Question 1162355: A large bakery regularly orders cartons of blueberries. Random sample of 50 cartons from each
supplier were weighted. The weights in grams of the cartons were shown in following frequency
distribution.
-------------Supplier A ---------------------Supplier B-----------
Weight of cartons ---Frequency ----Weight of cartons ----Frequency
...501 - 550 ............5 .................501 - 550 .............2
...551 - 600 ...........10................. 551 - 600 .............8
...601 - 650 ...........20 .................601 - 650 ............12
...651 - 700 ...........12 .................651 - 700 ............20
...701 - 750 ............3 .................701 - 750 .............8
a) State the variable of interest. Indicate whether the variable is quantitative discrete,
quantitative continuous or qualitative. (3 marks)
b) Construct suitable graph to represent the above frequency distribution of the weight of
cartons for supplier A and supplier B, respectively. The weight of cartons from which
supplier weight more? Justify. (12 marks)
c) Refer to the graph in (b), suggest the best measure of central tendency and a measure of
dispersion that is suitable to represent the set of data. Justify your answers. (4 marks)
d) Compute all the measures of central tendency and standard deviation for the weight of
cartons from supplier A and interpret the values. (18 marks)
e) Draw ogive and compute the quartile deviation for weight of cartons from supplier B.
(10 marks)
f) The bakery uses one carton of blueberries in each blueberry muffin recipe. It is important
that the cartons be of consistent weight so that the muffins turn out right. Which supplier
should the bakery choose? Explain why. (3 marks)
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! 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).
---
### 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.
---
### 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.
---
### 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 ($s$):** $\sqrt{\frac{\sum fx^2 - \frac{(\sum fx)^2}{n}}{n-1}} = \sqrt{\frac{19572412.5 - 19437812.5}{49}} = \sqrt{2746.94} \approx \mathbf{52.41\text{g}}$.
---
### 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:** $\frac{689.25 - 610.92}{2} = \mathbf{39.17\text{g}}$.
---
### 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?
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