Question

The Frequency Polygon 1. Represent a sample of values of a continuous random variable using a frequency table, frequency histogram and frequency polygon 2. Compare the effectiveness of a frequency polygon as an approximation of a frequency distribution for different sizes of the intervals — 3. Make connections between the normal distribution and the binomial and hypergeometric distributions for increasing number of trials of the discrete distributions 4. Identify the conditions required to reduce interval sizes to more accurately approximate probabilities

You have to make an example for the first one. And then answer the question about the first one

Answer 1

1) Lets us cosnider an example of where the distributions represent the marks scored by students in science test to construct frequency table ,frequency histogram and frequency polygon

MARKS	NUMBER OF STUDENTS
30-40	10
40-50	30
50-60	15
60-70	20
70-80	10

FREQUENCY TABLE :

MARKS	FREQUENCY
[30-40)	10
[40-50)	30
[50-60)	15
[60-70)	20
[70-80)	10

FREQUENCY HISTOGRAM :By plotting marks on x axis and number of students on y axis using bar graph we get frequency histogram, it just give the number of accourence of a particular instant.

3. 23 S Number of students 204 in 9 S o X 10 20 30 40 so 60 구이 35 8 90 marks

FREQUENCY POLYGON: it is also a graphical representation of frequency distribution it is constructed by joining the midpoints of the interval of each data.in the frequency histogram

Hence for each interval of marks calculate midpoint that is

$\frac{30+40}{2}= 35$$\frac{40+50}{2}= 45$   $\frac{50+60}{2}= 55$

$\frac{60+70}{2}= 65$$\frac{70+80}{2}= 75$

join all these points on histogram to get frequency polygon here frequency polygon approximates the frequency histogram.

R 30 Number of students 2 Jo lo7 o o 10 30 90 so 60 4o 20 70 marks

2) As the interval size decreases the frequency polygon can make a better approximation of the actual data and it reduces the area that the bars in a histogram are above or below the polygon.

But we can't very small interval also it should be selected in a reasonable manner based on how the data is spread and also depends on the amount of data and presicion of collection.

3) The normal distribution is the smooth approximated curve upon the frequency polygon

Binomial distribution is used when the trails of the discrete probability distributions are independent whereas hypergeometric distribution is trails dependent and the objects are not replaced.

4)The more effective interval size for more accurate approximation of probabilities is not too high or too low the smaller intervals approximates close to the actual data on the other hand it should not be too small also , size depends on the amount and how precisely the data is collected and how the data is spread.