Question

2. The owner of a factory that sells a particular bottled fruit juice claims that the average capacity of their product is 250 ml. To test the claim, a consumer group gets a sample of 100 such bottles, calculates the capacity of each bottle, and then finds the mean capacity to be 248 ml. The standard deviation is 5 ml. Is the claim true?

Answer 1

Here,

$\overline{x}=248$

$n=100$

$\alpha =0.5$

$\:s=5$

Hypothesis tested are,

$H_o:\mu=250$

$vs$

$H_1:\mu\:\not=$ $250$

The test statistic is,

$t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}=\frac{248-250}{\frac{5}{\sqrt{100}}}={-2\over{1\over2}}=-2\times2=-4.$

Therefore,

$t=-4$

and

$\left|t\right|=4$

The critical table value, $t_{{\alpha\over2},n-1}$ with $\alpha=0.05$ with $df=n-1=99$ is $t_{0.025,99}= 1.984217$.

The null hypothesis is rejected if $|t|\gt t_{0.025,99}$

Here $\left|t\right|=4>t_{0.025,99}=1.984217$

Therefore, we reject the null hypothesis at $5\%$% level of significance and conclude that there is no sufficient evidence to show that the average capacity of the product is 250 ml.