Sure, let's break down the question regarding the formulas used to measure the spread of radii of 18 randomly selected tennis balls. Here's a detailed explanation with the correct notations and formulas:
1. Sample Variance [tex]\( s^2 \)[/tex]:
The sample variance measures the spread of the radii values around the sample mean [tex]\(\bar{x}\)[/tex]. The formula for sample variance is:
[tex]\[
s^2 = \frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1}
\][/tex]
This formula divides by [tex]\( n - 1 \)[/tex] to account for the loss of a degree of freedom in estimating the population variance from the sample.
2. Sample Standard Deviation [tex]\( s \)[/tex]:
The sample standard deviation is the square root of the sample variance, providing a measure of spread in the same units as the original data:
[tex]\[
s = \sqrt{\frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1}}
\][/tex]
By taking the square root, we convert the variance back to the original unit of measurement (e.g., millimeters if the radii are measured in millimeters).
3. Population Variance [tex]\( \sigma^2 \)[/tex]:
When considering the entire population (i.e., all tennis balls from the factory), the population variance measures the spread of radii values around the population mean [tex]\( \mu \)[/tex]:
[tex]\[
\sigma^2 = \frac{\sum_{i=1}^n (a_i - \mu)^2}{n}
\][/tex]
Here, [tex]\( \mu \)[/tex] is the true mean of the entire population, and we divide by [tex]\( n \)[/tex] because we are not estimating based on a sample.
4. Population Standard Deviation [tex]\( \sigma \)[/tex]:
Similarly, the population standard deviation is the square root of the population variance, again providing a measure of spread in the same units as the original data:
[tex]\[
\sigma = \sqrt{\frac{\sum_{i=1}^n (a_i - \mu)^2}{n}}
\][/tex]
This value indicates the average distance of each data point from the population mean [tex]\( \mu \)[/tex].
To summarize, the formulas you have for measuring the spread of the radii of the tennis balls are:
1. Sample Variance:
[tex]\[
s^2 = \frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1}
\][/tex]
2. Sample Standard Deviation:
[tex]\[
s = \sqrt{\frac{\sum_{i=1}^n (a_i - \bar{x})^2}{n - 1}}
\][/tex]
3. Population Variance:
[tex]\[
\sigma^2 = \frac{\sum_{i=1}^n (a_i - \mu)^2}{n}
\][/tex]
4. Population Standard Deviation:
[tex]\[
\sigma = \sqrt{\frac{\sum_{i=1}^n (a_i - \mu)^2}{n}}
\][/tex]
