## Code

```
<- Clock_auction |> mutate(nbidders = ifelse(bidders >= 10, "10 or more", "9 or fewer"))
Clock_auction <- Clock_auction |> group_by(nbidders) |>
Stats summarize(mp = mean(price), mage = mean(age))
```

These are two graphs of the data from `Clock_auction`

showing the relationship between the winning price and the number of bidders. (I’ve simplified the number of bidders to two categories.) The age of the clock is a covariate. The large dots show the mean age and mean price of the clocks in those auctions with 10 or more bidders versus 9 or fewer bidders.

```
mod1 <- lm(price ~ nbidders, data=Clock_auction)
model_plot(mod1, x=age, color=nbidders) |>
gf_point(mp ~ mage, color=~nbidders, data=Stats, size=3)
```

**Part A**. In the model without `age`

as a covariate, what is the difference in mean prices for the 10-or-more-bidders group versus the 9-or-fewer-bidders group?

**Part B**. Now the picture when including `age`

as a covariate. Adjusting for `age`

, what is the difference in mean prices for the 10-or-more-bidders group versus the 9-or-fewer-bidders group?

```
mod2 <- lm(price ~ nbidders + age, data=Clock_auction)
model_plot(mod2, x=age, color=nbidders) |>
gf_point(mp ~ mage, color=~nbidders, data=Stats, size=3)
```

**Part C**. Here are confidence intervals for the two models graphed above. Explain what about these coefficients matches the conclusions you got in Parts (A) and (B)?

```
# A tibble: 2 × 4
term .lwr .coef .upr
<chr> <dbl> <dbl> <dbl>
1 (Intercept) 1226. 1420 1614.
2 nbidders9 or fewer -479. -221. 37.1
```

```
# A tibble: 3 × 4
term .lwr .coef .upr
<chr> <dbl> <dbl> <dbl>
1 (Intercept) -467. -56.3 354.
2 nbidders9 or fewer -490. -336. -182.
3 age 7.79 10.6 13.5
```

*This activity was inspired by schematic diagrams in Milo Schield’s *Statistical Literacy: Seeing the story behind the statistics*, 2011, pp. 224-5.*