Listed below are measured amounts of caffeine obtained in one can from each of 14 brands. Find the variance, data and leave the answer to the nearest thousandths. 57, 34, 30, 44, 33, 41, 60, 47, 56, 41, 31, 18, 12, 40
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![**Calculating the Variance of Caffeine Content**
Listed below are measured amounts of caffeine obtained in one can from each of 14 brands. Find the **variance**, for the given sample data and leave the answer to the nearest thousandths.
Caffeine Content Data:
57, 34, 30, 44, 33, 41, 60, 47, 56, 41, 31, 18, 12, 40
To calculate the variance, use the following formula:
\[ S^2 = \frac{SS}{n - 1} \]
where the Sum of Squares (SS) is given by:
\[ SS = \sum x^2 - \frac{(\sum x)^2}{n} \]
### Steps for Calculation
1. **Compute the sum \(\sum x\)**: Add all the data points together.
2. **Compute the sum of squares \(\sum x^2\)**: Square each data point and then add those squares together.
3. **Substitute into SS equation**: Plug the values from steps 1 and 2 into the SS formula.
4. **Compute the variance \(S^2\)**: Divide SS by \( n - 1 \), where \( n \) is the number of data points.
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By following these steps, students can accurately calculate the variance of the sample data, providing insight into the dispersion of caffeine content across different brands.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F027b99d7-4937-4ffe-8b7c-718538e4e3c7%2F115e9b9a-6bf6-4866-9f95-82cb1c16dd4a%2Ffotahz_processed.jpeg&w=3840&q=75)
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