SPS SPS FM Statistics (SPS FM Statistics) 2021 May

1. The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the \(1 \%\) level whether there has been a change in the mean crop yield.
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2. The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
a) Find the probability that the weight of a randomly selected peeled banana is at most 150 g . Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
b) Find the probability that the total weight of a smoothie is less than 700 g .
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3. A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\). The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method (continuity correction is not required), which should be justified, to find \(P ( \bar { X } \leq 6.10 )\).
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4. A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726 . Test at the \(5 \%\) significance level whether there is a difference in the ranks of the students taught by the two professors.
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5. The continuous random variable \(T\) has cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 ,
1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}$$

1. Find the cumulative distribution function of \(2 T\).
2. Show that, for constant \(k , \quad \mathrm { E } \left( \mathrm { e } ^ { k T } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
3. \(T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\).
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