October 2021 Paper 3 Q2
2. Marc took a random sample of 16 students from a school and for each student recorded
- the number of letters, \(x\), in their last name
- the number of letters, \(y\), in their first name
His results are shown in the scatter diagram on the next page.
Marc suggests that parents with long last names tend to give their children shorter first names.
The results from Marc’s random sample of 16 observations are given in the table below.
| \(x\) | 3 | 6 | 8 | 7 | 5 | 3 | 11 | 3 | 4 | 5 | 4 | 9 | 7 | 10 | 6 | 6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(y\) | 7 | 7 | 4 | 4 | 6 | 8 | 5 | 5 | 8 | 4 | 7 | 4 | 5 | 5 | 6 | 3 |
You should
- state your hypotheses clearly
- use a 5% level of significance
| Scheme | Marks | AO |
|---|---|---|
| Negative | B1 | 1.2 |
| (1) |
Notes
B1 for “negative” Allow “slight” or “weak” etc
Allow a description e.g. “as \(x\) increases \(y\) decreases” or in context e.g. “people with longer last names tend to have shorter first names”
A comment of “negative skew” is B0
Need to see distinct or separate responses for (a) and (b)
| Scheme | Marks | AO |
|---|---|---|
| Marc’s suggestion is compatible because it’s negative correlation | B1 | 2.4 |
| (1) |
Notes
B1 for a comment that suggests data is compatible with the suggestion and a suitable reason such as “there is negative correlation” or a description in \(x\) and \(y\) or in context
or the points lie close to a line with negative gradient
or draw line \(y = x\) and state that more points below the line so supports (or is compatible with) his suggestion
A reason based on just a single point is B0
e.g. “11 letters in last name has only 5 in first name”
| Scheme | Marks | AO |
|---|---|---|
| \((r =)\ -0.54458266\ldots\) awrt \(-\)0.545 | B1 | 1.1b |
| (1) |
Notes
B1 for awrt \(-0.545\)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : \rho = 0 \qquad \mathrm{H}_1 : \rho \lt 0\) | B1 | 2.5 |
| [5% 1-tail cv = ] \((\pm)\ 0.4259\) | M1 | 1.1a |
| (significant result / reject \(\mathrm{H}_0\)) There is evidence of negative correlation between the number of letters in (or length of) a student’s last name and their first name | A1 | 2.2b |
| (3) | ||
| (6 marks) |
Notes
B1 for both hypotheses correct in terms of \(\rho\)
M1 for a critical value compatible with their \(\mathrm{H}_1\):
1-tail: awrt \(\pm\) 0.426 (condone \(\pm\) 0.425) or 2-tail (B0 scored for \(\mathrm{H}_1\)) : awrt \(\pm\) 0.497
If hypotheses are in words and can deduce whether one or two-tail then use their words.
If no hypotheses or their \(\mathrm{H}_1\) is not clearly one or two tail assume one-tail
A1 for compatible signs between cv and \(r\) and a correct conclusion in context mentioning correlation and number of letters or length and name (ft their value from (c))
Do NOT award this A mark if contradictory comments or working seen e.g. “accept \(\mathrm{H}_0\)” or comparison of 0.426 with significance level of 0.05 etc
NB The M1A1 can be scored independently of the hypotheses