A stealth alarm system works by sending noise signals: A...

A stealth alarm system works by sending noise signals: A "situation normal" signal is sent by transmitting voltages that are Gaussian iid random variables with mean zero and variance 4; an "alarm" signal is sent by transmitting iid Gaussian voltages with mean zero and variance less than 4. (a) Find a $1 \%$ level hypothesis test to determine whether the situation is normal (null hypothesis) based on the calculation of the sample variance from $n$ voltage samples. (b) Find the power of the hypothesis test for $n=8,64,256$ as the variance of the alarm signal is varied.

A stealth alarm system works by sending noise signals: A "situation normal" signal is sent by transmitting voltages that are Gaussian iid random variables with mean zero and variance 4; an "alarm" signal is sent by transmitting iid Gaussian voltages with mean zero and variance less than 4.
(a) Find a $1 \%$ level hypothesis test to determine whether the situation is normal (null hypothesis) based on the calculation of the sample variance from $n$ voltage samples.
(b) Find the power of the hypothesis test for $n=8,64,256$ as the variance of the alarm signal is varied.

Key Concepts

Hypothesis Testing

This concept involves formulating a null hypothesis (in this case, that the voltage samples come from a Gaussian distribution with variance 4) and an alternative hypothesis (here, that the variance is less than 4). A statistical test is then designed to decide whether to reject the null hypothesis based on sample data, while controlling the probability of falsely rejecting it (Type I error).

Chi-Square Distribution

When data are drawn from a normal population, the statistic (n–1)S²/?², where S² is the sample variance and ?² is the true variance, follows a chi-square distribution with n–1 degrees of freedom. This property is exploited to test hypotheses about the variance by comparing the computed statistic to quantiles of the chi-square distribution.

Critical Region

The critical region is the set of test statistic values that lead to rejection of the null hypothesis. For a one?sided test at a 1% level, the critical region is determined by the lower tail of the chi-square distribution under the null hypothesis. Specifically, one rejects the null if the test statistic computed from the sample falls below the value corresponding to the 1% quantile of the chi-square distribution.

Power Analysis

The power of a test is the probability that the test will correctly reject the null hypothesis when the alternative hypothesis is true. In this context, it involves calculating the probability that the sample variance falls into the rejection region when the true variance is less than 4. The power depends on the true signal variance and the sample size, and it is evaluated by integrating over the chi-square distribution corresponding to the alternative variance.

Sample Variance Properties

The sample variance is a fundamental statistic that summarizes the variability in the data. Its distribution, particularly when the data are normally distributed, can be derived and is used to conduct inference about the population variance. The relation of sample variance to the chi-square distribution underpins statistical tests involving variance and is essential for constructing confidence intervals and hypothesis tests.

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