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 p with the heights of their main opponents. Complete parts (a) and (b) below. Height (cm) of President Height (cm) of Main Opponent 163 188 175 180 179 178 177 171 172 175 184 174 D a. 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 am 0 cm. In this example, p, 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 min opponent's height. What are the null and alternative hypotheses for the hypothesis test?

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**Question:**
What feature of the confidence interval leads to the same conclusion reached in part (a)?

**Statement:**
Since the confidence interval contains [dropdown] [dropdown] the null hypothesis.

**Explanation:**
This text is part of an exercise exploring the relationship between confidence intervals and hypothesis testing. The question asks what aspect of the confidence interval supports a conclusion made earlier in a problem.

**Instructional Note:**
The student is expected to choose options from the dropdowns that describe a specific feature of the confidence interval, such as including or excluding the null hypothesis value, which influences the decision to accept or reject the null hypothesis.
Transcribed Image Text:**Question:** What feature of the confidence interval leads to the same conclusion reached in part (a)? **Statement:** Since the confidence interval contains [dropdown] [dropdown] the null hypothesis. **Explanation:** This text is part of an exercise exploring the relationship between confidence intervals and hypothesis testing. The question asks what aspect of the confidence interval supports a conclusion made earlier in a problem. **Instructional Note:** The student is expected to choose options from the dropdowns that describe a specific feature of the confidence interval, such as including or excluding the null hypothesis value, which influences the decision to accept or reject the null hypothesis.
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) below.

| Height (cm) of President | 177 | 171 | 172 | 175 | 184 | 174 |
| Height (cm) of Main Opponent | 163 | 182 | 175 | 180 | 179 | 178 |

**a.** 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, \(\mu_d\) 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?

\(H_0: \mu_d \leq\) ____ cm

\(H_1: \mu_d >\) ____ cm

(Type integers or decimals. Do not round.)

**Identify the test statistic.**

\(t =\) ____ (Round to two decimal places as needed.)

**Identify the P-value.**

P-value = ____ (Round to three decimal places as needed.)

**What is the conclusion based on the hypothesis test?**

Since the P-value is ____ the significance level, ____ the null hypothesis. There ____ sufficient evidence to support the claim that presidents tend to be taller than their opponents.

**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)?

The confidence interval is ____ cm < \(\mu_d\) < ____ cm. (Round to one decimal place as needed.)
Transcribed Image Text: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) below. | Height (cm) of President | 177 | 171 | 172 | 175 | 184 | 174 | | Height (cm) of Main Opponent | 163 | 182 | 175 | 180 | 179 | 178 | **a.** 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, \(\mu_d\) 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? \(H_0: \mu_d \leq\) ____ cm \(H_1: \mu_d >\) ____ cm (Type integers or decimals. Do not round.) **Identify the test statistic.** \(t =\) ____ (Round to two decimal places as needed.) **Identify the P-value.** P-value = ____ (Round to three decimal places as needed.) **What is the conclusion based on the hypothesis test?** Since the P-value is ____ the significance level, ____ the null hypothesis. There ____ sufficient evidence to support the claim that presidents tend to be taller than their opponents. **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)? The confidence interval is ____ cm < \(\mu_d\) < ____ cm. (Round to one decimal place as needed.)
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