Consider the following distributions of the marks scored by 100 students in a quantitative
techniques activity. The median mark is known and given as 49.8.
Marks 0- 15 15- 30 30- 45 45- 60 60- 75 75 β 90 90-105 Total
No. of Students 10 12 20 f1 15 f2 8 100
Find the missing frequencies f1 and f2 and then compute the coefficient of variation and
comment.
b) Fit a linear trend curve by the least-squares method to the following data and determine
trend values and the find levels of production in 2012 and 2018
Year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Production
(Kg.)
30 50 60 60 80 100 110 120 130 150
To find the missing frequencies f1 and f2 in the given distribution, we can use the property that the median divides the data into two equal halves.
Given that the median mark is 49.8, it falls in the 45-60 interval. We know that 15 students have scored marks in the 60-75 interval, so the total number of students below the median is 20 + 15 = 35.
Since the total number of students is 100, the number of students above the median will also be 35. From the given data, we can see that 8 students have scored marks in the 90-105 interval. Therefore, the missing frequency f1 will be 35 - 8 = 27.
Similarly, to find the missing frequency f2, we need to subtract the number of students scored marks in the 0-15 and 15-30 intervals from the total. From the data, we can see that 10 and 12 students have scored marks in these intervals, respectively. Hence, f2 will be 100 - 10 - 12 = 78.
Now, let's compute the coefficient of variation (CV). The coefficient of variation is a measure of relative variability and is defined as the ratio of the standard deviation to the mean, expressed as a percentage.
To calculate the coefficient of variation, we need the mean and the standard deviation.
Mean (πΏΜ
):
To find the mean, we can use the midpoint of each interval and the number of students in that interval.
Mean (πΏΜ
) = (0.5 * (0+15) * 10 + 0.5 * (15+30) * 12 + 0.5 * (30+45) * 20 + 0.5 * (45+60) * f1 + 0.5 * (60+75) * 15 + 0.5 * (75+90) * f2 + 0.5 * (90+105) * 8) / 100
Standard Deviation (π):
To find the standard deviation, we can use the formula:
π = β((Ξ£(ππβπΏΜ
)Β²ππ) / π)
where ππ is the midpoint of the interval, π is the total number of observations, ππ is the frequency of the interval, and πΏΜ
is the mean.
Now, let's calculate the mean and standard deviation.
Once we have the mean and standard deviation, we can calculate the coefficient of variation (CV):
CV = (π / πΏΜ
) * 100
Now, let's move on to the second part of the question regarding fitting a linear trend curve using the least-squares method.
To fit a linear trend curve using the least-squares method, we need to find the equation of a straight line that best fits the given data.
The equation of a straight line is given by:
π = π + ππ
where π is the dependent variable (in this case, the production), π is the independent variable (in this case, the year), π is the intercept, and π is the slope of the line.
To find the values of π and π using the least-squares method, we need to calculate the following:
Ξ£π, Ξ£πΒ², Ξ£π, Ξ£ππ
where Ξ£ represents the summation (sum) of the respective variables.
Using the formulas:
π = (nΞ£ππ β Ξ£πΞ£π) / (nΞ£πΒ² β (Ξ£π)Β²)
π = (Ξ£π β πΞ£π) / n
where n is the number of data points, and Ξ£ππ represents the sum of the product of π and π values.
Once we determine the values of π and π, we can substitute the year values into the equation to find the trend values and determine the production levels in 2012 and 2018.