All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(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 \}

Nothing special about this question. Students still forget to introduce Central Limit Theorem when doing the hypothesis testing though. Lastly, the question asks for the set of values, so students should use precise set notation.

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