Solution — 2019 Question 12 Sampling| Hypothesis Testing

Let $X$ denote the random variable of the mass (in grams) of a randomly chosen cake
Let $\mu$ denote the population mean mass (in grams) of a randomly chosen cake

${\mathrm{H}}_0:\mu =100$
${\mathrm{H}}_1:\mu <100$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(100,\frac{{\frac{6568}{295}}^2}{60})$$ $$Z=\frac{\bar{X}-100}{\sqrt{\frac{{\frac{6568}{295}}^2}{60}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=60$ is large

From GC,
$p$-value = $0.024423\le 0.05\Rightarrow \text{Reject }{\mathrm{H}}_0$

Hence there is sufficient evidence at the $5\%$ level of significance to conclude that the mean mass of the cakes is less than 100 grams $∎$

${\mathrm{H}}_0:\mu =100$
${\mathrm{H}}_1:\mu \ne 100$

Under $H_0,$ $$\bar{X}\sim \mathrm{N}(100,\frac{{19}^2}{60})$$ $$Z=\frac{\bar{X}-100}{\sqrt{\frac{{19}^2}{60}}}\sim \mathrm{N}(\mathrm{0,1})$$ approximately by the Central Limit Theorem since $n=60$ is large

Since the null hypothesis is rejected,