16
256
2 The random variable \(T\) has an exponential distribution with mean 2
Find \(\mathrm { P } ( T \leq 1.4 )\)
Circle your answer.
\(\mathrm { e } ^ { - 2.8 }\)
\(\mathrm { e } ^ { - 0.7 }\)
\(1 - e ^ { - 0.7 }\)
\(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function
$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r }
0 & y < 2 \\
- \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\
1 & y \geq 5
\end{array} \right.$$
Find the median of \(Y\)
Circle your answer.
2
\(\frac { 10 - 3 \sqrt { 2 } } { 2 }\)
\(\frac { 7 } { 2 }\)
\(\frac { 10 + 3 \sqrt { 2 } } { 2 }\)
Turn over for the next question
4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20
Sandra records 162 volcanic eruptions during a period of one week.
Sandra claims that there has been an increase in the mean number of volcanic eruptions per week.
Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$
Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\)
6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old.
Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are
$$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$
6
- State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim.
6
- Investigate Imran's claim, using the 10\% level of significance.