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ESTIMATION

Estimation is a mathematical technique for producing information about a population based on a sample of units from that population. Different sampling techniques require different estimation techniques.

Estimation allows you to derive measures of location, spread, and totals for the whole population. This and following pages will outline the estimation techniques for the mean and total of a population from a simple random sample only.


ESTIMATE OF POPULATION MEAN

For a simple random sample, the estimate of the population mean is the same as the mean of the sample:




where:
x stands for an observed value,
stands for the estimate of population mean,
stands for the sum of all observed x values in the sample, and
n stands for the number of observations in the sample.

NOTE:
Lower case x and n should be used if you are referring to a sample survey, and upper case X and N if referring to a population.

If the sample results have been summarised in a frequency table then the estimate for the population mean is again the same as the sample:



where:
x stands for an observed value,
stands for the estimate of the population mean,
xf stands for the sum of all xf values in the sample, and
f stands for the sum of the frequencies in the sample.


EXAMPLE

1.		Give an example of a simple random sample and briefly describe why it is classed in this category.
2.	a)	If a company has a workforce of 2,700 people, and a sample of 300 people were to be systematically surveyed, what would the sampling interval be?
3.	b)	Choose a number at random as a starting point for the above sample. What would be the first 5 numbers in the sample? What would be the last 5 numbers in the sample?

	MELB.	ADEL.	PERTH	SYD.	BRIS.	HOB.	DAR.	CANB.
LABOR	85	65	81	127	74	40	22	37
LIBERAL	80	70	60	135	50	40	26	13
OTHER	10	31	6	22	13	8	18	6
UNDECIDED	25	14	13	16	13	12	4	4
TOTAL	200	180	160	300	150	100	70	60

	a)	In which city was the greatest percentage of people:
		i) in favour of Labor?
		ii) in favour of Liberal?
		iii) in favour of another political party?
		iv) undecided?
	b)	In which city was the least percentage of people:
		i) in favour of Labor?
		ii) in favour of Liberal?
		iii) in favour of another political party?
		iv) undecided?
	c)	Is it possible to estimate overall percentages for capital cities from the above table?
4.		In a school, the number of students in each year level from kindergarten to Year 12 is as follows:

	K	P	1	2	3	4	5	6	7	8	9	10	11	12
Males	9	8	9	9	13	20	23	28	78	74	69	71	60	48
Females	6	8	11	10	13	18	35	34	63	62	61	88	70	56
Total	15	16	20	19	26	38	58	62	141	136	130	159	130	104

Once the number of students in each category has been determined, the students will be selected randomly.

	a)	What type of sampling technique is this called?
	b)	If the Principal wishes to survey 200 students, how many students of each sex and in each year level should be surveyed?

1.		Use one of the random sampling methods described to obtain a random sample from your class or year level. Use this sample to find out one or more of the following:
	a)	average number of children in a family,
	b)	type of transport used to get to school,
	c)	number of students in favour of capital punishment,
	d)	amount of pocket money received,
	e)	type of pets kept,
	f)	number of people in a family who have had tertiary education.
2.		Obtain a list showing the name and gender of each student by year level in your school. Using the stratified sampling technique, survey 20% of the school’s population to find the students’ favourite subject. Use the strata of year level and gender.