An industrial plant dumps its waste into a nearby river, but claims that it is n
ID: 3131723 • Letter: A
Question
An industrial plant dumps its waste into a nearby river, but claims that it is not impacting the native fish that live in the river. You have measured the calcium concentration from a random sample of 18 locations in the river and calculated a mean of 97.4 mg/L, with a standard deviation of 4.1 mg/L. The fish are able to tolerate calcium concentrations up to 95 mg/L.
a. Assuming a normal distribution, calculate a 95% confidence interval for the mean calcium concentration in the river.
b. Do your data suggest that the fish are being negatively impacted? Explain your answer.
c. Assuming that you held everything else constant, how would the width of the confidence interval change if you doubled your sample size from 18 to 36? Would it become wider, narrower, or stay the same?
Explanation / Answer
a)
Note that
Margin of Error E = t(alpha/2) * s / sqrt(n)
Lower Bound = X - t(alpha/2) * s / sqrt(n)
Upper Bound = X + t(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.025
X = sample mean = 97.4
t(alpha/2) = critical t for the confidence interval = 2.109815578
s = sample standard deviation = 4.1
n = sample size = 18
df = n - 1 = 17
Thus,
Margin of Error E = 2.038882033
Lower bound = 95.36111797
Upper bound = 99.43888203
Thus, the confidence interval is
( 95.36111797 , 99.43888203 ) mg/L [ANSWER]
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b)
As the whole interval is over 95 mg/L, then YES, the data suggest that the fish are being negatively impacted. [ANSWER]
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c)
As we can see, the margin of error decreases with increasing n.
Hence, it will be NARROWER. [ANSWER]