Refer to Example 25 in Chapter 1 regarding the plant height (in 1/8 inches) of

Question

Refer to Example 25 in Chapter 1 regarding the plant height (in 1/8 inches) of cross-fertilized and self-fertilized plants. Denote by [latex]X_{i}[/latex]’s and [latex]Y_{i}[/latex]’s, respectively, the heights of cross-fertilized and self-fertilized plants. Then the observed values for the 15 pairs are given in Example 25 of Chapter 1, which are reproduced in the Table 15.2 for convenience. At the level of significance α = 0.05, test the null hypothesis [latex]H_{0}\colon F=G,[/latex] where F and G are the d.f.’s of the [latex]X_{i}[/latex]’s and [latex]Y_{i}[/latex]’s, respectively.

Accepted Answer

From Table 15.2,we have:
[latex]Z_{1}=0,~~Z_{2}=1,~~Z_{3}=0,~~Z_{4}=0,~~Z_{5}=0,~~Z_{6}=0,~~Z_{7}=0,~~Z_{8}=0,[/latex]

[latex]Z_{9}=0,~~Z_{10}=0,~~Z_{11}=0,~~Z_{12}=0,~~Z_{13}=0,~~Z_{14}=0,~~Z_{15}=1,[/latex]
so that Z = 2. Suppose first that the alternative is [latex]H_{A}^{\prime\prime}:\ F\ eq\ G.[/latex] Then [latex] extstyle H_{0}[/latex] is rejected in favor of [latex]H_{A}^{\prime\prime}[/latex] whenever [latex]Z\leq C_{1}\;\mathrm{or}\;Z\geq C_{2},[/latex] where:
[latex]P(Z\lt C_{1})+\gamma_{0}P(Z=C_{1})=0.025\;\mathrm{and}\;P(Z\leq C_{2})-\gamma_{0}P(Z=C_{2})=0.975,[/latex] and [latex]Z\sim B(15,1/2).[/latex]
From the Binomial tables, we find [latex]C_{1}=4,\,C_{2}=11,\mathrm{~and~}\gamma_{0}= extstyle{\frac{37}{208}}\simeq0.178.[/latex] Since [latex]Z=2\lt C_{1}(=4),[/latex] the null hypothesis is rejected. Next, test [latex] extstyle H_{0}[/latex] against the alternative [latex]H_{A}^{\prime}\colon p\lt { extstyle\frac12}[/latex] again at level α = 0.05. Then [latex] extstyle H_{0}[/latex] is rejected in favor of [latex]H_{A}^{\prime}[/latex] whenever [latex]Z\leq C^{\prime},{\mathrm{where}}\;C^{\prime}[/latex] is determined by:
[latex]P(Z\lt C^{\prime})+\gamma^{\prime}P(Z=C^{\prime})=0.05,\quad Z\sim B(15,~1/2).[/latex]
From the Binomial tables, we find [latex]C^{\prime}\,=\,4\,\operatorname{and}\,\gamma^{\prime}\,=\,{ extstyle\frac{81}{104}}\,\simeq\,0.779.[/latex] Since [latex]Z=2\lt C^{\prime}(=4),\,H_{0}[/latex] is rejected in favor of [latex]H_{A}^{\prime},[/latex] which is consistent with what the data say.
For the Normal approximation, we get from (15):

[latex]\left.\begin{array}{l}{{~C\simeq{\frac{n}{2}}+z_{\alpha}{\frac{\sqrt{n}}{2}}},~~~C^{\prime}\simeq{\frac{n}{2}}-z_{\alpha}{\frac{\sqrt{n}}{2}}}\ {{C_{1}\simeq{\frac{n}{2}}-z_{\frac{\alpha}{2}}\frac{\sqrt{n}}{2},~~~C_2\simeq\frac{n}{2}+z_{\frac{\alpha}{2}}\frac{\sqrt{n}}{2}.}}\end{array} ight\} [/latex] (15)

[latex]{\bf z}_{0.025}\:=\:1.96,[/latex] so that [latex]C_{1}=3.703,C_{2}\simeq11.297,[/latex] and [latex] extstyle H_{0}[/latex] is rejected again, since [latex]Z=2\lt C_{1}(\simeq3.703).[/latex] Also, [latex]z_{0.05}={1}.645,[/latex] and hence [latex]C^{\prime}\simeq4.314..[/latex] Again, [latex] extstyle H_{0}[/latex] is rejected in favor of [latex]H_{A}^{\prime},[/latex] since [latex]Z=2\lt C^{\prime}(\simeq4.314).[/latex]

Table 15.2
Pair	Cross-	Self-	Pair	Cross-	Self-
1	188	139	9	146	132
2	96	163	10	173	144
3	168	160	11	186	130
4	176	160	12	168	144
5	153	147	13	177	102
6	172	149	14	184	124
7	177	149	15	96	144
8	163	122

Question 15.4: Refer to Example 25 in Chapter 1 regarding the plant height ......

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