What Is The Decision Rule For A Two-Tailed Hypothesis Test?

What is the decision rule for a two-tailed hypothesis test at a 0.10 significance level

If you use a 0.10 level of significance in a two-tailed hypothesis test, the decision rule for rejecting the null hypothesis when conducting a Z test involves comparing the calculated test statistic to the critical Z values corresponding to the significance level. Specifically, the critical Z values are ±1.645 because the significance level split equally between the two tails in a two-tailed test is 0.10, leaving 0.05 (5%) in each tail. If the absolute value of the test statistic exceeds 1.645, then the null hypothesis is rejected. Conversely, if the test statistic falls within the range of -1.645 to 1.645, we fail to reject the null hypothesis. This decision rule ensures that the probability of making a Type I error—rejecting a true null hypothesis—is controlled at 10%. The process involves calculating the Z statistic based on sample data, then comparing this to the critical Z values: reject H₀ if |Z| > 1.645; otherwise, do not reject H₀.

Paper For Above instruction

The significance level in hypothesis testing is a crucial parameter that guides decision-making by defining the threshold for rejecting the null hypothesis. When using a significance level of 0.10 in a two-tailed test, the decision rule becomes a straightforward application of critical Z values derived from the standard normal distribution. Essentially, this means that if the calculated Z statistic falls beyond the critical boundary values of ±1.645, the null hypothesis is rejected; if it falls within these bounds, the null cannot be rejected. This approach maintains a controlled probability of 10% that a true null hypothesis will be incorrectly rejected, aligned with the specified significance level.

The process begins with calculating the Z statistic based on the sample data—specifically, the sample mean, the hypothesized population mean, the standard deviation (if known), and the sample size. Once the Z value is computed, the comparison against the critical Z values determines the outcome. For two-tailed tests at the 0.10 significance level, the critical zones are split between the two tails of the normal distribution, each accounting for 5% of the total probability. The critical values are therefore ±1.645, which correspond to cumulative probabilities of 0.025 in the lower tail and 0.975 in the upper tail.

In practice, the decision rule can be stated simply: reject the null hypothesis if the absolute value of the test statistic exceeds 1.645 (|Z| > 1.645). If the test statistic is less extreme, within the bounds of -1.645 to 1.645, then there is not enough evidence at the 10% significance level to reject H₀. This rule ensures that the stated probability of a Type I error—incorrectly rejecting a true null—is exactly 10%, evenly divided across the two tails of the distribution, thereby controlling the overall error rate.

Given this, the null hypothesis is rejected when results are sufficiently extreme in either tail of the distribution, indicating statistically significant evidence against H₀ at the 10% level. Conversely, with less extreme test statistics, the evidence is not enough to justify rejection, and the null hypothesis remains plausible. This framework provides a clear, statistically rigorous decision-making process while adhering to the predefined significance threshold, ensuring the integrity of inferences drawn from sample data in hypothesis testing scenarios.

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