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 0.0003 gram.

(a) The weight is measured three times. The mean result is 10.0023 grams. Give a 98% confidence interval for the mean of repeated measurements of the weight. (Round your answers to four decimal places.)

(b) How many measurements must be averaged to get a margin of error of ±0.0001 with 98% confidence? (Round your answer up to the nearest whole number.)

Answer 1

a)

sample mean, xbar = 10.0023
sample standard deviation, σ = 0.0003
sample size, n = 3

Given CI level is 98%, hence α = 1 - 0.98 = 0.02
α/2 = 0.02/2 = 0.01, Zc = Z(α/2) = 2.33

ME = zc * σ/sqrt(n)
ME = 2.33 * 0.0003/sqrt(3)
ME = 0

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (10.0023 - 2.33 * 0.0003/sqrt(3) , 10.0023 + 2.33 * 0.0003/sqrt(3))
CI = (10.0019 , 10.0027)

b)

The following information is provided,
Significance Level, α = 0.02, Margin or Error, E = 0.0001, σ = 0.0003

The critical value for significance level, α = 0.02 is 2.33.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.33 * 0.0003/0.0001)^2
n = 48.86

Therefore, the sample size needed to satisfy the condition n >= 48.86 and it must be an integer number, we conclude that the minimum required sample size is n = 49
Ans : Sample size, n = 49