2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by
$$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$
Calculate the mean and standard deviation of the marks.
The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.
3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.
Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\).
Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.
4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula
$$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6$$
Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.
\(r\)
1
2
3
4
5
6
\(\mathrm { P } ( X = r )\)
\(k\)
\(11 k\)
Find the mean of \(X\).
A fair six-sided die is rolled three times.
5 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution
$$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
Show that \(k = \frac { 1 } { 50 }\).
Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).