Chapter 23

The resistances of 50 resistors are measured and the results recorded as follows: $$ \begin{array}{cc} \hline \text { Resistance }(\Omega) & \text { Frequency } \\ \hline 5.0 & 17 \\ 5.5 & 12 \\ 6.0 & 10 \\ 6.5 & 6 \\ 7.0 & 5 \\ \hline \end{array} $$ Calculate the standard deviation of the measurements.

Components are made by machines A, B and C. Machine A makes \(35 \%\) of the components, machine B makes \(25 \%\) and machine C makes the rest. Two components are picked at random. Calculate the probability that (a) both are made by machine \(\mathrm{C}\) (b) one is made by machine A and one is made by machine B (c) exactly one is made by machine \(\mathrm{A}\) (d) at least one is made by machine B (e) both are made by the same machine.

A machine needs all five of its micro-chips to be functional in order to work correctly. The probability that a micro-chip works is \(0.99\). Calculate the probability that the machine works.

Precision components are made by machines A, B and C. Machines A and C each make \(30 \%\) of the components with machine \(\mathrm{B}\) making the rest. The probability that a component is acceptable is \(0.91\) when made by machine A, \(0.95\) when made by machine B and \(0.88\) when made by machine \(\mathrm{C}\). (a) Calculate the probability that a component selected at random is acceptable. (b) A batch of 2000 components is examined. Calculate the number of components you expect are not acceptable.

Classify the following variables as discrete or continuous: (a) the number of times a machine breaks down in 12 months (b) the time between breakdowns of a machine (c) the capacitance of a capacitor (d) the amount of money in your pocket (e) the number of hairs on your head.

The standard deviation of the values \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) is \(\sigma .\) Calculate the standard deviation of the values \(k x_{1}, k x_{2}, k x_{3}, \ldots, k x_{n}\) where \(k\) is a constant.

Table 1 shows the results of measuring the petrol consumption of a car over 90 trials. $$ \begin{array}{lc} \hline \text { Miles per gallon } & \text { Frequency } \\ \hline 42 & 17 \\ 43 & 18 \\ 44 & 12 \\ 45 & 20 \\ 46 & 23 \\ \hline \end{array} $$ (a) Calculate the mean consumption. (b) Calculate the standard deviation.

The diameters of ball bearings produced in a factory follow a normal distribution with mean \(6 \mathrm{~mm}\) and standard deviation \(0.04 \mathrm{~mm}\). Calculate the probability that a diameter is (a) more than \(6.05 \mathrm{~mm}\), (b) less than \(5.96 \mathrm{~mm}\), (c) between \(5.98\) and \(6.01 \mathrm{~mm}\).

A machine makes resistors of which \(96 \%\) are acceptable and \(4 \%\) are unacceptable. Three resistors are picked at random. Calculate the probability that (a) all are acceptable (b) all are unacceptable (c) at least one is unacceptable.

The probability that a bearing meets a specification is \(0.92\). Six bearings are picked at random. Calculate the probability that (a) all six meet the specification (b) more than four meet the specification (c) one or none meets the specification (d) exactly four meet the specification.