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

Null Hypothesis (H0): µ = 40 (The mean amperage of the fuses is 40 A)\nAlternative Hypothesis (H1): µ ≠ 40 (The mean amperage of the fuses is not 40 A)

Step-by-step solution

Understand the Situation

A fuse manufacturer wants to check if the mean amperage at which its fuses burn out is indeed 40 amp. If the mean amperage is lower or higher than 40 amp, there could be complaints from customers or potential liability issues. This needs to be tested statistically.

Formulate the Null Hypothesis

The null hypothesis (H0) is the status quo or the claim that the manufacturer wants to test. Here, it's the statement that the mean amperage of the fuses is 40 amp. So we get H0: µ = 40.

Formulate the Alternative Hypothesis

The alternative hypothesis (H1) is the statement against the null hypothesis. In this case, it's the statement that the mean amperage is not equal to 40 A. So, we get H1: µ ≠ 40.

Key concepts

Null Hypothesis

When it comes to hypothesis testing in statistics, the null hypothesis is a critical starting point. It represents the default statement or the current view that there is no effect or no difference. In essence, it's the assumption that any kind of variation or difference you observe in your data is merely due to chance, and not due to any actual effect.

In the case of the fuse manufacturer, the null hypothesis is set up to test whether the mean amperage of the fuses is exactly 40 amps, as claimed. So the formal statement of the null hypothesis is: \(H_0: \text{The mean amperage} = 40 \text{ amps}\). If the data we collect provides strong enough evidence to refute this hypothesis, only then we would consider alternatives.

Alternative Hypothesis

Contrasting with the null hypothesis, the alternative hypothesis signifies what we aim to demonstrate or support. This hypothesis is a statement that proposes there is a statistically significant effect or difference.

For our fuse manufacturer, the concern lies in whether the mean amperage is not actually 40 amps—it could be either lower or higher. Therefore, the alternative hypothesis is a two-sided test, which we can express as: \(H_1: \text{The mean amperage} eq 40 \text{ amps}\). Successfully proving this alternative hypothesis can lead to a change in business practices, improvements in the product, or the validation of concerns over product performance.

Mean Amperage

Mean amperage is the average current at which a batch of fuses is expected to burn out. This average value is crucial for manufacturers who need to ensure their fuses perform correctly within a safe and expected range. Think of it as ensuring the right balance — not burning out too soon (causing customer inconvenience) and not holding on too long (which could raise safety concerns).

For the discussed exercise, establishing whether the actual mean amperage aligns with the stated 40 amps is vital. It anchors the claims of the manufacturer and dictates customer satisfaction as well as legal compliance and safety.

Statistical Significance

Statistical significance is a measure of whether the effect or difference observed is likely to be due to chance or if it's a result of a specific cause. It's what determines the validity of our hypothesis test results. To establish statistical significance, we compare the p-value — the probability of obtaining an effect at least as extreme as what was observed, given that the null hypothesis is true — to a predefined significance level, often 0.05 or 5%.

If the p-value is less than the significance level, we conclude that our findings are statistically significant and consequently reject the null hypothesis. This is a pivotal moment in hypothesis testing because it can influence decisions and incite changes based on the data analysis. In the context of our example, finding statistical significance would mean that the manufacturer needs to reassess the fuse design or production process to ensure the mean amperage is consistently at the desired level.