6 A student believes that if you ask people to choose an integer between 1 and 10, not all integers are equally likely to be chosen. The student asks a random sample of 100 people to choose an integer between 1 and 10 inclusive. The observed frequencies \(O\), together with the values of \(\frac { ( O - E ) ^ { 2 } } { E }\) where \(E\) is the corresponding expected frequency, are shown in the table.
| Integer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| O | 7 | 8 | 20 | 8 | 7 | 6 | 19 | 7 | 8 | 10 |
| \(\frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } }\) | 0.9 | 0.4 | 10.0 | 0.4 | 0.9 | 1.6 | 8.1 | 0.9 | 0.4 | 0 |
- Show how the value of 8.1 for integer 7 is obtained.
- Show that there is evidence at the \(1 \%\) significance level that the student’s belief is correct.
The student wishes to suggest an alternative model for the probabilities associated with each integer. In this model, two of the integers have the same probability \(p _ { 1 }\) of being chosen and the other eight integers each have probability \(p _ { 2 }\) of being chosen.
- Suggest which two integers should have probability \(p _ { 1 }\) and suggest a possible value of \(p _ { 1 }\).