Use the listed paired sample data, and assume that the...

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 15 "Presidents"). a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than $0 \mathrm{cm} .$ b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)? $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Height (cm) of President } & 185 & 178 & 175 & 183 & 193 & 173 \\ \hline \text { Height (cm) of Main Opponent } & 171 & 180 & 173 & 175 & 188 & 178 \\ \hline \end{array}$$

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 15 "Presidents").
a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than $0 \mathrm{cm} .$
b. Construct the confidence interval that could be used for the hypothesis test described in part
(a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Height (cm) of President } & 185 & 178 & 175 & 183 & 193 & 173 \\
\hline \text { Height (cm) of Main Opponent } & 171 & 180 & 173 & 175 & 188 & 178 \\
\hline
\end{array}$$

Key Concepts

Significance Level

The significance level, denoted by alpha (?), is the probability threshold for rejecting the null hypothesis in a hypothesis test. It represents the risk of concluding that a difference exists when there is none. Commonly, a significance level of 0.05 is used, meaning there is a 5% risk of a Type I error, where the null hypothesis is wrongly rejected.

Confidence Interval for Mean Difference

A confidence interval for the mean difference provides a range of values that is likely to contain the true mean difference between two paired groups. This interval is constructed using the sample mean difference, the standard deviation of the differences, and the t-distribution critical value. If the entire interval lies above or below a particular value (such as zero), it supports the conclusion drawn from the hypothesis test.

t-Distribution

The t-distribution is used in hypothesis testing, particularly when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the extra uncertainty when estimating the standard deviation from the sample data. This distribution is fundamental to the paired t-test and the construction of confidence intervals in these cases.

Paired t-Test

The paired t-test is a statistical method used to determine whether the mean difference between paired observations is significantly different from zero (or another specified value). This test compares the average change in the paired data, accounting for the fact that the observations are related, and uses the t-distribution to assess significance.

Paired Sample Design

In a paired sample design, measurements are taken from the same subjects or matched subjects under two different conditions. This approach is used to eliminate the variability that exists between different subjects, allowing for a more accurate comparison of the two conditions by focusing on the differences within paired observations.

Hypothesis Testing

Hypothesis testing is a framework for making inferences about population parameters based on sample data. In this context, the null hypothesis typically states that there is no difference (or that the difference is equal to a specific value), and the alternative hypothesis states that there is a difference (for example, that the mean difference is greater than zero). The process involves calculating a test statistic and comparing it to a critical value, or obtaining a p-value, to decide whether to reject the null hypothesis.

Transcript

00:01 So the first thing we would need to do is take the president minus the opposing person and find that difference.

00:08 So i'm going to do the actual height of the president versus the opponent.

00:15 And let me click back here.

00:17 So president height in centimeters minus the opponent.

00:22 So make sure you label that what you're doing.

00:24 And this is again in centimeters.

00:26 And so i had that data as 14, negative 2.

00:32 285 and negative 5.

00:34 And so there are six pieces of data and i got the mean to be 3 .6 repeating and the standard deviation.

00:42 And so i let me call that the difference.

00:44 So this will be the mean difference and the standard deviation of those differences comes out to be 6 .8896.

00:52 And let's see.

00:54 I don't think we need anything else written down.

00:56 Now we are again assuming that the mean difference is zero.

01:03 So our null hypothesis would be that the mean difference in the president and his opponent is zero and alternately we think the president is higher in height or taller.

01:16 So we would use that as higher and we're getting the mean difference to be up here at 3 .6.

01:23 This will be our p -value.

01:25 So what's the likelihood if this distribution is true and it's really centered at zero and we're assuming these are random values, that we would get a mean difference that is greater than or equal to 3 .6 repeating.

01:38 So we change it into our test statistic, and that will be a t value with 5 degrees of freedom, and we take the 3 .6 repeating, minus the mean we're assuming, and then we take that standard deviation over the square root of n, and that test statistic that we end up getting is, let me find it here...

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