Stat-Interval estimation-2

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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. = 0.048 inch. Find the 95% confidence interval for
the average diameter of these 2000 ball bearings.

Solution: If a random sample of large size n is drawn without replacement
from a finite population of size N, then the 95% confidence limits
for the population mean µ are x_m ± 1.96(S.E. of x), where
x_m denotes the sample mean and
S.E. of x_m = (σ / √n ) * √ {(N – n) / (N – 1)}

σ denoting the standard deviation (s.d.) of the population.

Here,

Sample size (n) = 100

Population size (N) = 2000

Sample mean (x_m) = 0.354

Sample s.d. (S) = 0.048

Since σ is not known, an approximate value of S.E. is obtained on replacing
the population s.d. (σ) by the sample s.d. (S).

S.E. of x_m = (S / √n) * √ {(N – n) / (N – 1)}

= (0.048 / √ 100) * √ {(2000 – 100) / (2000 – 1)}

= 0.0047

The 95% confidence limits for the population mean µ are

x_m ± 1.96(S.E. of xm) = 0.0354 ± 1.96 * 0.0047

= 0.0354 ± 0.0092

= 0.3632 and 0.3448

Thus, the 95% confidence interval is (0.3632 and 0.3448) inch.