To test the accuracy of a lab scale, a standard weight of 10 g is repeatedly wei
ID: 3205800 • Letter: T
Question
To test the accuracy of a lab scale, a standard weight of 10 g is repeatedly weighed. Assume the scale readings are Normally distributed with unknown mean (the mean is 10 g if the scale is accurate). The standard deviation for the readings is 0.0002 g. The weight is measured 5 times. The sample mean is 10.0035 g. Give a 99% confidence interval for the mean of repeated measurements of the weight. How many measurements would need to be made in order to get a margin of error of 0.0001 g with 99% confidence? What is the power to detect a systematic weight error of 0.0002 g with 5 measurements and a = 0.01? How many measurements are necessary to detect a systematic weight error of 0.0001 g, with a = 0.02 and power = 0.8?Explanation / Answer
Let X be the random variable that weight.
Given that,
Xbar = 10 g
sigma = 0.0002 g
n = 5
a) Here we have to find confidence interval for mean (mu).
Here we use one sample z-interval because population standard deviation is known.
99% confidence interval for population mean is,
Xbar - E < mu < Xbar + E
where E is margin of error.
E = Zc * sigma/sqrt(n)
where Zc is critical value for normal distribution
Zc we can find by using EXCEL.
syntax :
=NORMSINV(probability)
where probability = 1-a/2
a = 1- C
C is confidence level =98% = 0.98
Confidence interval we can find by using TI-83 calculator.
steps :
STAT --> TESTS --> 7:ZInterval --> ENTER --> Highlight on Stats --> ENTER --> Input all the values --> Calculate --> ENTER
99% confidence interval for mu is (10.003, 10.004)
b) Here we have tond n for given margin of error 0.0001 with 99% confidence.
E = 0.0001
C = 99%
n we can find by using formula,
n = [(Zc*sigma)/E]2
Zc = 2.33
n = [(2.33*0.0002)/0.0001]2
n = 21.65 which is approximately equal to 22.
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