Question #345493

The owner of the factory that sells a particular bottled fruit juice claimed 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 s is 5 ml. Use α = 0.03. Will

you reject or not to reject the claim.


1
Expert's answer
2022-05-30T05:39:54-0400

The following null and alternative hypotheses need to be tested:

H0:μ=250H_0:\mu=250

H1:μ250H_1:\mu\not=250

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is α=0.03,\alpha = 0.03, df=n1=99df=n-1=99 and the critical value for a two-tailed test is tc=2.201819.t_c = 2.201819.

The rejection region for this two-tailed test is R={t:t>2.201819}.R = \{t:|t|> 2.201819\}.

The t-statistic is computed as follows:


t=xˉμs/n=2482505/100=4t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{248-250}{5/\sqrt{100}}=-4

Since it is observed that t=4>2.201819=tc,|t|=4> 2.201819=t_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, df=99df=99 degrees of freedom, t=4t=-4 is p=0.000122,p=0.000122, and since p=0.000122<0.03=α,p=0.000122<0.03=\alpha, it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ\mu

is different than 250, at the α=0.03\alpha = 0.03 significance level.


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