Measurement error modeling

1. Albert uses scales in his kitchen to weigh some fruit.

The random variable \(D\) represents, in grams, the weight of the fruit given by the scales minus the true weight of the fruit. The random variable \(D\) is uniformly distributed over the interval \([ - 2.5,2.5 ]\)

1. Specify the probability density function of \(D\)
2. Find the standard deviation of \(D\) Albert weighs a banana on the scales.
3. Write down the probability that the weight given by the scales equals the true weight of the banana.
4. Find the probability that the weight given by the scales is within 1 gram of the banana's true weight. Albert weighs 10 bananas on the scales, one at a time.
5. Find the probability that the weight given by the scales is within 1 gram of the true weight for at least 6 of the bananas.

2 The error, in minutes, made by Paul in estimating the time that he takes to complete a college assignment may be modelled by the random variable \(T\) with probability density function $$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 30 } & - 5 \leqslant t \leqslant 25
0 & \text { otherwise } \end{array} \right.$$

1. Find:
   1. \(\mathrm { E } ( T )\);
      (1 mark)
   2. \(\quad \operatorname { Var } ( T )\).
2. Calculate the probability that Paul will make an error of magnitude at least 2 minutes when estimating the time that he takes to complete a given assignment.

1 Josephine accurately measures the widths of A4 sheets of paper and then rounds the widths to the nearest 0.1 cm . The rounding error, \(X\) centimetres, follows a rectangular distribution. A randomly selected A4 sheet of paper is measured to be 21.1 cm in width.

1. Write down the limits between which the true width of this A4 sheet of paper lies.
   (1 mark)
2. Write down the value of \(\mathrm { E } ( X )\) and determine the exact value of the standard deviation of \(X\).
3. Calculate \(\mathrm { P } ( - 0.01 \leqslant X \leqslant 0.03 )\).

4 The error, \(X\) millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \begin{cases} k & - 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$

1. State the value of \(k\).
2. Write down the value of \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { P } ( X > 0 )\).
4. The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres.

4 A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2 . The error, \(X ^ { \circ } \mathrm { C }\), resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \left\{ \begin{array} { l c } k & 0 \leqslant x \leqslant 0.1
0 & \text { otherwise } \end{array} \right.$$

1. State the value of \(k\).
2. Find the probability that the error resulting from this rounding down is greater than \(0.03 ^ { \circ } \mathrm { C }\).
3. 1. State the value for \(\mathrm { E } ( X )\).
   2. Use integration to find the value for \(\mathrm { E } \left( X ^ { 2 } \right)\).
   3. Hence find the value for the standard deviation of \(X\).
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4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.

1. Given that the rounding error, \(X\) metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05
   0 & \text { otherwise } \end{array} \right.$$
2. Calculate \(\mathrm { P } ( - 0.01 < X < 0.02 )\).
3. Find the mean and the standard deviation of \(X\).