1 A summary of 24 observations of \(x\) gave the following information:
$$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$
The mean of these values of \(x\) is 8.95 .
Find the value of the constant \(a\).
Find the standard deviation of these values of \(x\).
2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).
Find \(p\).
Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
4
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The random variable \(X\) has a normal distribution with mean 4.5. It is given that \(\mathrm { P } ( X > 5.5 ) = 0.0465\) (see diagram).
Find the standard deviation of \(X\).
Find the probability that a random observation of \(X\) lies between 3.8 and 4.8.
5 The arrival times of 204 trains were noted and the number of minutes, \(t\), that each train was late was recorded. The results are summarised in the table.
Number of minutes late \(( t )\)
\(- 2 \leqslant t < 0\)
\(0 \leqslant t < 2\)
\(2 \leqslant t < 4\)
\(4 \leqslant t < 6\)
\(6 \leqslant t < 10\)
Number of trains
43
51
69
22
19
Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.
6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.
Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .
On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
On another day she performs the routine \(n\) times. Find the smallest value of \(n\) for which the expected number of correct performances is at least 8 .
7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).
Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
Find the probability that the paper clip taken from box \(B\) is red.
Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).