document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162355>Question 1162355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A large bakery regularly orders cartons of blueberries. Random sample of 50 cartons from each<BR>\n" );
document.write( "supplier were weighted. The weights in grams of the cartons were shown in following frequency<BR>\n" );
document.write( "distribution.<BR>\n" );
document.write( "-------------Supplier A ---------------------Supplier B-----------<BR>\n" );
document.write( "Weight of cartons ---Frequency ----Weight of cartons ----Frequency<BR>\n" );
document.write( "...501 - 550 ............5 .................501 - 550 .............2<BR>\n" );
document.write( "...551 - 600 ...........10................. 551 - 600 .............8<BR>\n" );
document.write( "...601 - 650 ...........20 .................601 - 650 ............12<BR>\n" );
document.write( "...651 - 700 ...........12 .................651 - 700 ............20<BR>\n" );
document.write( "...701 - 750 ............3 .................701 - 750 .............8<BR>\n" );
document.write( "a) State the variable of interest. Indicate whether the variable is quantitative discrete,<BR>\n" );
document.write( "quantitative continuous or qualitative. (3 marks)<BR>\n" );
document.write( "b) Construct suitable graph to represent the above frequency distribution of the weight of<BR>\n" );
document.write( "cartons for supplier A and supplier B, respectively. The weight of cartons from which<BR>\n" );
document.write( "supplier weight more? Justify. (12 marks)<BR>\n" );
document.write( "c) Refer to the graph in (b), suggest the best measure of central tendency and a measure of<BR>\n" );
document.write( "dispersion that is suitable to represent the set of data. Justify your answers. (4 marks)<BR>\n" );
document.write( "d) Compute all the measures of central tendency and standard deviation for the weight of<BR>\n" );
document.write( "cartons from supplier A and interpret the values. (18 marks)<BR>\n" );
document.write( "e) Draw ogive and compute the quartile deviation for weight of cartons from supplier B.<BR>\n" );
document.write( "(10 marks)<BR>\n" );
document.write( "f) The bakery uses one carton of blueberries in each blueberry muffin recipe. It is important<BR>\n" );
document.write( "that the cartons be of consistent weight so that the muffins turn out right. Which supplier<BR>\n" );
document.write( "should the bakery choose? Explain why. (3 marks) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854406 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854406\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This analysis compares the blueberry carton weights from two suppliers to determine which is better suited for a bakery's needs.\r<BR>\n" );
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document.write( "### a) Variable Analysis<BR>\n" );
document.write( "* **Variable of Interest:** The **weight** of the blueberry cartons (in grams).<BR>\n" );
document.write( "* **Classification:** This is a **quantitative continuous** variable (weight can be measured to infinitely precise decimal values, even though it is grouped here).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### b) Graphical Representation and Comparison<BR>\n" );
document.write( "To represent this data, a **Histogram** or **Frequency Polygon** is most suitable. By comparing the distributions:\r<BR>\n" );
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document.write( "* **Supplier A:** The peak (mode) is in the $601 - 650$g range ($20$ cartons).<BR>\n" );
document.write( "* **Supplier B:** The peak (mode) is in the $651 - 700$g range ($20$ cartons).\r<BR>\n" );
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document.write( "**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.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c) Suggested Measures<BR>\n" );
document.write( "* **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.<BR>\n" );
document.write( "* **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.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### d) Calculations for Supplier A<BR>\n" );
document.write( "To compute these, we use the midpoint ($x$) of each class.\r<BR>\n" );
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document.write( "| Weight (g) | Midpoint ($x$) | Freq ($f$) | $f \cdot x$ | $f \cdot x^2$ |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| 501 - 550 | 525.5 | 5 | 2627.5 | 1,380,751.25 |<BR>\n" );
document.write( "| 551 - 600 | 575.5 | 10 | 5755.0 | 3,312,002.50 |<BR>\n" );
document.write( "| 601 - 650 | 625.5 | 20 | 12510.0 | 7,825,005.00 |<BR>\n" );
document.write( "| 651 - 700 | 675.5 | 12 | 8106.0 | 5,475,603.00 |<BR>\n" );
document.write( "| 701 - 750 | 725.5 | 3 | 2176.5 | 1,579,050.75 |<BR>\n" );
document.write( "| **Total** | | **50** | **31,175** | **19,572,412.5** |\r<BR>\n" );
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document.write( "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.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "3.  **Mode:** The class with highest frequency ($601-650$). Midpoint = $\mathbf{625.5\text{g}}$. This is the most common weight.<BR>\n" );
document.write( "4.  **Standard Deviation ($s$):** $\sqrt{\frac{\sum fx^2 - \frac{(\sum fx)^2}{n}}{n-1}} = \sqrt{\frac{19572412.5 - 19437812.5}{49}} = \sqrt{2746.94} \approx \mathbf{52.41\text{g}}$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### e) Quartile Deviation for Supplier B<BR>\n" );
document.write( "The Quartile Deviation is $QD = \frac{Q_3 - Q_1}{2}$. For Supplier B:<BR>\n" );
document.write( "* **$Q_1$ (12.5th value):** Falls in $601-650$ class. $600.5 + [\frac{12.5-10}{12}]50 \approx \mathbf{610.92\text{g}}$.<BR>\n" );
document.write( "* **$Q_3$ (37.5th value):** Falls in $651-700$ class. $650.5 + [\frac{37.5-22}{20}]50 = \mathbf{689.25\text{g}}$.<BR>\n" );
document.write( "* **Quartile Deviation:** $\frac{689.25 - 610.92}{2} = \mathbf{39.17\text{g}}$.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### f) Decision and Justification<BR>\n" );
document.write( "The bakery should choose **Supplier A**.\r<BR>\n" );
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document.write( "**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.\r<BR>\n" );
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document.write( "How do the prices of these two suppliers compare in your scenario? </SPAN>\n" );
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