A beverage company produces thousands of cans of soda each day. The volume of a soda in a can follows a normal distribution with mean $355$ ml and standard deviation $1.2$ ml. 
Following the maintenance of the filling machine, there may have been a slight adjustment. To determine if the machine still produces $355$ ml of soda per can, $15$ randomly chosen cans are chosen. The volume, $x$ ml, of the $15$ cans of soda is summarised by 
$\sum{x}=5314.5$, $\sum{{{x}^{2}}=1892040.25}$ 
Assuming that the variance of the distribution is unaltered by the adjustment, test at the $6\%$ significance level whether there is any change in the mean volume of the soda.

A beverage company produces thousands of cans of soda each day. The volume of a soda in a can follows a normal distribution with mean $355$ ml and standard deviation $1.2$ ml.

Following the maintenance of the filling machine, there may have been a slight adjustment. To determine if the machine still produces $355$ ml of soda per can, $15$ randomly chosen cans are chosen. The volume, $x$ ml, of the $15$ cans of soda is summarised by

$\sum{x}=5314.5$, $\sum{{{x}^{2}}=1892040.25}$

Assuming that the variance of the distribution is unaltered by the adjustment, test at the $6\%$ significance level whether there is any change in the mean volume of the soda.

Normal Distribution

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

Worked Example 3

Let $X$ be the volume of a soda in a can.
Let $\mu$ be the population mean volume of soda in a can.

**Hypotheses:**
$H_0: \mu = 355 \text{ ml}$
$H_1: \mu \neq 355 \text{ ml}$

**Given:** $\sigma = 1.2$ ml, $n = 15$, $\bar{x} = \frac{5314.5}{15} = 354.3$ ml, $\alpha\% = 6\%$

**Under $H_0$, test statistic:**
$Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \sim N(0, 1)$

From GC, **p-value = 0.02346** < 6% (level of significance)

**Conclusion:** Since p-value < level of significance, we **reject $H_0$** and conclude that at 6% level of significance, we have sufficient evidence to state that there is a **change in the mean volume** of the soda in a can.

Worked Example 3

Worked Example 3

Video Solution: