## E.10 Answers to Lecture 10 tutorial

**Return to index of tutor information.**

### E.10.1 Answers to Sect. 10.1

The correlations:

**Plot 1**: \(0.94\) (correlation**A**);

**Plot 2**: \(-0.95\) (correlation**D**);

**Plot 3**: \(0.12\) (correlation**B**):

**Plot 6**: \(0.75\) (correlation**C**).Correlation is inappropriate for

**Plot 4**and**Plot 5**.Examples of the direction in

**Plot 1**: Any two variables moderately positively correlated, such as height and weight, distance lived from university and travel time, etc.Examples of direction in

**Plot 2**: Any two variables moderately negatively correlated, such as hours of weekly exercise and body weight, number of SCI110 tutorial missed and final mark, etc.These are:

**Plot 1**: \(88.4\)%;

**Plot 2**: \(90.3\)%;

**Plot 3**: \(1.4\)%;

**Plot 6**: \(56.3\)%.The answers:

- No answer. But a line is fine for
**Plot 1**,**Plot 2**,**Plot 3**(but very weak!) and**Plot 6**, but not for**Plot 4**and**Plot 6**. - My
**very rough**slope estimates are:

**Plot 1**: \((50-10)/10 \approx 4\);

**Plot 2**: \((20 - 50)/15 \approx -2\);

**Plot 3**: \(0\);

**Plot 6**: \((55 - 35)/5 = 4\). - My
**very rough**intercept estimates are:

**Plot 1**: \(8\);

**Plot 2**: \(40\);

**Plot 3**: \(32\);

**Plot 6**: \(10\). - My very rough estimates are:

**Plot 1**; \(\hat{y} = 8 + 4x\);

**Plot 2**: \(\hat{y} = 40 -2x\);

**Plot 3**: \(\hat{y} = 32\);

**Plot 6**: \(\hat{y} = 10 + 4x\).

- No answer. But a line is fine for

### E.10.2 Answers to Sect. 10.2

If you count the dots on the scatterplot, you won’t find \(n=38\) dots, because of overplotting. For example, there are two Corolla’s from 2006 selling for $9500.

If you have time, you can ask students about this, and even ask for suggestions to improve this (such as jittering).Approximately linear, negative, reasonably strong.

Condition, extras (air con, etc.), sedan/hatch, colour, when rego due, new/old tyres, location, etc.

No answer.

Looks to be expensive, as $15,000 would be above the line (at least for the line I’d draw).

Probably $3900.

My guess is about \(b_0\approx 17\) or $17,000. This would mean the average price of a 2014 second-hand Corolla can be expected to be about $17,000.

**Note**: Since all the cars in the sample are second-hand, technically the results only generalise to second-hand cars, so that \(b_0\) really is the estimated price of a second-hand 2014 Corolla.\(b_1 = (1 - 17)/(16-0)\approx -1\). That is, the price reduces by about $1000 each year older the Corolla gets.

Using the above, we have \(\hat{y} = 17 - x\) approximately. Guessing the regression line won’t, of course, produce this level of precision, so anything close-ish to this is fine.

\(\hat{y} = 16.54 - 0.96x\) (jamovi) or \(\hat{y} = 16.536 - 0.963x\) (SPSS), where \(y\) is the price in thousands of dollars, and \(x\) is the age in years.

\(r = -0.929\), and so \(R^2 = (-0.929)^2 = 0.863\), or about 86%, so about 86% of the variation in prices can be explained by age alone. The rest can be explained by the car’s condition, odometer reading, accessories, service history, etc.

jamovi: \(\hat{y} = 16.54 - (0.96\times 20) = -2.66\), or -$2660.

SPSS: \(\hat{y} = 16.536 - (0.963\times 20) = -2.72\), or -$2720.

This is clearly silly, as we are extrapolating.jamovi: \(\hat{y} = 16.54 - (0.96\times 6) = 10.78\), or $10,780.

SPSS: \(\hat{y} = 16.536 - (0.963\times 6) = 10.76\), or $10,760.

So the price seems a bit steep, unless it is highly specified and in great condition.Hardly need a test… \(H_0\): \(\beta_1=0\) vs \(H_1\): \(\beta_1<0\). From output, \(t=-15.059\), and \(P=0.000/2=0.000\): very strong evidence that older cars fetch lower second-hand prices, on average.

\(-0.963\pm(2\times 0.064)\), or \(-0.963\pm 0.128\), or from \(-1.091\) to \(-0.835\).

Looks the same really, just reflected left-to-right.

*Size*of \(r\) won’t change, sign from negative to positive; i.e. \(r=0.93\).- Value of \(R^2\) will be the same.
- Slope will be the same except sign will change (in both cases, the values on the horizontal axis are one year apart).
- Intercept will change a lot… it is the predicted value of the price if the line is extended to year 0 (which is, of course, meaningless).