SPS SPS SM Statistics (SPS SM Statistics) 2021 May

1. Three Bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.

\begin{displayquote} Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only \end{displayquote} Sasha selects at random one marble from \(\operatorname { Bag } A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from Bag \(C\).

1. Draw a tree diagram to represent this information.
2. Find the probability that Sasha selects 3 green marbles.
3. Find the probability that Sasha selects at least 1 marble of each colour.
4. Given that Sasha selects a red marble, find the probability that he selects it from Bag \(B\).

2.

1. The variable \(X\) has the distribution \(\mathrm { N } ( 20,9 )\).
   (a) Find \(\mathrm { P } ( X > 25 )\).
   (b) Given that \(\mathrm { P } ( X > a ) = 0.2\), find \(a\).
   (c) Find \(b\) such that \(\mathrm { P } ( 20 - b < X < 20 + b ) = 0.5\).
2. The variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \frac { \mu ^ { 2 } } { 9 } \right)\). Find \(\mathrm { P } ( Y > 1.5 \mu )\).

3. Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.

1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
2. Use the table below to carry out the test at the \(5 \%\) significance level. Critical values of Pearson's product-moment correlation coefficient.

   \cline{2-5}	1-tail test	\(5 \%\)	\(2.5 \%\)	\(1 \%\)
   2-tail test	\(10 \%\)	\(5 \%\)	\(2.5 \%\)	\(1 \%\)
   38	0.2709	0.3202	0.3760	0.4128
   39	0.2673	0.3160	0.3712	0.4076
   40	0.2638	0.3120	0.3665	0.4026
   41	0.2605	0.3081	0.3621	0.3978

1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml

Given that \(15 \%\) of bottles contain less than 24.63 ml

1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\)

5. Two events C and D are such that \(P ( C \mid D ) = 3 \times P ( C )\) where \(P ( C ) \neq 0\).

1. Explain whether or not events C and D could be independent events. Given also that $$P ( C \cap D ) = \frac { 1 } { 2 } \times P ( C ) \text { and } P \left( C ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }$$
2. find \(P ( C )\), showing your working clearly.

6. The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 .
This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the \(5 \%\) significance level, whether the mean level of pollutant has changed.