Question

1. In lecture 23, we learnt an objective way of testing The goal of this quiz is to derive the testing for the less-than type alternative hypothesis o s. (a) Imitate the proof in Lecture 23 to construct a test at 5% level of significance starting from a general test statistic θ for testing the hypothesis in (1). You can have a similar asymptotic normality assumption. Be careful in getting the Z-cutoff to find the appropriate threshold for e rejection region. (b) The output voltage for an electric circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean 128.6 and standard deviation 2.1 Test the hypothesis that the average output voltage is 130 against the alternative that it is less than 130. Use a test with level .05.

Answer 1

a)

We reject the null if

$P(\frac{\overline{X}-\theta }{\frac{\sigma }{\sqrt{n}}} < z_{\alpha }) = \alpha$

hence

${\overline{X} < \theta } &plus; z_{\alpha }\frac{\sigma }{\sqrt{n}}$

for 5 % level

z = -1.645

hence reject if

${\overline{X} < \theta } -1.645 \frac{\sigma }{\sqrt{n}}$

b)

critical value = 130 -1.645 *2.1/sqrt(40) = 129.4538

reject if Xbar < 129.4538

here Xbar = 128.6

since 128.6 < 129.4538

we reject the null hypothesis