www.downloadslide.com Exercises 365 for many years, he has discovered that these measurements (in square inches) are normally distributed with standard deviation approximately 4 square inches. If the forester samples n = 9 trees, find the probability that the sample mean will be within 2 square inches of the population mean. 7.12 Suppose the forester in Exercise 7.11 would like the sample mean to be within 1 square inch of the population mean, with probability .90. How many trees must he measure in order to ensure this degree of accuracy? 7.13 The Environmental Protection Agency is concerned with the problem of setting criteria for the amounts of certain 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 values contained in the natural logarithm of LC50 measurements are normally distributed, and, hence, the analysis is based on ln(LC50) data. Studies of the effects of copper on a certain species of fish (say, species A) show the variance of ln(LC50) measurements to be around .4 with concentration measurements in milligrams per liter. If n = 10 studies on 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. 7.14 If in Exercise 7.13 we want the sample mean to differ from the population mean by no more than .5 with probability .95, how many tests should be run? 7.15 Suppose that X 1 , X 2 ,...,X m and Y 1 , Y 2 ,...,Y n are independent random samples, with the variables X i normally distributed with mean µ 1 and variance σ1 2 and the variables Y i normally distributed with mean µ 2 and variance σ2 2 . The difference between the sample means, X − Y , is then a linear combination of m + n normally distributed random variables and, by Theorem 6.3, is itself normally distributed. a Find E(X − Y ). b Find V (X − Y ). c Suppose that σ1 2 = 2, σ2 2 = 2.5, and m = n. Find the sample sizes so that (X − Y ) will be within 1 unit of (µ 1 − µ 2 ) with probability .95. 7.16 Referring to Exercise 7.13, suppose that the effects of copper on a second species (say, species B) of fish show the variance of ln(LC50) measurements to be .8. If the population means of ln(LC50) for the two species are equal, find the probability that, with random samples of ten measurements from each species, the sample mean for species A exceeds the sample mean for species B by at least 1 unit. 7.17 Applet Exercise Refer to Example 7.4. Use the applet Chi-Square Probabilities and Quantiles ∑6 ) ( to find P( i=1 Z i 2 ≤ 6 . Recall that ∑ ) 6 i=1 Z i 2 has a χ 2 distribution with 6 df. 7.18 Applet Exercise Refer to Example 7.5. If σ 2 = 1 and n = 10, use the applet Chi-Square Probabilities and Quantiles to find P(S 2 ≥ 3). (Recall that, under the conditions previously given, 9S 2 has a χ 2 distribution with 9 df.) 7.19 Ammeters produced by a manufacturer are marketed under the specification that the standard deviation of gauge readings is no larger than .2 amp. One of these ammeters was used to make ten independent readings on a test circuit with constant current. If the sample variance of these ten measurements is .065 and it is reasonable to assume that the readings are normally distributed, do the results suggest that the ammeter used does not meet the marketing specifications? [Hint: Find the approximate probability that the sample variance will exceed .065 if the true population variance is .04.]
www.downloadslide.com 366 Chapter 7 Sampling Distributions and the Central Limit Theorem 7.20 a If U has a χ 2 distribution with ν df, find E(U) and V (U). b Using the results of Theorem 7.3, find E(S 2 ) and V (S 2 ) when Y 1 , Y 2 ,...,Y n is a random sample from a normal distribution with mean µ and variance σ 2 . 7.21 Refer to Exercise 7.13. Suppose that n = 20 observations are to be taken on ln(LC50) measurements and that σ 2 = 1.4. Let S 2 denote the sample variance of the 20 measurements. a Find a number b such that P(S 2 ≤ b) = .975. b Find a number a such that P(a ≤ S 2 ) = .975. c If a and b are as in parts (a) and (b), what is P(a ≤ S 2 ≤ b)? 7.22 Applet Exercise As we stated in Definition 4.10, a random variable Y has a χ 2 distribution with ν df if and only if Y has a gamma distribution with α = ν/2 and β = 2. a Use the applet Comparison of Gamma Density Functions to graph χ 2 densities with 10, 40, and 80 df. b What do you notice about the shapes of these density functions? Which of them is most symmetric? c In Exercise 7.97, you will show that for large values of ν, aχ 2 random variable has a distribution that can be approximated by a normal distribution with µ = ν and σ = √ 2ν. How do the mean and standard deviation of the approximating normal distribution compare to the mean and standard deviation of the χ 2 random variable Y ? d Refer to the graphs of the χ 2 densities that you obtained in part (a). In part (c), we stated that, if the number of degrees of freedom is large, the χ 2 distribution can be approximated with a normal distribution. Does this surprise you? Why? 7.23 Applet Exercise a b c Use the applet Chi-Square Probabilities and Quantiles to find P[Y > E(Y )] when Y has χ 2 distributions with 10, 40, and 80 df. What did you notice about P[Y > E(Y )] as the number of degrees of freedom increases as in part (a)? How does what you observed in part (b) relate to the shapes of the χ 2 densities that you obtained in Exercise 7.22? 7.24 Applet Exercise Refer to Example 7.6. Suppose that T has a t distribution with 5 df. a b c d Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is greater than 2. Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is less than −2. Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is between −2 and 2. Your answer to part (c) is considerably less than 0.9544 = P(−2 ≤ Z ≤ 2). Refer to Figure 7.3 and explain why this is as expected. 7.25 Applet Exercise Suppose that T is a t-distributed random variable. a If T has 5 df, use Table 5, Appendix 3, to find t .10 , the value such that P(T > t .10 ) = .10. Find t .10 using the applet Student’s t Probabilities and Quantiles. b Refer to part (a). What quantile does t .10 correspond to? Which percentile? c Use the applet Student’s t Probabilities and Quantiles to find the value of t .10 for t distributions with 30, 60, and 120 df.