Q4 The Environmental Protection Agency is responsible, inter alia, for setting c

ID: 3023395 • Letter: Q

Question

Q4 The Environmental Protection Agency is responsible, inter alia, for setting criteria for the amount of toxic chemicals to be allowed in freshwater lakes and rivers. A common measure of toxicity for any pollutant is the concentration of the pollutant that will kill half of the test species in a given amount of time (usually 96 hours for fish species). This measure is called LC50 lethal concentration killing 50% of the test species). In many studies, the observed values of the natural logarithm of LC50 measurements are Normally distributed, and hence the analysis is based on ln(LC50) data. (a) Studies of the effect of copper on a certain species of fish show the variance of ln(LC50) measurements to be around .4 with concentration measurements in MATH 324 McGill University, Winter Term 2016 Statistics Assignment 1: Due February 4, 2016 at 23:59 milligram per liter. If n 10 studies on ln(LC50) for copper are to be completed, find the probability that the sample mean of ln(LC50) will differ from the true population mean by no more than .5. (b) If we want the sample mean to differ from the population mean by no more than .5 with probability .95, what is the minimum mumber of tests that should be run? (c) Suppose that n 20 observations are to be taken on ln(LC50) and that now o2 1.4. Let S denote the sample variance of the 20 measurements. Find real numbers a and b such that Pr(a S SAS b) .95.

Explanation / Answer

a).

z=0.5/sqrt(0.4/10) =2.5

The required P=P( z< -2.5) +P( z >2.5) = 0.0062+0.0062 = 0.0124

b).

Sample size

For 95%, z=1.96

d=0.5

variance=0.4

Sample size = (z2*s2)/d2

= (1.962*0.4)/0.52

=6.147

The sample size required= 7

C).

Confidence Interval Estimate for the Population Variance

Data

Sample Size

Sample Standard Deviation

1.4

Confidence Level

95%

Intermediate Calculations

Degrees of Freedom

Sum of Squares

37.24

Single Tail Area

0.025

Lower Chi-Square Value

8.9065

Upper Chi-Square Value

32.8523

Results

Interval Lower Limit for Variance

1.1336

Interval Upper Limit for Variance

4.1812

a=1.1336

b=4.1812