(Confidence intervals of population mean mu when sigma is unknown.) Ishikawa et
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(Confidence intervals of population mean mu when sigma is unknown.) Ishikawa et al. (Journal of Bioscience and Bioengineering, 2012) studied the adhesion of various biofilms to solid surfaces for possible use in environmental technologies. Adhesion assay is conducted by measuring absorbance at A590. Suppose that for the bacterial strain Acinetobacter, five measurements gave readings of 2.69, 5.76, 2.67, 1.62 and 4.12 dyne-cm^2. Assume that the population distribution is normal. (a) Find a 95% confidence interval for the mean adhesion and give an interpretation. (b) Find a 95% confidence interval for the mean adhesion by using sample mean x = 3.372 and sample standard deviation s = 1.604. (c) If the scientists want the confidence interval to be no wider than 0.5 dyne-cm^2, how many observations should they take? (Note that, the width of the confidence interval is two times the margin error E, so E = 0.5/2). (d) Find the 95% lower and upper confidence bounds for the mean adhesion and give interpretations.Explanation / Answer
a) mean = 3.372 , s = 1.604 , n = 5
Z value at 95% confidence interval = 1.96
CI = mean + /- z * SE
= 3.372 + /- 1.96 * (1.604 / sqrt(5))
= (1.97 , 4.78)
(b)
mean = 3.372 , s = 1.604 , n = 5
Z value at 95% confidence interval = 1.96
CI = mean + /- z * SE
= 3.372 + /- 1.96 * (1.604 / sqrt(5))
= (1.97 , 4.78)
(c)
Margin of error = 0.25
ME = z*SE
0.25 = 1.96 * 1.604/sqrt(n)
n = (1.96*1.604/0.25)^2 = 158.1397
Hence the sample size should be 158.
(d)
Lower confidence bound = 1.97
upper confidence bound = 4.78
This means for every sample of size 5, 95% of the times mean lies between 1.97 and 4.78a
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