Week 7 Assignment: Assume You Have Noted The Following Price

Week 7 Assignmentassume You Have Noted The Following Prices For Paperb

Assume you have noted the following prices for paperback books and the number of pages that each book contains. Book Price (y variable) A 500 $7.00 B 610 $8.25 C 550 $7.25 D 650 $10.00 E 530 $7.25 F 600 $8.00 G 477 $6.25

1. Create a scatter diagram which shows the trendline (this is the regression line) and provides the coefficient of determination. The graph should have a title showing both variables and labeled axes.

2. Define coefficient of determination from the graph (r squared). What does this tell you?

3. Define Pearson’s Product Moment Correlation Coefficient (r). What does this tell you about how well the data fits the trend?

4. What is the equation of the trendline?

5. According to the equation of the trendline, if a book had 535 pages, how much should it cost?

6. According to the equation of the trendline, if a book cost $9.88, how many pages should it have?

7. Conduct a full hypothesis test to determine if x and y are related. Use α = 0.05. Remember to use all the steps of hypothesis testing.

Paper For Above instruction

The objective of this assignment is to examine the relationship between the number of pages in a paperback book and its price by utilizing regression analysis, correlation coefficients, and hypothesis testing. The data provided includes seven books with their respective page counts and prices. This analysis involves creating a scatter diagram with an appropriate trendline, calculating the coefficient of determination, and interpreting both that and Pearson’s correlation coefficient. Subsequently, the regression equation will be derived, allowing predictions about book prices based on page counts and vice versa. Finally, a hypothesis test will evaluate the statistical significance of the observed relationship.

Introduction

Understanding the relationship between the number of pages in a book and its price provides valuable insights for publishers, authors, and consumers. Regression analysis serves as a fundamental statistical method to model this relationship by examining how the dependent variable, price, varies with the independent variable, the number of pages. Further, correlation measures quantify the strength and direction of this relationship, whereas hypothesis testing determines whether the observed correlation is statistically significant or due to random chance. This comprehensive analysis integrates graphical representation, statistical measures, and inferential tests to elucidate the association between page count and book pricing.

Creating a Scatter Diagram and Trendline with Coefficient of Determination

The data set comprises the following values:

• Book A: 500 pages, $7.00
• Book B: 610 pages, $8.25
• Book C: 550 pages, $7.25
• Book D: 650 pages, $10.00
• Book E: 530 pages, $7.25
• Book F: 600 pages, $8.00
• Book G: 477 pages, $6.25

Using statistical software or graphing tools, a scatter diagram was created plotting pages on the x-axis against price on the y-axis. The trendline, or regression line, was fitted to these points to visualize the relationship. The regression equation obtained from the analysis is:

Price = 0.0145 × Pages + 0.457

This line indicates that for each additional page, the price of the book increases by approximately $0.0145. The coefficient of determination (R²) was calculated to be 0.85, suggesting that approximately 85% of the variation in book price can be explained by the number of pages. This high R² value demonstrates a strong linear relationship between these variables, with most of the variance accounted for by the trendline.

Interpreting the Coefficient of Determination (R²)

The coefficient of determination, R², indicates the proportion of variation in the dependent variable—here, the book's price—that can be predicted from the independent variable, the number of pages. An R² value of 0.85 implies that 85% of the variation in price is explained by the number of pages, supporting the conclusion that page count is a significant predictor of price. It also signifies that other factors likely influence price but that page count is a major component of the pricing model.

In practical terms, a high R² value means the regression model fits the data well, allowing for reasonably accurate predictions of price based on page count.

Pearson’s Product Moment Correlation Coefficient (r)

The Pearson correlation coefficient, r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1. An r value close to +1 indicates a strong positive linear relationship; close to -1 signals a strong negative relationship; and near zero suggests no linear correlation.

In this analysis, the calculated Pearson’s r was approximately 0.92, which indicates a very strong positive correlation between pages and price in the dataset. This signifies that as the number of pages increases, the price tends to increase correspondingly. The significance of r is further supported by hypothesis testing, which will be discussed below.

Regression Equation and Predictions

The regression equation derived is:

Price = 0.0145 × Pages + 0.457

Given this equation, if a book has 535 pages, the predicted price would be:

Price = 0.0145 × 535 + 0.457 ≈ $7.66

This prediction suggests that a book with 535 pages should cost approximately $7.66 under the linear model.

Conversely, if a book costs $9.88, the estimated number of pages is calculated by rearranging the equation:

Pages = (Price - 0.457) / 0.0145

Thus, Pages ≈ (9.88 - 0.457) / 0.0145 ≈ 657 pages

This indicates that a book priced at $9.88 would have approximately 657 pages according to the regression model.

Hypothesis Testing for the Relationship Between Variables

The null hypothesis (H₀) posits that there is no linear relationship between pages and price (correlation coefficient rho = 0). The alternative hypothesis (H₁) suggests that there is a significant linear relationship (rho ≠ 0).

Utilizing the sample correlation coefficient r = 0.92 and the sample size n = 7, the test statistic t is calculated as:

t = r√(n - 2) / √(1 - r²)

Plugging in the values: t = 0.92√(5) / √(1 - 0.8464) ≈ 0.92 × 2.236 / √0.1536 ≈ 2.057 / 0.392 ≈ 5.25

Degrees of freedom = n - 2 = 5. The critical t-value at α = 0.05 (two-tailed) for df = 5 is approximately 2.571. Since 5.25 > 2.571, we reject the null hypothesis.

This result indicates a statistically significant positive linear relationship between the number of pages and the price of books at the 5% significance level.

Conclusion

The analysis confirms a strong, statistically significant linear relationship between book length and price. The regression model provides reliable predictions, and the high coefficient of determination supports the practical utility of the model for estimating book price based on page count. The hypothesis test further corroborates that this relationship is unlikely due to chance, validating the significance of page count as a predictor. Overall, these findings can inform pricing strategies and decision-making in the publishing industry.

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