<p>A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br></p>
<p style="text-align: center">$\sum{\left( x-140 \right)}=-120$<br></br></p>
<ol class="i-list"><li><div class="list-item-paragraph">State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.</div></li><li><div class="list-item-paragraph">Test at the $10\%$ significance level whether the bakery’s claim was valid.</div></li></ol>

A bakery claimed that the average number of loaves of bread sold per day is more than $150$, with variance $2500$. A random sample of $50$days was taken and the sales, $x$ loaves, for each day were recorded. It was found that<br></br>

$\sum{\left( x-140 \right)}=-120$<br></br>

State, with a reason, whether it is necessary to assume that the bread sales follow a normal distribution in order to test this claim at the $10\%$ significance level.

Test at the $10\%$ significance level whether the bakery’s claim was valid.

Default solution

Not necessary. As $n = 50$ is large, by Central Limit Theorem, the mean number of bread sold per day will be normally distributed approximately.

Let $X$ be the number of loaves of bread sold per day.

Let $\mu$ be the population mean number of loaves of bread sold per day.

\[n = 50, \quad \sigma^2 = 2500, \quad \bar{x} = \frac{\sum(x-140)}{50} + 140 = \frac{-120}{50} + 140 = 137.6, \quad \alpha\% = 10\%\]

Under $H_0$, test statistic $Z = \displaystyle\frac{\bar{X} - \mu}{\sqrt{\sigma^2/n}} \sim N(0,1)$ approximately by Central Limit Theorem.

From GC, p-value $= 0.03974 < 10\%$ (level of significance)

Since p-value $<$ level of significance, we reject $H_0$ and conclude at 10% level of significance, we have sufficient evidence to claim that the bakery overstated its claim. Thus, the bakery's claim was not valid.

Worked Example 4

Worked Example 4

Video Solution:

Default solution