Question
Marissa went with her friends to the amusement park on a beautiful spring day. The park was crowded. Marissa wondered if there was an association between the weather and attendance. From data she received at the theme park office, Marissa randomly picked ten Saturdays and analyzed the data.
a. Marissa calculated the least squares regression line a=-14+0.41t where a is the attendance (in 1000s ) and t is the high temperature (Fº) that day. Interpret the slope in this context.
b. The residual plot Marissa created is shown at right. On days when temperatures were in the
,80s would you expect the predictions made by Marissa’s model to be too high, too low, or pretty accurate?
A residual plot with x axis labeled as temperature in fahrenheit scaled from 70 to 100 with y axis labeled Residuals in 100s of people scaled from negative 8 to 8. The points closely follow an upside down parabola.
c.What was the actual attendance on the day when the temperature was 95º?
D. Marissa drew the upper boundary line at a=-7+0.41t and the lower boundary line at a = -21+0.41t
. What are the upper and lower bounds for the predicted number of people attending when the temperature is 80º F?
Would you rely on this model to make predictions? Why or why not?
a. Marissa calculated the least squares regression line a=-14+0.41t where a is the attendance (in 1000s ) and t is the high temperature (Fº) that day. Interpret the slope in this context.
b. The residual plot Marissa created is shown at right. On days when temperatures were in the
,80s would you expect the predictions made by Marissa’s model to be too high, too low, or pretty accurate?
A residual plot with x axis labeled as temperature in fahrenheit scaled from 70 to 100 with y axis labeled Residuals in 100s of people scaled from negative 8 to 8. The points closely follow an upside down parabola.
c.What was the actual attendance on the day when the temperature was 95º?
D. Marissa drew the upper boundary line at a=-7+0.41t and the lower boundary line at a = -21+0.41t
. What are the upper and lower bounds for the predicted number of people attending when the temperature is 80º F?
Would you rely on this model to make predictions? Why or why not?
Answers
GPT-4o mini
To analyze Marissa's findings, let’s break down each question step-by-step.
### a. Interpret the slope in context.
The least squares regression line Marissa calculated is \( a = -14 + 0.41t \), where \( a \) represents attendance in thousands, and \( t \) is the high temperature in degrees Fahrenheit. The slope of the regression line is \( 0.41 \).
**Interpretation of the slope:**
The slope of \( 0.41 \) means that for each 1°F increase in temperature, the predicted attendance increases by approximately 410 people (since attendance is measured in thousands). Therefore, as the temperature rises, we can expect more people to visit the amusement park.
### b. Expectations for predictions in the 80s.
In the residual plot, if the points follow an upside-down parabola shape, this suggests that the model's predictions deviate in a systematic way depending on temperature.
If temperatures are in the 80s:
- **General observation:** If the residuals are positive in this range (above the x-axis), predictions are too low; if negative (below the x-axis), they are too high.
- Since we have an upside-down parabola, we'd expect predictions to be **too low** for temperatures in the 80s, as this pattern suggests that around this temperature range, the model underestimates attendance.
### c. Actual attendance when the temperature was 95°F.
To find the predicted attendance \( a \) when \( t = 95 \):
\[
a = -14 + 0.41(95)
\]
\[
a = -14 + 38.95
\]
\[
a = 24.95
\]
Thus, the predicted attendance when the temperature was 95°F is approximately 24.95 * 1000 = **24,950 people**.
### d. Upper and lower bounds for attendance at 80°F.
Using the boundary lines provided:
1. For the **upper boundary line**:
\[
a = -7 + 0.41(80)
\]
\[
a = -7 + 32.8 = 25.8
\]
Predicted attendance would be \( 25.8 * 1000 = 25,800 \) people.
2. For the **lower boundary line**:
\[
a = -21 + 0.41(80)
\]
\[
a = -21 + 32.8 = 11.8
\]
Predicted attendance would be \( 11.8 * 1000 = 11,800 \) people.
Therefore, the upper and lower bounds for the predicted number of people attending when the temperature is 80°F are **11,800 to 25,800** people.
