OCR S3 (Statistics 3) 2010 January

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 2 } { 5 } & - a \leqslant x < 0
\frac { 2 } { 5 } \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$ Find

1. the value of the constant \(a\),
2. \(\mathrm { E } ( X )\).

2 The amount of tomato juice, \(X \mathrm { ml }\), dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is \(Y \mathrm { ml }\).

1. Find \(\mathrm { P } ( Y < 2000 )\). The random variable \(V\) is defined as \(Y - 4 X\).
2. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
3. What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?

3 It is given that \(X _ { 1 }\) and \(X _ { 2 }\) are independent random variables with \(X _ { 1 } \sim \mathrm {~N} \left( \mu _ { 1 } , 2.47 \right)\) and \(X _ { 2 } \sim \mathrm {~N} \left( \mu _ { 2 } , 4.23 \right)\). Random samples of \(n _ { 1 }\) observations of \(X _ { 1 }\) and \(n _ { 2 }\) observations of \(X _ { 2 }\) are taken. The sample means are denoted by \(\bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\).

1. State the distribution of \(\bar { X } _ { 1 } - \bar { X } _ { 2 }\), giving its parameters. For two particular samples, \(n _ { 1 } = 5 , \Sigma x _ { 1 } = 48.25 , n _ { 2 } = 10\) and \(\Sigma x _ { 2 } = 72.30\).
2. Test at the \(2 \%\) significance level whether \(\mu _ { 1 }\) differs from \(\mu _ { 2 }\). A student stated that because of the Central Limit Theorem the sample means will have normal distributions so it is unnecessary for \(X _ { 1 }\) and \(X _ { 2 }\) to have normal distributions.
3. Comment on the student's statement.

4 The continuous random variable \(V\) has (cumulative) distribution function given by $$\mathrm { F } ( v ) = \begin{cases} 0 & v < 1
1 - \frac { 8 } { ( 1 + v ) ^ { 3 } } & v \geqslant 1 \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { 1 + V }\).

1. Show that the (cumulative) distribution function of \(Y\) is \(8 y ^ { 3 }\), over an interval to be stated, and find the probability density function of \(Y\).
2. Find \(\mathrm { E } \left( \frac { 1 } { Y ^ { 2 } } \right)\).

5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.

1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
2. Test the claim, at the \(10 \%\) significance level.

6 The deterioration of a certain drug over time was investigated as follows. The drug strength was measured in each of a random sample of 8 bottles containing the drug. These were stored for two years and the strengths were then re-measured. The original and final strengths, in suitable units, are shown in the following table.

1. Stating any required assumption, test at the \(5 \%\) significance level whether the mean strength has decreased by more than 0.2 over the two years.
2. Calculate a 95\% confidence interval for the mean reduction in strength over the two years.

7 A chef wished to ascertain her customers' preference for certain vegetables. She asked a random sample of 120 customers for their preferred vegetable from asparagus, broad beans and cauliflower. The responses, classified according to the gender of the customer, are shown in the table.

1. Test, at the \(5 \%\) significance level, whether vegetable preference and gender are independent.
2. Determine whether, at the \(10 \%\) significance level, the vegetables are equally preferred.