Question

The number of raisins in each 14 miniboxes (1/2-ounce size) was counted for a generic brand and for Sunmaid brand raisins. The two data sets are shown here: $$ \begin{array}{llll|llll} \multicolumn{3}{l|} {\text { Generic Brand }} & \multicolumn{3}{|c} {\text { Sunmaid }} \\ \hline 25 & 26 & 25 & 28 & 25 & 29 & 24 & 24 \\ 26 & 28 & 28 & 27 & 28 & 24 & 28 & 22 \\ 26 & 27 & 24 & 25 & 25 & 28 & 30 & 27 \\ 26 & 26 & & & 28 & 24 & & \end{array} $$ a. What are the mean and standard deviation for the generic brand? b. What are the mean and standard deviation for the Sunmaid brand? c. Compare the centers and variabilities of the two brands using the results of parts a and \(b\).

Short answer

Answer: The centers for both brands are similar, with Generic brand having a slightly higher mean (26.7273) than Sunmaid brand (26.5882). However, Generic brand has a smaller standard deviation (1.3438) than Sunmaid brand (2.1049), indicating a lower variability and more consistency in the number of raisins per minibox for the Generic brand.

Step-by-step solution

Organize the data

Create two separate lists for Generic brand and Sunmaid brand, representing their raisin counts per half-ounce minibox respectively. Generic Brand: 25, 26, 25, 26, 28, 28, 26, 27, 24, 26, 26 Sunmaid Brand: 28, 25, 29, 24, 24, 27, 28, 24, 28, 22, 25, 25, 28, 30, 27, 28, 24

Calculate means

To calculate the means for each brand, we sum the raisin counts and divide by the total number of miniboxes for each brand. Mean for Generic brand: \(\frac{25 + 26 + 25 ... + 24 + 26 + 26}{11} = 26.7273\) Mean for Sunmaid brand: \(\frac{28 + 25 + 29 ... + 28 + 30 + 27...+28+24}{17} = 26.5882\)

Calculate Deviations

To calculate the deviation for each data value, subtract the mean from the corresponding data value. Then square the result. Deviations are as follows: Generic brand deviations: \((25-26.7273)^2, (26-26.7273)^2, ...\) Sunmaid brand deviations: \((28-26.5882)^2, (25-26.5882)^2, ...\)

Calculate Standard Deviations

To calculate the standard deviation for each brand, sum up the squared deviations and divide by the total number of miniboxes in that brand minus 1, then take the square root of the quotient. Standard Deviation for Generic brand: \(\sqrt{\frac{(25-26.7273)^2 + (26-26.7273)^2 + ...}{11-1}} = 1.3438\) Standard Deviation for Sunmaid brand: \(\sqrt{\frac{(28-26.5882)^2 + (25-26.5882)^2 + ...}{17-1}} = 2.1049\)

Compare the means and standard deviations

The means for the two brands are very close, with Generic brand having a slightly higher mean at 26.7273 and Sunmaid brand at 26.5882. This suggests that the centers for both distributions are similar. However, the standard deviations for the two brands are different. Generic brand has a smaller standard deviation (1.3438) than Sunmaid brand (2.1049). This shows that the variability in the raisin counts for Generic brand is lower, indicating more consistency in the number of raisins per minibox than Sunmaid brand.

Key concepts

Mean Calculation

The mean is a fundamental concept in statistics that helps us understand the average of a data set. To calculate the mean, sum up all the values and divide by the number of values. This calculation gives an insight into the central tendency or 'average'.
For example, in the raisin data sets, the means were calculated for both brands.

• For the Generic brand, the sum of raisin counts across 11 boxes was calculated, then divided by 11, resulting in a mean of approximately 26.73 raisins per box.
• For the Sunmaid brand, a similar process was followed across 17 boxes, giving a mean of approximately 26.59 raisins per box.

The mean helps in understanding how many raisins you expect to find in a minibox, on a typical basis.

Data Variability

Data variability refers to how much the data points in a dataset differ from each other and from the mean. This is where concepts like standard deviation come into play. A higher variability implies that the data points are spread out over a larger range of values.
To measure variability, one often uses standard deviation. It shows how much individual data points deviate from the mean.

• The generic brand had a standard deviation of approximately 1.34, which suggests its raisin counts are quite close to the mean.
• The Sunmaid brand's standard deviation was about 2.10, indicating greater spread or inconsistency in raisin counts compared to the generic brand.

A small standard deviation can imply consistency, while a larger value might show more variation or unpredictability.

Data Comparison

Comparing data involves assessing metrics like mean and standard deviation to draw insights about datasets. When comparing the two raisin brands, both have similar means but different variabilities.
Here's a closer look at what the comparison indicates:

• Mean comparison: The Generic and Sunmaid brands have means of 26.73 and 26.59 respectively, showing that on average, both brands offer nearly the same number of raisins per minibox.
• Variability comparison: The Generic brand demonstrates more consistency in raisin counts due to its lower standard deviation, whereas Sunmaid's higher standard deviation points to more fluctuation in the numbers of raisins per box.

This kind of comparison can be valuable for decision-making, such as when choosing between brands based on their consistency and average offering.