6 An eight-sided dice has its faces numbered \(1,2 , \ldots , 8\).
- In this part of the question you should assume that the dice is fair.
- State the probability that, when the dice is rolled once, the score is at least 6 .
- Show that the probability that the score is within 2 standard deviations of its mean is 1 .
- A student thinks that the dice may be biased. To investigate this, the student decides to roll the dice 80 times and then carry out a \(\chi ^ { 2 }\) goodness of fit test of a uniform distribution. The spreadsheet below shows the data for the test, where some of the values have been deliberately omitted.
| \multirow[b]{2}{*}{1} | A | B | C | D |
| Score | Observed frequency | Expected frequency | Chi-squared contribution |
| 2 | 1 | 14 | 10 | 1.6 |
| 3 | 2 | 4 | 10 | 3.6 |
| 4 | 3 | 10 | 10 | 0 |
| 5 | 4 | 15 | 10 | |
| 6 | 5 | 6 | 10 | 1.6 |
| 7 | 6 | 11 | 10 | 0.1 |
| 8 | 7 | 7 | 10 | 0.9 |
| 9 | 8 | | 10 | 0.9 |
- Explain why all of the expected frequencies are equal to 10 .
- Determine the missing values in each of the following cells.
- D5
(iii) In this question you must show detailed reasoning.
Carry out the \(\chi ^ { 2 }\) test at the \(5 \%\) significance level.