7 In 1761, James Short took measurements of the parallax of the sun based on the transit of Venus. The mean and standard deviation of a random sample of 50 of these measurements are 8.592 and 0.7534 respectively, in suitable units.

1. Show that if \(X \sim \mathrm {~N} \left( 8.592,0.7534 ^ { 2 } \right)\), then $$\mathrm { P } ( X \leqslant 8.084 ) = \mathrm { P } ( 8.084 < X \leqslant 8.592 ) = \mathrm { P } ( 8.592 < X \leqslant 9.100 ) = \mathrm { P } ( X > 9.100 ) = 0.25 \text {. }$$ The following table summarises the 50 measurements using these intervals.

   Measurement \(( x )\)	\(x \leqslant 8.084\)	\(8.084 < x \leqslant 8.592\)	\(8.592 < x \leqslant 9.100\)	\(x > 9.100\)
   Frequency	8	22	11	9

2. Carry out a test, at the \(\frac { 1 } { 2 } \%\) significance level, of whether a normal distribution fits the data.
3. Obtain a 99\% confidence interval for the mean of all similar parallax measurements.