Math Problem Statement

The cases that make up this dataset are types of cars. The data include the engine size or displacement​ (in liters) and horsepower​ (HP) of 67 vehicles sold in a certain country in 2011. Use the SRM of the horsepower on the engine displacement to complete parts​ (a) through ​(c).

Displacement (liters) Horsepower 4.1 427 5.3 515 2.9 326 4.3 358 2.1 201 2.1 268 3.1 228 3.1 301 4.5 409 2.2 310 3.8 417 1.7 123 1.6 175 4.5 556 2.5 176 2.5 171 2.4 223 1.9 151 2.5 299 3.6 253 6.8 298 1.9 230 3.6 296 3.7 303 5.2 406 5.4 555 3.4 262 2.9 265 6.4 427 2.4 180 1.4 85 1.7 138 1.5 89 1.5 115 3.9 303 3.5 277 3.6 297 2.5 199 3.5 288 3.8 309 1.7 142 2.1 142 1.7 152 4.9 434 1.9 216 3.8 312 3.9 330 2.1 275 1.9 222 2.7 296 1.4 101 1.9 145 2.5 164 2.5 169 2.3 265 2.1 166 1.4 92 2.7 159 3.9 242 2.5 185 1.4 104 6.3 511 2.6 173 2.1 200 2.1 142 2.1 114 3.7 275

​(a) A manufacturer offers 2.8 and 3.4 liter engines in a particular model car. Based on these​ data, how much more horsepower should one expect the larger engine to​ produce? Give your answer as a​ 95% confidence interval. [___ to ___]    ​(Round to the nearest integer as​ needed.)

Solution

We need to perform a simple regression model (SRM) of horsepower (HP) on engine displacement to answer part (a). The SRM equation generally takes the form:

HP=β0+β1×DisplacementHP = \beta_0 + \beta_1 \times Displacement

where:

  • β0\beta_0 is the y-intercept,
  • β1\beta_1 is the slope of the regression line, representing the change in horsepower for each unit change in displacement.

Using the regression coefficients, we can predict the difference in horsepower between the two displacements (3.4 L and 2.8 L). Additionally, we can construct a 95% confidence interval for this difference.

Steps:

  1. Fit the Simple Linear Regression (SLR): Find the slope (β1\beta_1) and intercept (β0\beta_0).
  2. Predict the Horsepower: Use the regression equation to estimate the horsepower at 2.8 and 3.4 liters.
  3. Compute the Difference: Subtract the predicted horsepower for the 2.8 L engine from that of the 3.4 L engine.
  4. Construct the 95% Confidence Interval (CI): We calculate the standard error of the regression prediction and use it to form the confidence interval for the horsepower difference.

Let me calculate these values for you.Based on the simple regression model (SRM), the predicted increase in horsepower when moving from a 2.8-liter engine to a 3.4-liter engine is approximately 45 HP.

The 95% confidence interval for this difference is from 38 HP to 52 HP (rounded to the nearest integers).

Let me know if you'd like more details on the regression model or the calculations.

Here are 5 questions to expand on this:

  1. What is the formula for calculating horsepower based on engine displacement?
  2. How does one interpret the slope (β1\beta_1) in a simple regression model?
  3. What is the importance of a 95% confidence interval in statistical predictions?
  4. How can we check if the regression model assumptions are valid?
  5. How could other factors besides engine displacement affect horsepower?

Tip: The slope in a linear regression model represents the rate of change of the dependent variable (horsepower) for each unit change in the independent variable (displacement).

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Math Problem Analysis

Mathematical Concepts

Simple Linear Regression
Statistical Predictions
Confidence Intervals

Formulas

HP = β_0 + β_1 × Displacement
95% Confidence Interval formula: Prediction ± (t-value × SE)

Theorems

Linear Regression Theorem
Central Limit Theorem (for Confidence Intervals)

Suitable Grade Level

Undergraduate or Advanced High School (Statistics)

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