NYJC Hypothesis Testing Tutorial Q8

Experimental data concerning a variable \(X\), which measures the reliability of a certain electronic component, is as follows: \(\sum{x}=1164.2\), \(\sum{{{x}^{2}}}=13911.6\), \(n=100\). Calculate the unbiased estimates of the population mean and variance from these figures. Explain whether, on the evidence of this sample, you would reject the hypothesis that the mean value of \(X\) is \(12\) at \(5\%\) level of significance.

A second random sample of \(50\) electronic components of this type is to be tested and the measurement, \(y\), are summarized by

\(\sum{y}=582.1\), \(\sum{{{y}^{2}}}=7055.9\).

Combining the two samples into a single sample, carry out a test, at the \(5\%\) significance level, of the same null and alternative hypotheses.

Figures collected over a long period have established that the mean and standard deviation of \(X\) are \(12\) and \(2\) respectively. After a change in the manufacturing process, it is expected that the mean will have increased, but it may be assumed that the standard deviation remains equal to \(2\). A sample of \(n\) values of \(X\) is taken, with sample mean \(m\); if \(m\) is greater than some critical value, it will be accepted that the mean has in fact increased, but if \(m\) is less than the critical value, the increase is not established. State carefully appropriate null and alternative hypotheses for this situation, and find, in terms of \(n\), the critical value for a \(1\%\) significance level.