8 The continuous random variable \(Y\) has a distribution with mean \(\mu\) and variance 20. A random sample of 50 observations of \(Y\) is selected and these observations are summarised in the following grouped frequency table.
| Values | \(y < 20\) | \(20 \leqslant y < 25\) | \(25 \leqslant y < 30\) | \(y \geqslant 30\) |
| Frequency | 3 | 27 | 12 | 8 |
- Assuming that \(Y \sim \mathrm {~N} ( 25,20 )\), show that the expected frequency for the interval \(20 \leqslant y < 25\) is 18.41, correct to 2 decimal places, and obtain the remaining expected frequencies.
- Test, at the \(5 \%\) significance level, whether the distribution \(\mathrm { N } ( 25,20 )\) fits the data.
- Given that the sample mean is 24.91 , find a \(98 \%\) confidence interval for \(\mu\).
- Does the outcome of the test in part (ii) affect the validity of the confidence interval found in part (iii)? Justify your answer.