for these students. First find the mean:
\[ \bar{x} = \frac{490 + 580 + 450 + 570 + 650}{5} = \frac{2740}{5} = 548 \]
#### Detailed Explanation
**Figure 2.3** displays the data as points above the number line, with their mean marked by an asterisk (*). The arrows mark two of the deviations from the mean. The deviations show how variable the data are about their mean. They are the starting point for calculating the variance and the standard deviation.
#### Data Breakdown
| **Observations** (\( x_i \)) | **Deviations** (\( x_i - \bar{x} \)) | **Squared Deviations** (\( (x_i - \bar{x})^2 \)) |
|------------------------------|--------------------------------------|-------------------------------------------------|
| 490 | 490 - 548 = -58 | \((-58)^2 = 3,364\) |
| 580 | 580 - 548 = 32 | \(32^2 = 1,024\) |
| 450 | 450 - 548 = -98 | \((-98)^2 = 9,604\) |
| 570 | 570 - 548 = 22 | \(22^2 = 484\) |
| 650 | 650 - 548 = 102 | \(102^2 = 10,404\) |
This table provides a detailed breakdown of the observations, their respective deviations from the mean, and the squared deviations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fdaeb71-b2d8-4fdd-a245-4eedebfac750%2F4825e5af-4a8c-4b40-be78-6d7cd80ad1ff%2Fxankv1d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example 2: Calculating the Standard Deviation
#### Overview
Georgia Southern University had 2786 students with regular admission in its freshman class of 2015. For each student, data are available on their SAT and ACT scores (if taken), high school GPA, and the college within the university to which they were admitted. In Exercise 3.49 (page 98), the full data set for the SAT Mathematics scores will be examined. Here are the first five observations from that data set:
\[ 490, 580, 450, 570, 650 \]
We will compute \( \bar{x} \) and \( s \) for these students. First find the mean:
\[ \bar{x} = \frac{490 + 580 + 450 + 570 + 650}{5} = \frac{2740}{5} = 548 \]
#### Detailed Explanation
**Figure 2.3** displays the data as points above the number line, with their mean marked by an asterisk (*). The arrows mark two of the deviations from the mean. The deviations show how variable the data are about their mean. They are the starting point for calculating the variance and the standard deviation.
#### Data Breakdown
| **Observations** (\( x_i \)) | **Deviations** (\( x_i - \bar{x} \)) | **Squared Deviations** (\( (x_i - \bar{x})^2 \)) |
|------------------------------|--------------------------------------|-------------------------------------------------|
| 490 | 490 - 548 = -58 | \((-58)^2 = 3,364\) |
| 580 | 580 - 548 = 32 | \(32^2 = 1,024\) |
| 450 | 450 - 548 = -98 | \((-98)^2 = 9,604\) |
| 570 | 570 - 548 = 22 | \(22^2 = 484\) |
| 650 | 650 - 548 = 102 | \(102^2 = 10,404\) |
This table provides a detailed breakdown of the observations, their respective deviations from the mean, and the squared deviations.
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