16. Suppose the true average growth μ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is σ², whereas for the second type the variance is 40². Let X₁,..., X be m independent growth observations on the first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y be n independent growth observations on the second type [E(Y) =μ, V(Y) = 4σ²]. a. Show that the estimator û = 8X + (1 - 8) is unbi- ased for μ (for 0 < 8 <1, the estimator is a weighted average of the two individual sample means). b. For fixed m and n, compute V(u), and then find the value of 8 that minimizes V(i). [Hint: Differentiate V(u) with respect to 8.]
16. Suppose the true average growth μ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is σ², whereas for the second type the variance is 40². Let X₁,..., X be m independent growth observations on the first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y be n independent growth observations on the second type [E(Y) =μ, V(Y) = 4σ²]. a. Show that the estimator û = 8X + (1 - 8) is unbi- ased for μ (for 0 < 8 <1, the estimator is a weighted average of the two individual sample means). b. For fixed m and n, compute V(u), and then find the value of 8 that minimizes V(i). [Hint: Differentiate V(u) with respect to 8.]
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
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![16. Suppose the true average growth μ of one type of plant
during a 1-year period is identical to that of a second
type, but the variance of growth for the first type is σ²,
whereas for the second type the variance is 40². Let
X₁,..., X be m independent growth observations on the
first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y
be n independent growth observations on the second
type [E(Y) =μ, V(Y) = 4σ²].
a. Show that the estimator û = 8X + (1 - 8) is unbi-
ased for μ (for 0 < 8 <1, the estimator is a weighted
average of the two individual sample means).
b. For fixed m and n, compute V(u), and then find the
value of 8 that minimizes V(i). [Hint: Differentiate
V(u) with respect to 8.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe60e14a8-856f-447c-ac90-795ae43e00b4%2Ff3acb084-f1de-4ea9-bffb-97ddd56b972b%2Fp9vy7fe_processed.png&w=3840&q=75)
Transcribed Image Text:16. Suppose the true average growth μ of one type of plant
during a 1-year period is identical to that of a second
type, but the variance of growth for the first type is σ²,
whereas for the second type the variance is 40². Let
X₁,..., X be m independent growth observations on the
first type [so E(X;) = µ, V(X;) = σ²], and let Y₁,..., Y
be n independent growth observations on the second
type [E(Y) =μ, V(Y) = 4σ²].
a. Show that the estimator û = 8X + (1 - 8) is unbi-
ased for μ (for 0 < 8 <1, the estimator is a weighted
average of the two individual sample means).
b. For fixed m and n, compute V(u), and then find the
value of 8 that minimizes V(i). [Hint: Differentiate
V(u) with respect to 8.]
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