Question

A manufacturer fills soda bottles. Periodically they test to see if there is a difference in the amount of soda put in cola and diet cola bottles. A random sample of 14 cola bottles contains an average of 502 mL of cola with a standard deviation of 4 mL. A random sample of 16 diet cola bottles contains an average of 499 mL of cola with a standard deviation of 5 mL. Test the claim that there is a difference in the fill levels of the two types of soda using a 0.01 level of significance. Assume that the population variances are different since different machines are used for the filling process.

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

and

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha:

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the degrees of freedom are df=27.804. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal.

Hence, it is found that the critical value for this two-tailed test is tc​=2.765, for α=0.01 and df=27.804.

The rejection region for this two-tailed test is R={t:∣t∣>2.765}.

(3) Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=1.824≤tc​=2.765, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0789, and since p=0.0789≥0.01, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.01 significance level.

Graphically

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