Question

Birth rates (births per 1000 population) for 30 randomly selected countries are given below: 44.6 39.1 46.5 20.6 15.5 12.1 41.5 12.1 36.2 30.6 12.9 47.3 33.8 47.3 39.8 42.2 14.5 21.4 19.4 16.9 27.7 17.2 14.1 11.2 33.5 13.5 34.7 37.4 36.7 45.5 a) Construct a relative frequency histogram for the data (use k = 5 intervals !!!), b) Construct a cumulative relative frequency histogram (an empirical distribution function), c) Using grouped data estimate the unknown population mean µ and the population variance. d) What is the estimate of probability that a random variable X that has generated those data, will take the value of 12.1? Explain.

Answer 1

(a) Sorting the data into class intervals we have the following distribution:

Class Interval	Frequency	Relative Frequency
10-15	7	7/30 = 0.2333
15-20	4	4/30 = 0.1333
20-30	3	3/30 = 0.1
30-40	9	9/30 = 0.3
40-50	7	7/30 = 0.2333
Total	30	1

The relative frequency histogram is given as:

(b) The cumulative relative frequency of a class interval is the sum of the relative frequency upto the particular class interval.

Class Interval	Frequency	Relative frequency	Cumulative Relative Frequency
10-15	7	0.23333	0.2333
15-20	4	0.13333	0.3667
20-30	3	0.10000	0.4667
30-40	9	0.30000	0.7667
40-50	7	0.23333	1.0000
Total	30

The relative frequency histogram is given as:

(c) The mean and variance are calculated as follows:

Class Interval	Frequency(f)	Mid-point(x)	fx
10-15	7	12.5	87.5
15-20	4	17.5	70.0
20-30	3	25.0	75.0
30-40	9	35.0	315.0
40-50	7	45.0	315.0
Total	30	Total	863

Class Interval	Frequency(f)	Mid-point(x)	fx	fx2
10-15	7	12.5	87.5	1093.75
15-20	4	17.5	70.0	1225
20-30	3	25.0	75.0	1875
30-40	9	35.0	315.0	11025
40-50	7	45.0	315.0	14175
Total	30	Total	863	29394

(d) Since 12. 1 falls in the interval 10-15 we have the proabilty that the random variable takes tha value greater than 12.1 given as 1-0.2333=0.7667.