91Ó°ÊÓ

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9 - 1 along with "Table" answers based on Table \(A\) - 3 with df equal to the smaller of \(\boldsymbol{n}_{I}-\boldsymbol{I}\) and \(\boldsymbol{n}_{2}-\boldsymbol{I} .\) ) Coke and Pepsi Data Set 26 "Cola Weights and Volumes" in Appendix B includes volumes of the contents of cans of regular Coke \((n=36, \bar{x}=12.19 \text { oz, } s=0.11\) oz ) and volumes of the contents of cans of regular Pepsi \((n=36, \bar{x}=12.29 \text { oz, } s=0.09\) oz). a. Use a 0.05 significance level to test the claim that cans of regular Coke and regular Pepsi have the same mean volume. b. Construct the confidence interval appropriate for the hypothesis test in part (a). c. What do you conclude? Does there appear to be a difference? Is there practical significance?

Step by step solution

01

Title - State the Hypotheses

First, define the null and alternative hypotheses. The null hypothesis \(H_0\) states that there is no difference in the mean volumes of Coke and Pepsi cans. The alternative hypothesis \(H_a\) states that there is a difference. That is, \(\boldsymbol{H_0: \mu_{Coke} = \mu_{Pepsi}}\) and \(H_a: \mu_{Coke} \eq \mu_{Pepsi}\).

02

Title - Calculate the Test Statistic

Use the t-test for two independent samples. The formula to calculate the test statistic is: \(\boldsymbol{t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}}\). Substituting the given values: \(t = \frac{(12.19 - 12.29)}{\sqrt{(0.11)^2/36 + (0.09)^2/36}} = \-4.262\).

03

Title - Determine the Degrees of Freedom

The degrees of freedom (df) can be approximated as the smaller of \(n_1 - 1\) and \(n_2 - 1\). In this case, \(df = 36 - 1 = 35\).

04

Title - Find the Critical Value

For a two-tailed test at the \alpha = 0.05\ significance level with \(df = 35\), the critical t-value from the t-distribution table is approximately \(\boldsymbol{t_{critical} = \pm 2.0301}\).

05

Title - Compare Test Statistic to Critical Value

Since the calculated test statistic \(t = -4.262\) is less than \(t_{critical} = -2.0301\), we reject the null hypothesis.

06

Title - Construct the Confidence Interval

The confidence interval for the difference in means is given by \(\boldsymbol{(\bar{x}_1 - \bar{x}_2) \pm t_{critical} \sqrt{s_1^2/n_1 + s_2^2/n_2}}\). Substituting the values, the margin of error is \(2.0301 \sqrt{(0.11)^2/36 + (0.09)^2/36} = 0.0357\). The confidence interval is then \(12.19 - 12.29 \pm 0.0357 = (-0.100, -0.0286)\).

07

Title - Conclude

Since the confidence interval \((-0.100, -0.0286)\) does not include 0, there is statistical evidence to conclude that there is a difference in the mean volumes of Coke and Pepsi cans. The difference is also practically significant, as even small differences in beverage volumes could matter to consumers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test

A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups. In our problem, we are comparing the mean volumes of Coke and Pepsi cans. We use the t-test for two independent samples, which means the data from the Coke and Pepsi samples are not related. The formula for the t-test statistic is:

\( t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} \)
Here:

• \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means for Coke and Pepsi respectively.
• \(s_1\) and \(s_2\) are the standard deviations for the Coke and Pepsi samples.
• \(n_1\) and \(n_2\) are the sample sizes for Coke and Pepsi.

Using the values given in the problem, our test statistic was calculated to be approximately \(-4.262\). This tells us how many standard deviations our observed difference in means is away from the null hypothesis of no difference. This result is then compared to a critical value from the t-distribution table to determine whether it is statistically significant.

confidence interval

A confidence interval is a range of values derived from sample statistics that is likely to contain the true population parameter. In our problem, we constructed a confidence interval for the difference in mean volumes between Coke and Pepsi cans. The formula for the confidence interval is:

\( (\bar{x}_1 - \bar{x}_2) \pm t_{\text{critical}} \sqrt{s_1^2/n_1 + s_2^2/n_2} \)
This interval helps us understand the range within which the true difference in means lies, with a certain level of confidence (in this case, 95%). Given our values, we found the confidence interval to be \((-0.100, -0.0286)\). This means we are 95% confident that the true difference in mean volumes between Coke and Pepsi cans falls within this range. Since 0 is not within this interval, it suggests a significant difference exists.

statistical significance

Statistical significance helps us decide whether the result of a statistical test is meaningful. In our problem, we tested the claim that the mean volumes of Coke and Pepsi cans are the same using a significance level of 0.05.

The critical value from the t-distribution table for 35 degrees of freedom (since each sample had 36 observations) was approximately \(\pm 2.0301\). Our calculated t-statistic was \(-4.262\), which is less than \(-2.0301\).

• This means our test statistic falls in the rejection region.
• We reject the null hypothesis and conclude there is a statistically significant difference between the mean volumes of Coke and Pepsi cans.

Statistical significance indicates that the observed difference in sample means is not likely due to random chance but rather suggests a real difference in population means.

independent samples

In hypothesis testing, independent samples refer to two sets of data that are not connected or related. Each sample is randomly selected from its own population. In our problem, the Coca-Cola and Pepsi volume data are treated as independent samples.

Important properties of independent samples include:

• Data from one sample does not affect or inform the data from the other sample.
• Each sample is drawn randomly from its respective normally distributed population.
• We do not assume that the population standard deviations are equal.

By treating Coke and Pepsi volumes as independent samples, we can apply the two-sample t-test to compare their means without any bias introduced by potential interdependencies in the data. This allows for more robust and reliable hypothesis testing.