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(i)
Let X denotes the length of a leaf in cm.

$H_0: \mu = 7$

$latex H_0: \mu < 7$ Under $latex H_0, \bar{X} ~\sim~ N(7, \frac{4.4}{50})$ approximately by Central Limit Theorem since n is large. From Graphing Calculator, p-value $latex = 0.0459 < 0.05$. Thus we reject $latex H_0$, and conclude with sufficient evidence at 5% level of significant that the mean length of the leaf is less than 7 cm. (ii) Unbiased estimate of $latex \mu = \frac{310.4}{50} = 6.208$ Unbiased estimate of $latex {\sigma}^{2} = \frac{1}{49}[2209.2 - \frac{310.4^{2}}{50}] = 5.76$ (iii) $latex H_0: \mu = 7$ $latex H_0: \mu \ne 7$ Under $latex H_0, \bar{X} ~\sim~ N(7, \frac{5.79935}{50})$ approximately by Central Limit Theorem since n is large. From Graphing Calculator, p-value $latex = 0.019624$. Since $latex H_0$ is rejected, p-values $latex \le \alpha \%$ Required set of values \$latex = \{ \alpha \in \mathbb{R}: 1.97 \le \alpha \le 100 \}