### e. Would you rely on this model to make predictions? Why or why not?
**Reliability of the model:**
Whether to rely on the model for predictions depends on several factors:
1. **Residual Plot Analysis:** The upside-down parabola in the residual plot suggests that the model does not adequately describe the relationship between temperature and attendance across the entire range of temperatures. It indicates possible non-linearity in the relationship, which could mean that predictions using the model may not be accurate.
2. **Data Range:** If Marissa's sample size was limited (only 10 Saturdays), and if it didn't capture extreme temperatures or different weather conditions, it may not generalize well to all Saturdays.
3. **Outliers:** The effectiveness of the model could be impacted by outliers or anomalies in the data. The residuals indicate that predictions may be less reliable around certain temperature ranges.
**Conclusion:** Given these points, you might hesitate to rely on the model for critical decision-making about staffing, budgeting, or planning at the amusement park without additional adjustments or consideration of more complex modeling techniques. It's advisable to further analyze the relationship or collect more data for better accuracy.
### a. Interpret the slope in context.
The least squares regression line Marissa calculated is \( a = -14 + 0.41t \), where \( a \) represents attendance in thousands, and \( t \) is the high temperature in degrees Fahrenheit. The slope of the regression line is \( 0.41 \).
**Interpretation of the slope:**
The slope of \( 0.41 \) means that for each 1°F increase in temperature, the predicted attendance increases by approximately 410 people (since attendance is measured in thousands). Therefore, as the temperature rises, we can expect more people to visit the amusement park.
### b. Expectations for predictions in the 80s.
In the residual plot, if the points follow an upside-down parabola shape, this suggests that the model's predictions deviate in a systematic way depending on temperature.
If temperatures are in the 80s:
- **General observation:** If the residuals are positive in this range (above the x-axis), predictions are too low; if negative (below the x-axis), they are too high.
- Since we have an upside-down parabola, we'd expect predictions to be **too low** for temperatures in the 80s, as this pattern suggests that around this temperature range, the model underestimates attendance.
### c. Actual attendance when the temperature was 95°F.
To find the predicted attendance \( a \) when \( t = 95 \):
\[
a = -14 + 0.41(95)
\]
\[
a = -14 + 38.95
\]
\[
a = 24.95
\]
Thus, the predicted attendance when the temperature was 95°F is approximately 24.95 * 1000 = **24,950 people**.
### d. Upper and lower bounds for attendance at 80°F.
Using the boundary lines provided:
1. For the **upper boundary line**:
\[
a = -7 + 0.41(80)
\]
\[
a = -7 + 32.8 = 25.8
\]
Predicted attendance would be \( 25.8 * 1000 = 25,800 \) people.
2. For the **lower boundary line**:
\[
a = -21 + 0.41(80)
\]
\[
a = -21 + 32.8 = 11.8
\]
Predicted attendance would be \( 11.8 * 1000 = 11,800 \) people.
Therefore, the upper and lower bounds for the predicted number of people attending when the temperature is 80°F are **11,800 to 25,800** people.
### e. Would you rely on this model to make predictions? Why or why not?
**Reliability of the model:**
Whether to rely on the model for predictions depends on several factors:
1. **Residual Plot Analysis:** The upside-down parabola in the residual plot suggests that the model does not adequately describe the relationship between temperature and attendance across the entire range of temperatures. It indicates possible non-linearity in the relationship, which could mean that predictions using the model may not be accurate.
2. **Data Range:** If Marissa's sample size was limited (only 10 Saturdays), and if it didn't capture extreme temperatures or different weather conditions, it may not generalize well to all Saturdays.
3. **Outliers:** The effectiveness of the model could be impacted by outliers or anomalies in the data. The residuals indicate that predictions may be less reliable around certain temperature ranges.
**Conclusion:** Given these points, you might hesitate to rely on the model for critical decision-making about staffing, budgeting, or planning at the amusement park without additional adjustments or consideration of more complex modeling techniques. It's advisable to further analyze the relationship or collect more data for better accuracy.