Question

A manufacturer fills soda botties. Periodically the company tests to see if there is a difference between the soda put in bottles of regular cola and diet cola. A random sample of 12 bottles of regular cola hetween the mean amounts of with a standard deviation of 2.6 mL. A random sample of 19 bottles of diet cola regular cola has a mean of 500.6 mL of soda standard deviation of 4.4 mL. Test the claim that there is a difference between has a mean of 498.4 mL of soda with a using a 0.05 level of significance. Assume that both populations are approximately merm fill levels for the two types of soda not equal since different machines are used to fill bottles of regular cola and diet cormal and that the population variances are and let bottles of diet cola be Population 2 . You were asked to draw a conclusion and interpret the decision. From Step 1 we know that the test is a two-tailed test, thus the p value is computed as p-value =P(∣t∣≥1.749). We are also given that the population variances are assumed to be not equal. Thus, find the pvalue by either calculating the number of degrees of freedom and using the appropriate technology, or by inputting the given data into technology and performing the test. Note that the results from these two methods should be similar, but may differ slightly due to some technologies using an alternate formula for the number of degrees of freedom. Round your answer to four decimal places.

Answer 1

To test the claim of a difference in mean soda amounts between regular cola and diet cola, use a two-sample t-test and follow these steps: state null and alternative hypotheses, calculate the test statistic, determine degrees of freedom, find critical value(s), compare the test statistic to critical value(s), and interpret the decision. The conclusion will determine if there is enough evidence to support the claim.

Step-by-step explanation:

The question is asking you to test the claim that there is a difference between the mean amounts of soda filled in bottles of regular cola and diet cola. To do this, you can use a two-sample t-test. Here are the steps:

1. State the null and alternative hypotheses. The null hypothesis is that the mean amount of soda in regular cola bottles is equal to the mean amount of soda in diet cola bottles. The alternative hypothesis is that the mean amount of soda in regular cola bottles is different from the mean amount of soda in diet cola bottles.
2. Calculate the test statistic, which is the t-value. You can use the formula t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2)), where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
3. Determine the degrees of freedom, which is calculated as df = (s1^2 / n1 + s2^2 / n2)^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1)).
4. Find the critical value(s) for the desired level of significance (0.05 in this case).
5. Compare the test statistic to the critical value(s) and make a decision. If the test statistic is outside the critical value(s) range, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
6. Interpret the decision in the context of the problem. If the null hypothesis is rejected, it means that there is enough evidence to suggest that there is a difference between the mean amounts of soda in regular cola and diet cola bottles. If the null hypothesis is not rejected, there is not enough evidence to support the claim of a difference in the mean amounts of soda.

Answer 2

The question involves conducting a Welchs t-test to determine if there is a significant difference between the mean fill levels of regular cola and diet cola. The process includes calculating the test statistic, evaluating the degrees of freedom, and computing the p-value to compare with the significance level.

Step-by-step explanation:

We are tasked with determining whether there is a statistical difference in the mean fill levels of soda between regular cola and diet cola. To test this claim, we employ a two-sample t-test with unequal variances, which is more commonly known as Welchs t-test. Given the sample sizes, means, and standard deviations, we first calculate the test statistic using the formula for Welchs t-test, which accounts for the unequal variances.

Afterward, we find the degrees of freedom using the formula for Welchs t-test df approximation. Lastly, we could utilize technology or a t-distribution table to find the p-value associated with the computed t-statistic over the calculated degrees of freedom. A p-value smaller than the alpha level of 0.05 would lead to the rejection of the null hypothesis, indicating a significant difference between the mean fill levels.