Chi-squared test then t-test sequential

4. The length \(X \mathrm {~mm}\) of a spring made by a machine is normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 20 springs is selected and their lengths measured in mm . Using this sample the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) are $$\bar { x } = 100.6 , \quad s ^ { 2 } = 1.5 .$$ Stating your hypotheses clearly test, at the \(10 \%\) level of significance,

1. whether or not the variance of the lengths of springs is different from 0.9 ,
2. whether or not the mean length of the springs is greater than 100 mm .

1. A machine produces components whose lengths are normally distributed with mean 102.3 mm and standard deviation 2.8 mm . After the machine had been serviced, a random sample of 20 components were tested to see if the mean and standard deviation had changed. The lengths, \(x \mathrm {~mm}\), of each of these 20 components are summarised as

$$\sum x = 2072 \quad \sum x ^ { 2 } = 214856$$

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a change in standard deviation.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the mean length of the components has changed from the original value of 102.3 mm using
   1. a normal distribution,
   2. a \(t\) distribution.
3. Comment on the mean length of components produced after the service in the light of the tests from part (a) and part (b). Give a reason for your answer.

6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a \(95 \%\) confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the \(t\) distribution to obtain a \(95 \%\) confidence interval of \(( 1.193,1.367 )\) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

1. Test, at the \(10 \%\) level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.

The length \(X\) mm of a spring made by a machine is normally distributed N(\(\mu, \sigma^2\)). A random sample of 20 springs is selected and their lengths measured in mm. Using this sample the unbiased estimates of \(\mu\) and \(\sigma^2\) are \(\bar{x} = 100.6\), \(s^2 = 1.5\). Stating your hypotheses clearly test, at the 10\% level of significance,

1. whether or not the variance of the lengths of springs is different from 0.9, [6]
2. whether or not the mean length of the springs is greater than 100 mm. [6]