AP STATS 2.07 EXPLORING RELATIONSHIPS PT 2

Prior to fitting a linear regression model with x and y variables, both variables were transformed using logarithms. The following computer output was produced. (10 points) Part A: Interpret the value of r2. (5 points)Part B: Assuming the residual plot of the transformed data shows a pattern, comment on the appropriateness of this linear regression for modeling the relationship between the transformed variables. (5 points)

Part A: 82.14% of variance in y is explained by the LSRL. The model should fit the data well because r^2 is high. Part B: If the residual plot shows a pattern, then the linear model is not the best model for the data.

The percentage of children ages 1 to 14 living in a single-parent home in 1985 compared to 1991 for 13 states was gathered. (10 points)Part A: Determine and interpret the LSRL. (3 points)Part B: Predict the percent of children living in single-parent homes in 1991 for State 14 if the percentage in 1985 was 18.3. Show your work. (3 points)Part C: Calculate and interpret the residual for State 14 if the observed percent of children living in single-parent homes in 1991 was 21.5. Show your work. (4 points)

Part A: The LSRL is y^= 7.41 + 0.7x. Part B: y^= 7.41 + 0.7(18.3) y^= 20.22% Part C: 21.5 - 20.22 = 1.28. The residual is 1.28. This means that the value was underpredicted.

Data were gathered on how high dolphins jump from the surface of the water based on body length and graphed on a scatterplot. (10 points)

Part A: The scatter plot has a strong, positive linear correlation. There are no outliers. Part B: This correlation coefficient does make sense. It shows a strong positive correlation, like the interpretation in Part A, because the value is very close to 1, meaning the correlation is strong.

A random sample of online makeup tutorial videos and sales of makeup was measured, and the computer output for regression was obtained: (10 points) R-sq. = 99.53%R-sq. (adj) = 99.44% Part A: Use the computer output to determine the LSRL. Identify all the variables used in the equation. (3 points)Part B: What proportion of the variation in makeup sales is explained by its linear relationship to makeup tutorial videos? Explain. (4 points)Part C: Determine if increased makeup sales were caused by online makeup tutorial videos. Explain your reasoning. (3 points)

Part A: y^= -7.57 + 0.92x Part B: The R-sq value of 99.53% tells us that 99.53% of increased makeup sales are explained by its linear relationship to makeup tutorial videos. Part C: We cannot determine if increased makeup sales were caused by online makeup tutorials because correlation does not mean causation.

The mean unemployment rate in February of each year from 1990 to 2000 is x̄ = 5.65, with a standard deviation of sx = 0.92. The mean unemployment rate in July of each year for the same time frame is ȳ = 5.62, with a standard deviation of sy = 1.3. The correlation coefficient is r = 0.91. (10 points)Part A: Find the equation of the least-squares regression line for predicting July unemployment rate from February's unemployment rate. Show your work. (3 points)Part B: Use the regression line to predict the unemployment rate if February's rate is 3.8. Show your work. (3 points)Part C: Find and interpret r-squared. (4 points)

b= (1.3/0.92)(0.91) = 1.29 a= 5.62 - (1.29)(5.65) = -1.67 y^= -1.67 +1.29x Part B: -1.67 + 1.29(3.8) = 3.23 Part C: (0.91)^2 = 0.83. This means that 83% of the unemployment rate in July can be explained by the unemployment rate in February. Because r^2 is high, the model fits the data well.