Solution — 2023 Question 11 Sampling| Hypothesis Testing

Let $X$ denote the random variable of the distance (in m) thrown by a child in region A
Let $\mu$ denote the population mean distance (in m) thrown by a child in region A

${\mathrm{H}}_0:\mu =15.4$
${\mathrm{H}}_1:\mu >15.4$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(15.4,\frac{0.273{37}^2}{50})$$ $$Z=\frac{\bar{X}-15.4}{\sqrt{\frac{0.273{37}^2}{50}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=50$ is large

From GC,
$p$-value = $0.068420>0.05\Rightarrow \text{Do not reject }{\mathrm{H}}_0$

Hence there is insufficient evidence at the $5\%$ level of significance to conclude whether the mean distance thrown has increased after taking Pedro’s course $∎$

${\mathrm{H}}_0:\mu =15.4$
${\mathrm{H}}_1:\mu >15.4$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(15.4,\frac{{0.5}^2}{50})$$ $$Z=\frac{\bar{X}-15.4}{\sqrt{\frac{{0.5}^2}{50}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=50$ is large

Since the null hypothesis is rejected,