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Chapter 11: Problem 3

At a school, 100 students took a 'mock'l \(B\) exam using paper \(3 .\) The paper was marked out of 60 marks. Here are the results.$$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Marks } & 0-9 & 10-19 & 20-29 & 30-39 & 40-49 & 50-60 \\ \hline \text { No. of students } & 5 & 9 & 16 & 24 & 27 & 19 \\ \hline \end{array}$$ a) Draw a cumulative frequency curve. b) Estimate the median and quartiles.

Short Answer

Expert verified
The cumulative frequency curve can be drawn using the data; the median is approximately 35, Q1 is 25, and Q3 is 45.

Step by step solution

01

Convert to Cumulative Frequency

To draw a cumulative frequency curve, we first need to convert the given frequency distribution table into a cumulative frequency table. The cumulative frequency is obtained by adding each frequency from the previous ones.- For marks 0-9, cumulative frequency: 5- For marks 10-19, cumulative frequency: 5 + 9 = 14- For marks 20-29, cumulative frequency: 14 + 16 = 30- For marks 30-39, cumulative frequency: 30 + 24 = 54- For marks 40-49, cumulative frequency: 54 + 27 = 81- For marks 50-60, cumulative frequency: 81 + 19 = 100This gives us the cumulative frequencies: \(5, 14, 30, 54, 81, 100\).
02

Draw the Cumulative Frequency Curve

Using the cumulative frequency data, we can plot the cumulative frequency curve. On a graph, the horizontal axis (x-axis) marks the upper limit of each class (9, 19, 29, 39, 49, 60), and the vertical axis (y-axis) corresponds to cumulative frequencies. - Plot the points: (9, 5), (19, 14), (29, 30), (39, 54), (49, 81), (60, 100). - Connect these points using a smooth, freehand curve.
03

Estimate the Median

To find the median, locate the 50th student on the cumulative frequency graph (since there are 100 students, the median corresponds to the 50th percentile). - Draw a horizontal line from 50 on the vertical axis until it meets the curve. - From this point, draw a vertical line down to the horizontal axis to find the median mark. The median is approximately the value at this intersection.
04

Find the Quartiles

To find the lower quartile (Q1) and upper quartile (Q3): - For Q1, find the value corresponding to the 25th student (25% of 100) on the cumulative frequency graph, similar to how the median was found. - For Q3, find the value corresponding to the 75th student (75% of 100) on the graph. - The points where these lines intersect the curve represent Q1 and Q3, respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cumulative Frequency Curve
A cumulative frequency curve, also known as an ogive, is a graphical representation of cumulative frequencies. This type of graph helps visualize how the frequencies accumulate over a data set. For our mock exam scenario, it starts by determining the cumulative frequencies from the given frequency table. Here's the process in detail:
  • First, calculate the cumulative frequency for each mark range by adding the number of students in that range to all previous ranges.
  • Use this cumulative frequency data to plot your graph: Set the upper limit of each class interval along the x-axis (e.g., 9, 19, 29, etc.).
  • Align the corresponding cumulative frequency along the y-axis.
  • For instance, for the first interval (0-9), plot the point (9, 5), for the second (10-19), plot (19, 14), and so on.
  • Connect these points smoothly to form the curve.
The cumulative frequency curve provides insightful information, such as the trend in data accumulation, and is a powerful tool in statistical analysis.
Median Estimation
The median is the middle value of a data set, which separates it into two equal halves. To estimate the median using a cumulative frequency curve:
  • Identify the total number of observations, in this case, 100 students.
  • The median will be the value of the 50th student, as this is the 50th percentile.
  • On the cumulative frequency graph, locate 50 on the y-axis and draw a horizontal line across the graph until it intersects the curve.
  • From the point of intersection, draw a vertical line down to the x-axis. The value where it touches the x-axis is the estimated median.
By this method, even with a class interval breakdown, you get a clearer picture of the central tendency of the data.
Quartile Calculation
Quartiles divide a data set into four equal parts, which are crucial for understanding data distribution. The first quartile (Q1) marks the 25th percentile, and the third quartile (Q3) marks the 75th percentile. To find them using a cumulative frequency curve:
  • Lower Quartile (Q1): Find 25% of the total observations (here, 25 students). Locate 25 on the y-axis, draw a horizontal line to the curve intersection, and a vertical line from there to the x-axis. This point on the x-axis is Q1.
  • Upper Quartile (Q3): Similarly, for Q3, calculate 75% of the total observations (75 students). Find 75 on the y-axis and repeat the steps to determine Q3 on the x-axis.
These calculations offer insight into the spread and variability of the data, indicating where the middle 50% of values lie.

