Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 12.CR.2

Chapter 10, Problem 12.CR.2

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that there is no significant difference between the mean of presidents and the mean of popes (H₀: μ₁ = μ₂). The alternative hypothesis states that there is a significant difference (H₁: μ₁ ≠ μ₂).

Step 2: Choose the appropriate statistical test. Since we are comparing two means, use a two-sample t-test. Determine whether the test should be independent or paired based on the nature of the data. In this case, it is likely an independent two-sample t-test.

Step 3: Calculate the test statistic. Use the formula for the t-test statistic:

t = \frac{(x_{1} - x_{2})}{\sqrt{\frac{s_{1} ² / n_{1} + s_{2} ² / n_{2}}{1}}}

, where

x_{1}

and

x_{2}

are the sample means,

s_{1}

and

s_{2}

are the sample standard deviations, and

n_{1}

and

n_{2}

are the sample sizes.

Step 4: Determine the degrees of freedom (df). For an independent two-sample t-test, use the formula:

df = \frac{(s_{1} ² / n_{1} + s_{2} ² / n_{2}) ²}{\frac{(s_{1} ² / n_{1}) ²}{n_{1} - 1} + \frac{(s_{2} ² / n_{2}) ²}{n_{2} - 1}}

Step 5: Compare the calculated t-statistic to the critical t-value from the t-distribution table at the chosen significance level (e.g., α = 0.05). If the absolute value of the t-statistic exceeds the critical t-value, reject the null hypothesis and conclude that there is a significant difference between the means. Otherwise, fail to reject the null hypothesis.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample of data to support a particular claim about a population. In this context, it involves formulating a null hypothesis (no difference between means) and an alternative hypothesis (a significant difference exists) and using sample data to test these hypotheses.

t-Test

A t-test is a statistical test used to compare the means of two groups to see if they are significantly different from each other. It is particularly useful when the sample sizes are small and the population standard deviations are unknown. In this case, a t-test would help assess whether the means of presidents and popes differ significantly.

P-Value

The p-value is a measure that helps determine the significance of the results obtained from a statistical test. It represents the probability of observing the test results, or something more extreme, assuming the null hypothesis is true. A low p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting a significant difference between the means being compared.

Recommended video:

Guided course

06:50

Step 3: Get P-Value

Related Practice

Textbook Question

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

131

views

Textbook Question

Clusters Refer to the Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

165

views

Textbook Question

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions Repeat the preceding exercise assuming that the linear correlation coefficient is r = 0.132.

123

views

Textbook Question

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added in the last column. Is there sufficient evidence to conclude that there is a linear correlation between lottery jackpot amounts and numbers of tickets sold? Comment on the effect of the added pair of values in the last column. Compare the results to those obtained in Example 4.

[IMAGE]

213

views