https://file.aiquickdraw.com/mathbot/file/1728523887040p2x1ehn6.csv

https://file.aiquickdraw.com/mathbot/file/1728523887041ub41hflq.jpg

I see you've uploaded both a CSV and an image file. The CSV file likely contains the data needed for analysis, while the image provides context for the problem at hand.

Let's walk through the steps based on the instructions in the image:

1. **Task Objective**: Perform linear regression to predict the final unit mark (`unitMark`) based on the number of polling quizzes answered (`numQuizzes`).

2. **Extract Key Information**:
   - You'll need to import the data from the CSV file to analyze it using Excel or any statistical tool.
   - The goal is to calculate:
     - Estimated slope for the linear regression equation.
     - Test statistic and p-value.
     - Percentage of variance explained by the model (i.e., R-squared).
     - Decision of the test (whether the regression is statistically significant).

### Let me first open and analyze the CSV file you've provided. I'll begin with extracting the data and performing the linear regression analysis.The dataset contains the following columns:

- **SubID**: Subject ID (identifying each student).
- **liveStream**: Whether the student mostly watched lectures via live stream ("True") or attended in-person ("False").
- **numQuizzes**: The number of polling quizzes the student answered.
- **unitMark**: The final unit mark the student achieved.

To address the research question, we will focus on predicting **unitMark** using **numQuizzes** via linear regression.

Let's now perform the linear regression and compute the required values: slope, test statistic, p-value, and the percentage of variance explained by the model (R-squared).Here are the results of the linear regression analysis:

1. **Estimated slope**: \(0.5789\) (to 4 decimal places) – this means that for each additional quiz answered, the final unit mark increases by approximately 0.579 units.
2. **Test statistic (t-value)**: \(9.27\) (to 2 decimal places).
3. **p-value**: \(1.205 \times 10^{-15}\) – this extremely small p-value indicates strong evidence that the number of quizzes is a significant predictor of the final unit mark.
4. **Unexplained variance**: \(57.43\%\) – this percentage of the variation in the final unit mark cannot be explained by the model (i.e., 42.57% is explained by the model).

### Decision of the test:
Since the p-value is much smaller than any common significance level (e.g., 0.05), we reject the null hypothesis. Therefore, **there is a statistically significant relationship between the number of quizzes answered and the final unit mark**.

Would you like more details or to explore any specific part of the analysis?

Here are some follow-up questions you might be interested in:
1. How can we interpret the intercept in this regression model?
2. What would happen if we included the `liveStream` variable in the model?
3. Can we visualize the regression line on a scatter plot of `numQuizzes` vs. `unitMark`?
4. What are the assumptions behind linear regression, and how can we test them in this case?
5. Could we calculate confidence intervals for the slope?

**Tip**: When analyzing regression models, always check the assumptions like linearity, independence, and homoscedasticity to ensure the validity of the results.

Can linear regression be used to predict the final unit mark from the number of polling quizzes answered?

Explore the use of linear regression to predict final unit marks based on the number of polling quizzes answered. This analysis covers estimated slope, test statistics, p-value, and R-squared to assess the model's significance and unexplained variance. Ideal for undergraduate statistics studies.

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