Math Problem Statement

A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights ( in centimeters ) of randomly selected presidents along with the heights of their main opponents. Complete parts (a) and (b) with the data here. The height (cm) of president is : 183, 179, 168, 189, 189, 170. the height ( cm) of main opponent is : 177,171,173,174,195,183. Part A asks : use the sample data with a 0.05 significance level to test the claim that for the population of heights for presidents and their main opponents, the differences have a mean greater than 0 cm. In this example, ųd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the president’s height minus their main opponent’s height. what are the null and alternative hypotheses for the hypothesis test?

Solution

To solve this hypothesis test problem, we'll first identify the null and alternative hypotheses based on the claim provided. The claim is that the differences in heights between presidents and their main opponents have a mean greater than 0 cm.

Let μd\mu_d represent the population mean of the differences in heights (president’s height minus opponent’s height).

Step 1: Define the Hypotheses

  • Null Hypothesis H0H_0: The null hypothesis will reflect the assumption that there is no advantage in terms of height difference. In this case, the mean difference μd\mu_d is 0. H0:μd=0H_0: \mu_d = 0

  • Alternative Hypothesis H1H_1: The alternative hypothesis will reflect the claim that the mean difference in heights is greater than 0. H1:μd>0H_1: \mu_d > 0

This is a right-tailed test, since the claim is that the mean difference is greater than zero.

Summary of the hypotheses:

  • Null Hypothesis H0H_0: The mean difference in heights is 0 (i.e., μd=0\mu_d = 0).
  • Alternative Hypothesis H1H_1: The mean difference in heights is greater than 0 (i.e., μd>0\mu_d > 0).

Would you like to proceed with calculating the test statistic and p-value for the next part? Let me know if you need further clarification!

Here are five related questions:

  1. How do you calculate the test statistic in a hypothesis test involving paired differences?
  2. What is the critical value for a one-tailed test at a 0.05 significance level?
  3. How can you interpret the p-value in hypothesis testing?
  4. How is the standard deviation of differences calculated for paired data?
  5. How do we check if the normality assumption holds for small sample sizes in paired t-tests?

Tip: In a paired t-test, it's important to keep track of the differences between paired data points, as it simplifies the analysis.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Paired t-test
Significance Level
Mean Difference

Formulas

Null Hypothesis (H0): μd = 0
Alternative Hypothesis (H1): μd > 0
Test Statistic for Paired t-test: t = (mean difference - hypothesized mean) / (standard deviation of differences / sqrt(n))

Theorems

Central Limit Theorem
T-distribution

Suitable Grade Level

Undergraduate Statistics