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Chapter 10: Q33E (page 435)

Suppose that \({X_{ij}}\) is a binomial variable with parameters n and \({P_i}\) (so approximately normal when \(n{p_i} \ge 10\)and \(n{q_i} \ge 10\)).Then since \({\mu _i} = n{p_i}\), \(V({X_{ij}}) = \sigma _i^2 = n{p_i}(1 - {p_i}) = {\mu _i}(1 - {\mu _i}/n)\).How should the \({X_{ij}}\)鈥檚 be transformed so as to stabilize the variance?(Hint: \(g({\mu _i}) = {\mu _i}(1 - {\mu _i}/n)).\)

Short Answer

Expert verified

\(h(x) = 2\sqrt n \arcsin \left( {\sqrt {\frac{x}{n}} } \right)\)

Step by step solution

01

Solving h(x) using g(x)

From given

\(g({\mu _i}) = {\mu _i}\left( {1 - \frac{{{\mu _i}}}{n}} \right)\)

The function that yield appropriate transformations is

\(h(x) = \int {\frac{1}{{\sqrt {g(x)} }}} dx\)

Since

\(g(x) = x\left( {1 - \frac{x}{n}} \right)\)

The h(x) becomes

\(h(x) = \int {\frac{1}{{\sqrt {x \cdot \left( {1 - \frac{x}{n}} \right)} }}} dx = \int {\frac{1}{{\sqrt x }}} \cdot \frac{1}{{\sqrt {1 - \frac{x}{n}} }}dx\)

02

Solving h(x) using substitution

By substituting

Let \(t = \sqrt {\frac{x}{n}} ;\)

Differentiating t with respect to x

\(dt = \frac{1}{{2\sqrt {n \cdot x} }}dx\)

Substituting dt in h(x)

\(h(x) = 2 \cdot \sqrt n \int {\frac{1}{{\sqrt {1 - {t^2}} }}} dt = 2 \cdot \sqrt n .\arcsin (t) = 2\sqrt n \arcsin \left( {\sqrt {\frac{x}{n}} } \right)\)

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Most popular questions from this chapter

Apply the modified Tukey鈥檚 method to the data in Exercise 22 to identify significant differences among the\({\mu _i}\)鈥檚.

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  1. Perform an F at level \(\alpha = .05\)
  2. What happens when Tukey鈥檚 procedure is applied?
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