Determine a normal mean and standard deviation from two percentile conditions.

The weights of players in a sports league are normally distributed with a mean of \(76.6\) kg, (correct to three significant figures). It is known that \(80 \%\) of the players have weights between \(68\) kg and \(82\) kg. The probability that a player weighs less than \(68\) kg is \(0.05\).

1. Find the probability that a player weighs more than \(82\) kg.

   [2]

2. 1. Write down the standardized value, \(z\), for \(68\) kg.

      [4]

   2. Hence, find the standard deviation of weights. To take part in a tournament, a player’s weight must be within \(1.5\) standard deviations of the mean.

3. 1. Find the set of all possible weights of players that take part in the tournament.

      [5]

   2. A player is selected at random. Find the probability that the player takes part in the tournament. Of the players in the league, \(25 \%\) are women. Of the women, \(70 \%\) take part in the tournament.

4. Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman.[4]