--- title: "Lab 6: Reading the Line" author: "Your Name Here" date: "`r Sys.Date()`" format: docx editor_options: chunk_output_type: console markdown: wrap: 72 --- # Lab 6 — Reading the Line Render this document to Word and submit it to myCourses. (If you prefer to answer in a Word doc directly, that is fine — keep the question numbers.) All of the output you need is printed below. You do not have to run anything, except for the optional bonus at the end. Your readings for this week (ROS Ch. 6, ST 14.1, LSR Ch. 11) answer most of these questions in the authors' own words — that is the fastest way through. Write your answers under each **Your Answer:** line. Answers in complete sentences, please; "0.18" by itself will not earn the points. ------------------------------------------------------------------------ # Part 1 — The study strategies data (14 points) Same study as Lab 5. 120 graduate students were **randomly assigned** to *Rereading* or *Retrieval Practice*. The researchers also **measured** study time, sleep, and test anxiety. Everyone took the same stats quiz (`exam_score`, 0–100). This time the predictor is `study_minutes` — minutes spent studying, which ranged from **40 to 146** minutes (M = 89.79). Here is the model: ``` lm(formula = exam_score ~ study_minutes, data = study) Residuals: Min 1Q Median 3Q Max -23.445 -7.295 -0.092 6.602 26.616 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 43.92291 3.45312 12.720 < 2e-16 *** study_minutes 0.17766 0.03724 4.771 5.31e-06 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 9.436 on 118 degrees of freedom Multiple R-squared: 0.1617, Adjusted R-squared: 0.1546 F-statistic: 22.76 on 1 and 118 DF, p-value: 5.31e-06 ``` ## Q1. Write the fitted regression equation (2 points) Use the variable names, not generic `x` and `y`. Round to two decimals. **Your Answer:** ## Q2. Interpret the slope (3 points) What does the slope mean, in the units of this study? Then: two students differ by **30 minutes** of studying. How many points does the model predict separates them? **Your Answer:** ## Q3. Interpret the intercept (3 points) What does the intercept mean here? Then answer the harder half: is that a meaningful value in this study? Why or why not? (Look again at the range of `study_minutes`.) **Your Answer:** ## Q4. Predict and find a residual (3 points) a) What exam score does the model predict for a student who studied **100 minutes**? Show the arithmetic. b) Student **S062** actually studied 100 minutes and scored **63**. What is that student's residual? c) Did the model **over-predict** or **under-predict** this student's score? How do you know from the sign? **Your Answer:** ## Q5. Connect it back to Lab 5 (2 points) In Lab 5, the correlation between `study_minutes` and `exam_score` was *r*(118) = .40, *t* = 4.77, *p* \< .001. Look at the output above and find **two** numbers that are not a coincidence. Name them and explain the connection. **Your Answer:** ## Q6. Causal or not? (1 point) Does this model let you tell a student that studying 30 more minutes *will raise* their score by that amount? One or two sentences. **Your Answer:** ------------------------------------------------------------------------ # Part 2 — What does a LEGO set cost? (10 points) A different dataset, to show the skill travels. We have **59** current LEGO sets (104 to 10,001 pieces; \$9.99 to \$999.99). The model predicts retail `price` (US dollars) from the number of `pieces`. ``` lm(formula = price ~ pieces, data = lego) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 18.77300 16.14237 1.163 0.25 pieces 0.08252 0.00478 17.277 < 2e-16 *** --- Residual standard error: 86.71 on 57 degrees of freedom Multiple R-squared: 0.8397, Adjusted R-squared: 0.8369 F-statistic: 298.5 on 1 and 57 DF, p-value: < 2.2e-16 ``` ## Q7. Write the fitted equation (2 points) **Your Answer:** ## Q8. Interpret the slope in dollars (2 points) What does the slope mean here? What does the model say **100 additional pieces** are worth? **Your Answer:** ## Q9. Interpret the intercept (2 points) What is the intercept saying, and should you take it seriously? Note its p-value (.25) in your answer. **Your Answer:** ## Q10. Compare the two models (2 points) R² is **.16** for the study model and **.84** for the LEGO model. Explain what R² means in plain language, and what the difference between these two values tells you. Does the bigger R² mean the LEGO model is the better piece of *science*? Why or why not? **Your Answer:** ## Q11. The set the model gets wrong (2 points) The **Eiffel Tower** set has **10,001 pieces** and sells for **\$629.99** — the largest set in the data. a) What price does the model predict for it? b) What is the residual? c) Give one plausible reason the model misses this set by so much. **Your Answer:** ------------------------------------------------------------------------ # Part 3 — Would you trust it? (4 points) Back to the study strategies data. Two more one-predictor models: ``` Model C: exam_score ~ test_anxiety Estimate Std. Error t value Pr(>|t|) (Intercept) 70.00480 3.78190 18.511 < 2e-16 *** test_anxiety -0.33480 0.12130 -2.760 0.0067 ** Multiple R-squared: 0.0606 Model D: exam_score ~ sleep_hours Estimate Std. Error t value Pr(>|t|) (Intercept) 50.92040 6.32630 8.049 7.42e-13 *** sleep_hours 1.33380 0.93200 1.431 0.155 Multiple R-squared: 0.0171 ``` ## Q12. Model C: small but significant (2 points) The anxiety slope is statistically significant (*p* = .007) and R² is .06. Write the one sentence you would put in a results section, and then say what a reader should **not** conclude from it. **Your Answer:** ## Q13. Model D: not significant (2 points) A classmate says: "*p* = .155, so sleep has no effect on exam scores." Rewrite their claim so that it is accurate, and give one reason a real relationship could still exist here even with this result. **Your Answer:** ------------------------------------------------------------------------ # Bonus (+1 point) Pick any **one** number from Part 1 and reproduce it in R. For example, compute the predicted score for a student who studied 100 minutes, or confirm that `sqrt(0.1617)` is the correlation from Lab 5. Paste your code and its output. ```{r} # Your code here (optional) ``` **Your Answer:** ------------------------------------------------------------------------ **Total: 28 points (+1 bonus)** *Remember to render to Word before submitting.*