Assume that the two samples are independent simple random...

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?

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?

Key Concepts

Independent Simple Random Samples

This concept refers to samples that are drawn in such a way that every individual in the population has an equal chance of being selected, and the selection of one individual does not affect the selection of another. It is a foundational assumption in many statistical procedures, ensuring that the sample represents the population without bias, which is critical for valid inference.

Normally Distributed Populations

Many inferential statistical methods assume that the populations from which samples are drawn are normally distributed. This assumption is essential for the validity of tests concerning means, particularly when sample sizes are small, though it is less crucial with larger sample sizes due to the Central Limit Theorem.

Unequal Population Variances

When comparing two means, assuming unequal variances means that the variability in the two populations is not necessarily the same. This condition requires the use of specific tests, such as Welch’s t-test, which adjusts the degrees of freedom to account for the differences in variability, ensuring a more accurate statistical inference.

Hypothesis Testing for Two Means

This statistical process is used to determine whether there is a significant difference between the means of two independent groups. It involves formulating a null hypothesis that the two means are equal and an alternative hypothesis that they are not, followed by calculating a test statistic and p-value to decide whether to reject the null hypothesis.

Confidence Interval for Difference of Means

A confidence interval provides a range of plausible values for the difference between two population means. Constructed alongside a hypothesis test, it offers additional insight by quantifying the uncertainty in the estimate and helps determine whether the observed difference is statistically significant.

Statistical vs. Practical Significance

While statistical significance assesses whether an observed difference is likely due to chance, practical significance considers the real-world importance or impact of the difference. It is possible for a result to be statistically significant yet have minimal practical implications, making this distinction crucial when interpreting results in applied contexts.

Transcript

00:01 So we're going to assume that the volume of regular coke is equal to the volume of regular pepsi, and alternately that the volumes are different.

00:15 So we're going to do a two -tail test, which means picture -wise, we're assuming that this difference between, and let's say i take regular coke, regular coke minus the regular pectoral peasant, pepsi could do the opposite way too.

00:34 We're going to assume that that mean is zero.

00:37 And then we will look at the actual difference we're getting is 0 .1 ounces.

00:45 And because we're doing a two -tail test, we'll also look at the point one.

00:49 And this is really a difference of negative point one, and this is 0 .1.

00:52 But these aren't test statistics.

00:53 These are the actual numbers.

00:55 And these combined together will be our p value.

00:58 So we want to find what that test statistic is.

01:01 And we will use the conservative estimate and say that the degrees of freedom is 35.

01:06 And we're going to take that 12 .19 minus the 12 .29, which again is that 0 .1 ounces.

01:14 And then we will have this square root of our standard deviation square to the first sample divided by sample size and the standard deviation or the finding the variance, basically of that one, and then divided by the sample size.

01:29 And when we get that test statistic, we end up getting negative 4 .22.

01:35 So we want to find what's the likelihood of having that come up or more unusual.

01:43 So this value, again, corresponds to a test statistic of negative 4 .22 for a t value, and this one would be positive 4 .22.

01:53 And so we're just going to double this probability to find our p value.

01:57 And when we do this, i use some software to find this, and this comes out to be pretty close to zero.

02:04 It comes out to be 0 .000, like, 16.

02:08 So it's very small.

02:10 And so this is smaller than the 5 % significance level.

02:14 So we would have strong evidence, because it's so much stronger, strong to reject the null and conclude that their volumes are different.

02:28 So volumes are different.

02:33 That's what we would conclude.

02:35 Now, for the confidence interval, the confidence interval, we would use for part b, we would use a 95 % confidence interval because our 5 % significance level has already been split into those two tails.

02:51 Now, on my book that i'm using, i didn't look up online this book...

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