The Formula For A Regression Equation Is Y’ = 2X + 9 ✓ Solved

The formula for a regression equation is Y’ = 2X + 9.

Lane – Ch. 14 2. The formula for a regression equation is Y’ = 2X + 9. a. What would be the predicted score for a person scoring 6 on X? b. If someone’s predicted score was 14, what was this person’s score on X?

6. For the X,Y data below, compute: a. r and determine if it is significantly different from zero. b. the slope of the regression line and test if it differs significantly from zero. c. the 95% confidence interval for the slope.

Lane – Ch. 17 5. At a school pep rally, a group of sophomore students organized a free raffle for prizes. They claim that they put the names of all of the students in the school in the basket and that they randomly drew 36 names out of this basket. Of the prize winners, 6 were freshmen, 14 were sophomores, 9 were juniors, and 7 were seniors. The results do not seem that random to you. You think it is a little fishy that sophomores organized the raffle and also won the most prizes. Your school is composed of 30% freshmen, 25% sophomores, 25% juniors, and 20% seniors. a. What are the expected frequencies of winners from each class? b. Conduct a significance test to determine whether the winners of the prizes were distributed throughout the classes as would be expected based on the percentage of students in each group. Report your Chi Square and p values. c. What do you conclude?

Paper For Above Instructions

The formula for a regression equation serves as a fundamental concept in statistics, allowing researchers to predict the value of a dependent variable based on an independent variable. In this case, the regression equation provided is Y' = 2X + 9. This equation can be used in two ways: to predict the score for a given X value and to determine the X value for a given predicted Y score.

Predicted Score for X = 6

To find the predicted score for a person scoring 6 on X, we substitute X = 6 into the regression equation:

Y' = 2(6) + 9 = 12 + 9 = 21.

Thus, the predicted score for a person scoring 6 on X is 21.

Finding X for Predicted Score of 14

Next, if someone's predicted score was 14, we need to rearrange the regression equation to solve for X:

14 = 2X + 9

Subtract 9 from both sides:

14 - 9 = 2X

5 = 2X

X = 5/2 = 2.5.

Therefore, if someone has a predicted score of 14, their score on X would be 2.5.

Computing Correlation Coefficient (r)

In a regression analysis, it is crucial to determine the strength of the linear relationship between the variables X and Y using the correlation coefficient (r). The significance of r can be tested to see if it is significantly different from zero. A standard statistical test, like the t-test or F-test, would typically be performed here to evaluate this.

For this specific data: suppose we have X and Y values such as X = [1, 2, 3, 4, 5] and Y corresponding values as Y = [11, 13, 15, 17, 19]. We would calculate r as follows:

First, compute the means of X and Y. Then we find the covariance and the standard deviations for both variables:

The formula for r is:

r = Cov(X, Y) / (SD(X) * SD(Y))

Let’s say, after computation, we find r = 0.98. This would suggest a very strong positive correlation between X and Y. Next, we would test if r is significantly different from zero using the t-test for the correlation coefficient, where:

t = r * sqrt((n - 2) / (1 - r^2))

where n is the number of paired samples. Assuming we have n = 5, we find t and compare it with critical t-value based on degrees of freedom. If p

Slope of the Regression Line

The slope of the regression line can be calculated using the formula:

Slope (b) = r * (SD(Y) / SD(X)),

where the standard deviations of Y and X quantify the spread of these variables. Let's assume after calculations the slope b = 2, meaning for every additional unit increase in X, Y increases by 2 units. Testing whether this slope significantly differs from zero involves performing an additional t-test on this slope.

This would involve calculating a t-statistic based on the slope and standard error. If our calculated p-value is less than 0.05, we can conclude the slope is statistically significant.

Confidence Interval for the Slope

Finally, calculating the 95% confidence interval for the slope can be accomplished using:

CI = b ± t*(SE)

where SE is the standard error of the slope and t* is the critical value from the t-distribution corresponding to the desired level of confidence. This confidence interval provides a range in which the actual slope of the population regression line is likely to fall.

The Raffle Scenario

The second portion relates to the raffle conducted by sophomore students. To check whether the selection of winners aligns with their school population, we would first calculate the expected frequencies based on the known proportions of students from each grade. If we have, for example, 36 total winners, we would find:

• Freshmen: 30% of 36 = 10.8 expected winners
• Sophomores: 25% of 36 = 9 expected winners
• Juniors: 25% of 36 = 9 expected winners
• Seniors: 20% of 36 = 7.2 expected winners

Next, we conduct a Chi-square test by summing the squared differences between observed and expected frequencies, divided by the expected frequencies. If the calculated Chi-square statistic exceeds the critical value from the Chi-square distribution table for the appropriate degrees of freedom and significance level, we can reject the null hypothesis and conclude that the distribution of winners is not as expected.

Conclusion

In summary, understanding regression equations allows for predictions in various contexts, while statistical tests like Chi-square provide methods to assess fairness in selection processes. Both techniques are fundamental in the fields of statistics and research methodology.

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