A manufacturer is interested in the output voltage of a...

A manufacturer is interested in the output voltage of a power supply used in a $\mathrm{PC}$. Output voltage is assumed to be normally distributed, with standard deviation 0.25 volt, and the manufacturer wishes to test $H_{0}: \mu=5$ volts against $H_{1}: \mu \neq 5$ volts, using $n=8$ units. (a) The acceptance region is $4.85 \leq \bar{x} \leq 5.15 .$ Find the value of $\alpha$ (b) Find the power of the test for detecting a true mean output voltage of 5.1 volts.

A manufacturer is interested in the output voltage of a power supply used in a $\mathrm{PC}$. Output voltage is assumed to be normally distributed, with standard deviation 0.25 volt, and the manufacturer wishes to test $H_{0}: \mu=5$ volts against $H_{1}: \mu \neq 5$ volts, using $n=8$ units.
(a) The acceptance region is $4.85 \leq \bar{x} \leq 5.15 .$ Find the value of $\alpha$
(b) Find the power of the test for detecting a true mean output voltage of 5.1 volts.

Key Concepts

Power of a Test

The power of a test is the probability that it correctly rejects a false null hypothesis. It reflects the test’s sensitivity to detect an effect when one truly exists. High power is desirable, as it means that the test is less likely to make a Type II error (failing to reject a false null hypothesis). The power of a test is influenced by factors such as sample size, effect size, variability in the data, and the significance level.

Standard Error and Sampling Distribution of the Mean

The standard error of the mean (SEM) describes the distribution of sample means around the population mean and is calculated as the population standard deviation divided by the square root of the sample size. This concept is vital in hypothesis testing because it quantifies the variability inherent in the sampling process, allowing for the construction of confidence intervals and the execution of tests regarding the population mean.

Acceptance and Rejection Regions

The acceptance region (or non-rejection region) is the set of all values of the test statistic that would lead to the retention of the null hypothesis, whereas the rejection region consists of values that would result in the rejection of the null hypothesis. Defining these regions is essential in hypothesis testing to control the error rates and determine the outcome of the statistical test based on the sample data.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis (typically representing no change or no effect) and an alternative hypothesis, then using sample data to determine if the observed results are statistically significant with respect to these hypotheses.

Normal Distribution

The normal distribution is a key concept in statistics characterized by its bell-shaped curve. It is used to model continuous data, and many statistical tests assume that the data or errors are normally distributed. This concept allows for the calculation of probabilities and critical values, and it is fundamental in hypothesis testing and confidence interval estimation.

Type I Error (Significance Level)

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. The significance level, denoted by alpha (?), is the probability of making a Type I error. This concept is critical in the context of hypothesis testing as it quantifies the risk of a false positive result, guiding the choice of acceptance or rejection regions for test statistics.

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