Question

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40 -amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40 . If the mean amperage is lower than 40 , customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40 , the manufacturer might be liable for damage to an electrical system as a result of fuse malfunction. To verify the mean amperage of the fuses, a sample of fuses is selected and tested. If a hypothesis test is performed using the resulting data, what null and alternative hypotheses would be of interest to the manufacturer?

Short answer

The null hypothesis \(H_0\) for the manufacturer would be: The mean amperage at which the fuses burn out is 40 (\(H_0: \mu = 40\)). The alternative hypothesis \(H_a\) would be: The mean amperage at which the fuses burn out is not 40 (\(H_a: \mu \neq 40\)).

Step-by-step solution

Formulate the Null Hypothesis

The null hypothesis, denoted by \(H_0\), is a statement of no difference. In this context, it would state that the mean amperage at which the fuses burn out is 40, i.e. \(H_0: \mu = 40\). This represents the status quo or the manufacturer's requirement.

Formulate the Alternative Hypothesis

The alternative hypothesis, denoted by \(H_a\) or \(H_1\), is a statement that suggests a change. For this manufacturer, the alternative hypothesis would be that the mean amperage at which the fuses burn out is not 40, i.e. \(H_a: \mu \neq 40\). This would mean that there is a problem with the amperage at which the fuses burn out, causing either customer complaints or potential legal issues.

Key concepts

Null Hypothesis

When we talk about the null hypothesis, denoted by H₀, we're referring to a basic presumption that there is no effect or no difference in the context of a statistical test. Think of it as the 'status quo' or a baseline to compare against.

In the case of our fuse manufacturer, the null hypothesis asserts that the mean amperage at which the fuses blow is exactly 40 amperes (40A). Mathematically, this would be written as H₀: μ = 40. Here, μ symbolizes the population mean for the amperage. The null hypothesis serves as a starting point for hypothesis testing. If data gathered from testing fuses suggest that the null hypothesis is unlikely to be true, the hypothesis may be rejected, leading to further investigation.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis, labeled as H_a or H₁, represents a statement that indicates some statistical effect or difference that we want to test for. It's the hypothesis that researchers typically hope to provide evidence for.

For our example, the alternative hypothesis posits that the mean amperage isn't 40A - it could be lower or higher, thus it's a two-tailed test. Expressed in formula, it's H_a: μ ≠ 40. If the mean is indeed not 40A, then the fuses are not performing as intended. This could result in either customer dissatisfaction or a risk of product liability, both outcomes the manufacturer aims to avoid. Validating the alternative hypothesis would guide the manufacturer to make necessary changes to the production process.

Mean Amperage

The term 'mean amperage' describes the average current at which a batch of fuses is expected to burn out. This average is crucial for manufacturers, as it's a benchmark for product consistency and safety. The importance of this cannot be overstated; a fuse with too low of a mean amperage rating might blow frequently, causing an annoyance to consumers. Conversely, a fuse with a too high mean amperage rating creates a risk of electrical damage or fire.

In our context, the sample of fuses that the manufacturer tests will provide an empirical mean amperage. This observed mean can then be compared to the hypothesized mean (40A) to see if it falls within an acceptable range of variation, or if it significantly differs, implying potential issues with the fuses' design or manufacture.

Statistical Significance

The notion of 'statistical significance' is used to determine if the observed effect in a study is unlikely to have occurred by random chance. In other words, it helps us decide whether to accept or reject the null hypothesis. Statistical significance is generally determined using a p-value, which is calculated from hypothesis testing. A low p-value (<0.05 is a common threshold) indicates that the observed data is unlikely under the assumption that the null hypothesis is true, thus leading to the rejection of the null hypothesis.

In the example of the fuse manufacturer, if the testing of the sample fuses results in a mean amperage significantly different from 40A, and this difference is statistically significant, it compels the manufacturer to review their processes and product specifications. Conversely, if the result is not statistically significant, it suggests that the sampled fuses are burning out at the expected mean amperage, supporting the null hypothesis.