Question

The basis for rejecting any null hypothesis is arbitrary. The researcher can set more or less stringent standards by deciding to raise or lower the critical \(p\) value. Would the use of a standard of \(p=0.10\) be more or less stringent in failing to reject the null hypothesis? Explain.

Short answer

Answer: Using a critical p-value of 0.10 would be less stringent than using lower critical p-values like 0.05 or 0.01. This means that there is a higher chance of committing a Type I error (falsely rejecting the null hypothesis) as we're allowing for weaker evidence against the null hypothesis.

Step-by-step solution

Understanding the Null Hypothesis and p-Values

The null hypothesis (\(H_0\)) is a statement that there is no significant difference, relationship, or effect between variables under investigation. The p-value is a probability value used to decide whether there is enough evidence to reject the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis, and a larger p-value indicates weaker evidence against the null hypothesis.

Critical p-Values and Stringency

The critical p-value is a predetermined threshold below which we reject the null hypothesis. Common critical p-values are \(p=0.05\) or \(p=0.01\). The lower the critical p-value, the more stringent the criteria for rejecting the null hypothesis, meaning that there's a smaller chance of falsely rejecting the null hypothesis (Type I error).

Impact of Using a p-value of 0.10

Using a critical p-value of \(p=0.10\) would be less stringent than using \(p=0.05\) or \(p=0.01\). It allows for a higher probability of making a Type I error, which means falsely rejecting the null hypothesis when it is actually true. By using a p-value of 0.10, we would be more likely to reject the null hypothesis even when there isn't a significant difference, relationship, or effect.

Conclusion

The use of a standard of \(p=0.10\) would be less stringent in failing to reject the null hypothesis compared to using a lower critical p-value such as \(p=0.05\) or \(p=0.01\). It means that there is a higher chance of committing a Type I error (falsely rejecting the null hypothesis), as we're allowing for weaker evidence against the null hypothesis.