Page 375 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes

(i) The ‘probable limits’ (without any reference to the degree of confidence) may be taken
to be ‘almost sure limits’ in all the above cases.
(ii) The formulae for S.E. involve population parameters. If these parameters are not
known, an approximate value of S.E. may be obtained by substituting the statistic
for the corresponding parameter.]

Example 7: A sample of 6500 screws is taken form a large consignment and 75 are found to be
defective. Estimate the percentage of defectives in the consignment and assign limits within which
the percentage lies.
Solution: There are 75 defectives in a sample of size n = 600. Therefore, the sample proportion of
defectives is

75 1
p = = = 12.5%
600 8
This may be taken as an estimate of the percentage of defectives (P) in the whole consignment (‘Point
estimation’).
The ‘limits’ to the percentage of defectives refer to the confidence limits, which may be given as p ± 3
(S.E. of p).

PQ
S.E. of p =
n

pq
= approximately;
n
(since the population proportion P is not known).

1 ⎛ − ⎟ 1 1 ⎞
⎠
= 8 ⎜ ⎝ 8 = 1 7 = .0135
600 80 6
Thus, the limits for P are

1 3
8 ±×.0135 = .125 ± .0405

= .1655 and .0845 = 16.55% and 8.45%
The limits to the percentage of defectives in the consignment are 8.45% to 16.55% (‘Interval estimation’).
Ans. 12.5%; 8.45% to 16.55%.
Example 8: A random sample of 100 ball bearings selected from a shipment of 2000 ball bearings has
an average diameter of 0.354 inch with a S.D. = .048 inch. Find 95% confidence interval for the average
diameter of these 2000 ball bearings.
Solution: Theory: If a random sample of large size n is drawn without replacement from a finite population
of size N, hen the 95% confidence limits for the population mean μ are x ± 1.96 (S.E. of x), where x
denotes the sample mean and

σ N − n
S.E. of x = .
−
n N1

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