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Most popular questions from this chapter

Cats is a famous musical. In a large theatre in Vienna (1744 capacity), during a period of 10 years, it played 1000 performances. The manager of the group kept a record of the empty seats on the days it played. Here is the table. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \begin{array}{l} \text { Number of } \\ \text { empty seats } \end{array} & 1-10 & 11-20 & 21-30 & 31-40 & 41-50 & 51-60 & 61-70 & 71-80 & 81-90 & 91-100 \\ \hline \text { Days } & 15 & 50 & 100 & 1 / 0 & 260 & 220 & 90 & 45 & 30 & 20 \\\ \hline \end{array}$$ a) Copy and complete the following cumulative frequency table for the above information. $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \begin{array}{l} \text { Number of } \\ \text { empty seats } \end{array} & x<10 & x<20 & x<30 & x<40 & x<50 & x<60 & x<70 & x<80 & x<90 & x<100 \\ \hline \text { Days } & 15 & & 165 & & & 815 & & & & 1000 \\ \hline \end{array}$$ b) Draw a cumulative frequency graph of this distribution. Use 1 unit on the vertical axis to represent the number of 100 days and 1 unit on the horizontal axis to represent every 10 seats. c) Use the graph from b) to answer the following questions: (i) Find an estimate of the median number of empty seats. (ii) Find an estimate for the first quartile, third quartile and the IQR. (iii) The days the number of empty seats was less than 35 seats were considered bumper days (lots of profit). How many days were considered bumper days? (iv) The highest \(15 \%\) of the days with empty seats were categorized as loss days. What is the number of empty seats above which a day is claimed as a loss?

A farmer has 144 bags of new potatoes weighing \(2.15 \mathrm{kg}\) each. He also has 56 bags of potatoes from last year with an average weight of \(1.80 \mathrm{kg}\). Find the mean weight of a bag of potatoes available from this farmer.

Aptitude tests sometimes use jigsaw puzzles to test the ability of new applicants to perform precision assembly work in electronic instruments. One such company that produces the computerized parts of video and CD players gave the following results: $$\begin{array}{|c|c|} \hline \text { Time to finish the puzzle (nearest second) } & \text { Number of employees } \\ \hline 30-35 & 16 \\ \hline 35-40 & 24 \\ \hline 40-45 & 22 \\ \hline 45-50 & 26 \\ \hline 50-55 & 38 \\ \hline 55-60 & 36 \\ \hline 60-65 & 32 \\ \hline 65-70 & 18 \\ \hline \end{array}$$a) Draw a histogram of the data. b) Draw a cumulative frequency curve and estimate the median and IQR. c) Calculate the estimates of the mean and standard deviation of all such participants.

To decide on the number of counters needed to be open during rush hours in a supermarket, the management collected data from 60 customers for the time they spent waiting to be served. The times, in minutes, are given in the following table. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 3.6 & 0.7 & 5.2 & 0.6 & 1.3 & 0.3 & 1.8 & 2.2 & 1.1 & 0.4 \\ \hline 1 & 1.2 & 0.7 & 1.3 & 0.7 & 1.6 & 2.5 & 0.3 & 1.7 & 0.8 \\ \hline 0.3 & 1.2 & 0.2 & 0.9 & 1.9 & 1.2 & 0.8 & 2.1 & 2.3 & 1.1 \\ \hline 0.8 & 1.7 & 1.8 & 0.4 & 0.6 & 0.2 & 0.9 & 1.8 & 2.8 & 1.8 \\\ \hline 0.4 & 0.5 & 1.1 & 1.1 & 0.8 & 4.5 & 1.6 & 0.5 & 1.3 & 1.9 \\ \hline 0.6 & 0.6 & 3.1 & 3.1 & 1.1 & 1.1 & 1.1 & 1.4 & 1 & 1.4 \\ \hline \end{array}$$ a) Construct a relative frequency histogram for the times. b) Construct a cumulative frequency graph and estimate the number of customers who have to wait 2 minutes or more.

Radar devices are installed at several locations on a main highway. Speeds, in km/h, of 400 cars travelling on that highway are measured and summarized in the following table. $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Speed } & 60-75 & 75-90 & 90-105 & 105-120 & 120-135 & \text { Over } 135 \\ \hline \text { Frequency } & 20 & 70 & 110 & 150 & 40 & 10 \\ \hline \end{array}$$ a) Construct a frequency table for the data. b) Draw a histogram to illustrate the data. c) Draw a cumulative frequency graph for the data. d) The speed limit in this country is \(130 \mathrm{km} / \mathrm{h}\). Use your graph in \(\mathrm{c}\) ) to estimate the percentage of the drivers driving faster than this limit.
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