5. A researcher believes that parents with a short family name tended to give their children a long first name. A random sample of 10 children was selected and the number of letters in their family name, \(x\), and the number of letters in their first name, \(y\), were recorded.
The data are summarised as:
$$\sum x = 60 , \quad \sum y = 61 , \quad \sum y ^ { 2 } = 393 , \quad \sum x y = 382 , \quad \mathrm {~S} _ { x x } = 28$$
- Find \(\mathrm { S } _ { y y }\) and \(\mathrm { S } _ { x y }\)
- Calculate the product moment correlation coefficient, \(r\), between \(x\) and \(y\).
- State, giving a reason, whether or not these data support the researcher's belief.
The researcher decides to add a child with family name "Turner" to the sample.
- Using the definition \(\mathrm { S } _ { x x } = \sum ( x - \bar { x } ) ^ { 2 }\), state the new value of \(\mathrm { S } _ { x x }\) giving a reason for your answer.
Given that the addition of the child with family name "Turner" to the sample leads to an increase in \(\mathrm { S } _ { y y }\)
- use the definition \(\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )\) to determine whether or not the value of \(r\) will increase, decrease or stay the same. Give a reason for your answer.