To assess the accuracy of a laboratory scale, a standard weight known to weigh 1

Question

To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are normally distributed with unknown mean (this mean is 10 grams if the scale has no bias). The standard deviation of the scale readings is known to be 0.0002 gram. Use 6 decimals for this question.

(a) The weight is measured five times. The mean result is 10.0023 grams. Give a 95% confidence interval for the mean of repeated measurements of the weight.

Xbar = 10.0023

(b ) Give a 98% confidence interval for the mean of repeated measurements of the weight.

(c ) How many measurements must be averaged to get a margin of error of ±0.0001 with 95% confidence

(d) How many measurements must be averaged to get a margin of error of ±0.0001 with 98% confidence?

Explanation / Answer

(a)
xbar = 10.0023
std. dev. = 0.0002
n = 5
SE = std.dev. / sqrt(n) = 0.0002/sqrt(5) = 0.000089

For 95% CI, t-value = 2.77645
ME = t*SE = 2.77645*0.000089 = 0.000248

CI = (mean +/- ME)
= (10.0023 - 0.000248, 10.0023 + 0.000248)
= (10.002052, 10.002548)

(b)
For 98% CI, t-value = 3.74695
ME = t*SE = 3.74695*0.000089 = 0.000335

CI = (mean +/- ME)
= (10.0023 - 0.000335, 10.0023 + 0.000335)
= (10.001965 , 10.002635)

(c)
ME = 0.0001
z-value = 1.95996

n = (z * sigma / ME)^2
n = (1.95996 * 0.0002/0.0001)^2
n = 15.36577281
i.e. n = 15

(d)
ME = 0.0001
z-value = 2.32635

n = (z * sigma / ME)^2
n = (2.32635 * 0.0002/0.0001)^2
n = 21.6476
i.e. n = 22

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