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Research by the Food and Drug Administration (FDA) shows that acrylamide (a possible cancer-causing substance) forms in high-carbohydrate foods cooked at high temperatures and that acrylamide levels can vary widely even within the same brand of food (Associated Press, December 6, 2002). FDA scientists analyzed McDonald's French fries purchased at seven different locations and found the following acrylamide levels: \(\begin{array}{lllllll}497 & 193 & 328 & 155 & 326 & 245 & 270\end{array}\) a. Compute the mean acrylamide level and the seven deviations from the mean. b. Verify that, except for the effect of rounding, the sum of the deviations from mean is equal to 0 for this data set. (If you rounded the sample mean or the deviations, your sum may not be exactly zero, but it should be close to zero if you have computed the deviations correctly.) c. Calculate the variance and standard deviation for this data set.

Step by step solution

01

Calculate the Mean

To find the mean, sum up all the data points and divide by the number of data points. The data points given are: 497, 193, 328, 155, 326, 245, and 270. So, the mean (\( \mu \)) is \( \frac{497+193+328+155+326+245+270}{7} \).

02

Find the Deviations

The deviation from the mean is simply the difference between each data point and the mean. Subtract the mean from each data point (e.g., \(497 - \mu\), \(193 - \mu\), etc.) to get each deviation.

03

Verification of the Sum of Deviations

Add all the calculated deviations. In statistical theory, the sum of the deviations from the mean should be equal to 0. This is because the deviations above the mean cancel out the deviations below the mean.

04

Compute Variance

Variance (\( \sigma^2 \)) is the average of the squared deviations. To get this, square each deviation, sum them up, and divide the result by the number of data points minus 1 (used because this is a sample, not a whole population). The formula for the sample variance is \( \sigma^2 = \frac{\Sigma (x-\mu)^2}{n-1} \).

05

Compute Standard Deviation

The standard deviation (\( \sigma \)) is the square root of variance. So, \( \sigma = \sqrt{\sigma^2} \).

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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 a measure of central tendency. It's often termed as the 'average' of a data set. To find the mean, you add together all the numbers in your data set and then divide by the number of data points. This is a straightforward calculation, but it's crucial as it forms the basis for many other analyses.

For example, if you have the acrylamide levels from the data set as: 497, 193, 328, 155, 326, 245, and 270, the mean is found as follows:

• First, sum the acrylamide levels: 497 + 193 + 328 + 155 + 326 + 245 + 270 = 2014.
• Next, divide by the number of data points, which is 7: \(\mu = \frac{2014}{7} \approx 287.71\)

Thus, the mean acrylamide level is approximately 287.71, providing a central value around which the levels tend to cluster.

Variance and Standard Deviation

Variance and standard deviation are statistical measures that tell us how much the values in a data set vary or deviate from the mean.

Variance, denoted as \( \sigma^2 \), is the average of the squared differences from the mean. This accounts for all variability in data. When data points are spread out, the variance will be large; when they are close together, the variance will be small.

• To calculate variance, first, find each deviation from the mean and square it.
• Then, sum all those squared deviations.
• Finally, divide by the number of data points minus one (since it's a sample): \[\sigma^2 = \frac{\Sigma (x-\mu)^2}{n-1}\]

The standard deviation, \( \sigma \), is the square root of the variance, which gives a measure of spread in the same units as the original data. This is easier to interpret and helps compare different data sets.

Sum of Deviations from Mean

The sum of deviations is a foundational concept in statistics showing how data values vary with respect to the mean. If you take each value in your data set, subtract the mean, and then sum these differences, the total should be zero.

This occurs because positive deviations (data points above the mean) counterbalance the negative deviations (below the mean). But due to rounding, this sum can sometimes slightly differ from zero in practical calculations. As seen in the exercise, verifying this sum helps ensure calculations of mean and deviations have been done correctly. It's a basic but an essential check on statistical calculations.

Sample Data Analysis

Sample data analysis involves calculating statistics from a sample and using them to infer properties of the larger population. The methods used for data analysis in samples differ slightly from whole-population analysis due to the smaller size and increased variability.

The advantage of analyzing a sample, as with the acrylamide levels from McDonald's fries at seven locations, is efficiency. It's often impractical to gather complete data from an entire population,
so working with a sample gives us a glimpse of the population's characteristics.

• The sample mean offers insight into the population mean.
• Lastly, measures like variance and standard deviation show how much spread or variability there is within the data.

Ultimately, understanding sample data analysis helps translate smaller observations into larger insights about trends and variability in a population.