Question 1162355
This analysis compares the blueberry carton weights from two suppliers to determine which is better suited for a bakery's needs.

### a) Variable Analysis
* **Variable of Interest:** The **weight** of the blueberry cartons (in grams).
* **Classification:** This is a **quantitative continuous** variable (weight can be measured to infinitely precise decimal values, even though it is grouped here).

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### b) Graphical Representation and Comparison
To represent this data, a **Histogram** or **Frequency Polygon** is most suitable. By comparing the distributions:

* **Supplier A:** The peak (mode) is in the $601 - 650$g range ($20$ cartons).
* **Supplier B:** The peak (mode) is in the $651 - 700$g range ($20$ cartons).

**Justification:** **Supplier B** weighs more on average. Visually, its distribution is shifted to the right compared to Supplier A. The majority of Supplier B’s cartons ($28$ out of $50$) weigh over $650$g, whereas only $15$ of Supplier A’s cartons fall into those heavier categories.

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### c) Suggested Measures
* **Measure of Central Tendency:** The **Mean** is suggested because the data is relatively symmetric (bell-shaped) for both suppliers, making the mean a reliable balance point.
* **Measure of Dispersion:** The **Standard Deviation** is most suitable as it accounts for every data point in the set and is the standard accompaniment to the mean for symmetric distributions.

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### d) Calculations for Supplier A
To compute these, we use the midpoint ($x$) of each class.

| Weight (g) | Midpoint ($x$) | Freq ($f$) | $f \cdot x$ | $f \cdot x^2$ |
| :--- | :--- | :--- | :--- | :--- |
| 501 - 550 | 525.5 | 5 | 2627.5 | 1,380,751.25 |
| 551 - 600 | 575.5 | 10 | 5755.0 | 3,312,002.50 |
| 601 - 650 | 625.5 | 20 | 12510.0 | 7,825,005.00 |
| 651 - 700 | 675.5 | 12 | 8106.0 | 5,475,603.00 |
| 701 - 750 | 725.5 | 3 | 2176.5 | 1,579,050.75 |
| **Total** | | **50** | **31,175** | **19,572,412.5** |

1.  **Mean ($\bar{x}$):** $\frac{\sum fx}{n} = \frac{31175}{50} = \mathbf{623.5\text{g}}$. On average, a carton from A weighs $623.5$g.
2.  **Median:** Located at the $25^{\text{th}}$ value (in $601-650$ class). $L + [\frac{n/2 - F}{f}]C = 600.5 + [\frac{25-15}{20}]50 = \mathbf{625.5\text{g}}$. Half the cartons are below $625.5$g.
3.  **Mode:** The class with highest frequency ($601-650$). Midpoint = $\mathbf{625.5\text{g}}$. This is the most common weight.
4.  **Standard Deviation ($s$):** $\sqrt{\frac{\sum fx^2 - \frac{(\sum fx)^2}{n}}{n-1}} = \sqrt{\frac{19572412.5 - 19437812.5}{49}} = \sqrt{2746.94} \approx \mathbf{52.41\text{g}}$.

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### e) Quartile Deviation for Supplier B
The Quartile Deviation is $QD = \frac{Q_3 - Q_1}{2}$. For Supplier B:
* **$Q_1$ (12.5th value):** Falls in $601-650$ class. $600.5 + [\frac{12.5-10}{12}]50 \approx \mathbf{610.92\text{g}}$.
* **$Q_3$ (37.5th value):** Falls in $651-700$ class. $650.5 + [\frac{37.5-22}{20}]50 = \mathbf{689.25\text{g}}$.
* **Quartile Deviation:** $\frac{689.25 - 610.92}{2} = \mathbf{39.17\text{g}}$.

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### f) Decision and Justification
The bakery should choose **Supplier A**.

**Reasoning:** Although Supplier B provides more weight per carton, the prompt emphasizes that **consistency** is vital for the recipe. By looking at the frequency distributions, Supplier A's weights are more tightly clustered around its center ($40$ out of $50$ cartons are between $551$g and $700$g). Supplier B has more spread, particularly with a higher frequency of cartons in the extreme $701-750$g range, which could lead to inconsistent muffin quality.

How do the prices of these two suppliers compare in your scenario?