Edexcel S2 (Statistics 2) 2023 June

1. In a large population \(40 \%\) of adults use online banking.

A random sample of 50 adults is taken.
The random variable \(X\) represents the number of adults in the sample that use online banking.

1. Find
   1. \(\mathrm { P } ( X = 26 )\)
   2. \(\mathrm { P } ( X \geqslant 26 )\)
   3. the smallest value of \(k\) such that \(\mathrm { P } ( X \leqslant k ) > 0.4\) A random sample of 600 adults is taken.
2. 1. Find, using a normal approximation, the probability that no more than 222 of these 600 adults use online banking.
   2. Explain why a normal approximation is suitable in part (b)(i)

1. (a) State one characteristic of a population that would make a census a practical alternative to sampling.

A leisure centre has 2500 members.
It asks a sample of 300 members for their opinions on the fees it charges for using the centre. For the sample,
(b) (i) identify a suitable sampling frame,
(ii) identify a sampling unit. The leisure centre has the following pieces of information.
\(A\) is the list of the different types of membership that can be paid for by members.
\(B\) is the mean of the membership fees paid by all 2500 members.
\(C\) is the number in the sample of 300 members who are satisfied with the fees they pay.
(c) State the piece of information that is a statistic. Give a reason for your answer.

1. The continuous random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 48 } \left( x ^ { 2 } - 8 x + c \right) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Show that \(c = 31\)
2. Find \(\mathrm { P } ( 2 < X < 3 )\)
3. State whether the lower quartile of \(X\) is less than 3, equal to 3 or greater than 3 Give a reason for your answer. Kei does the following to work out the mode of \(X\) $$\begin{aligned} f ^ { \prime } ( x ) & = \frac { 1 } { 48 } ( 2 x - 8 )
   0 & = \frac { 1 } { 48 } ( 2 x - 8 )
   x & = 4 \end{aligned}$$ Hence the mode of \(X\) is 4 Kei's answer for the mode is incorrect.
4. Explain why Kei's method does not give the correct value for the mode.
5. Find the mode of \(X\) Give a reason for your answer.

1. (a) Given \(n\) is large, state a condition for which the binomial distribution \(\mathrm { B } ( n , p )\) can be reasonably approximated by a Poisson distribution.

A manufacturer produces candles. Those candles that pass a quality inspection are suitable for sale. It is known that 2\% of the candles produced by the manufacturer are not suitable for sale. A random sample of 125 candles produced by the manufacturer is taken.
(b) Use a suitable approximation to find the probability that no more than 6 of the candles are not suitable for sale. The manufacturer also produces candle holders.
Charlie believes that 5\% of candle holders produced by the factory have minor defects.
The manufacturer claims that the true proportion is less than \(5 \%\)
To test the manufacturer's claim, a random sample of 30 candle holders is taken and none of them are found to contain minor defects.
(c) (i) Carry out a test of the manufacturer's claim using a \(5 \%\) level of significance. You should state your hypotheses clearly.
(ii) Give a reason why this is not an appropriate test. Ashley suggests changing the sample size to 50
(d) Comment on whether or not this change would make the test appropriate. Give a reason for your answer.

1. A continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c r } 0 & y < 3
\frac { 1 } { 16 } \left( y ^ { 2 } - 6 y + a \right) & 3 \leqslant y \leqslant 5
\frac { 1 } { 12 } ( y + b ) & 5 < y \leqslant 9
\frac { 1 } { 12 } \left( 100 y - 5 y ^ { 2 } + c \right) & 9 < y \leqslant 10
1 & y > 10 \end{array} \right.$$ where \(a\), \(b\) and \(c\) are constants.

1. Find the value of \(a\) and the value of \(c\)
2. Find the value of \(b\)
3. Find \(\mathrm { P } ( 6 < Y \leqslant 9 )\) Show your working clearly.
4. Specify the probability density function, f(y), for \(5 < y \leqslant 9\) Using the information $$\int _ { 3 } ^ { 5 } ( 6 y - 5 ) f ( y ) d y + \int _ { 9 } ^ { 10 } ( 6 y - 5 ) f ( y ) d y = 26.5$$
5. find \(\mathrm { E } ( 6 Y - 5 )\) You should make your method clear.

1. Akia selects at random a value from the continuous random variable \(W\), which is uniformly distributed over the interval \([ a , b ]\)

The probability that Akia selects a value greater than 17 is \(\frac { 1 } { 5 }\) The probability that Akia selects a value less than \(k\) is \(\frac { 53 } { 60 }\)

1. Find the probability that Akia selects a value between 17 and \(k\) It is known that \(\operatorname { Var } ( W ) = 75\)
2. 1. Find the value of \(a\) and the value of \(b\)
   2. Find the value of \(k\)
3. Find \(\mathrm { P } ( - 5 < W < 5 )\)
4. Find \(\mathrm { E } \left( W ^ { 2 } \right)\)

1. A bakery sells muffins individually at an average rate of 8 muffins per hour.
   1. Find the probability that, in a randomly selected one-hour period, the bakery sells at least 4 but not more than 8 muffins.
   A sample of 5 non-overlapping half-hour periods is selected at random.
2. Find the probability that the bakery sells fewer than 3 muffins in exactly 2 of these periods. Given that 4 muffins were sold in a one-hour period,
3. find the probability that more muffins were sold in the first 15 minutes than in the last 45 minutes.