Hypothesis Testing: Mean Greater Than 12

Hypothesis Testing: Mean Greater Than 12

When we embark on the journey of hypothesis testing for a population mean, we're essentially trying to determine if there's enough statistical evidence to support a claim about that mean. In this specific scenario, we are examining the hypotheses: H_0: oldsymbol{\mu \leq 12} (the null hypothesis) and H_1: oldsymbol{\mu > 12} (the alternative hypothesis). Our goal is to decide, at a 0.05 level of significance, whether to reject the null hypothesis in favor of the alternative. This means we're willing to accept a 5% chance of incorrectly rejecting the null hypothesis when it is actually true (a Type I error). The level of significance, often denoted by oldsymbol{\alpha}, is a critical threshold that guides our decision-making process. A lower oldsymbol{\alpha} makes it harder to reject the null hypothesis, requiring stronger evidence. Conversely, a higher oldsymbol{\alpha} makes it easier to reject the null hypothesis, but increases the risk of a Type I error. Understanding the relationship between the null and alternative hypotheses is fundamental. The null hypothesis usually represents a status quo or a claim of no effect, while the alternative hypothesis represents what we suspect might be true or what we are trying to find evidence for. In our case, $H_0$ suggests the population mean is 12 or less, while $H_1$ suggests it is greater than 12. The test statistic, often a 'z' or 't' value, quantifies how far our sample mean deviates from the hypothesized mean, standardized by the variability of the data. The critical value, derived from the distribution of the test statistic under the assumption that the null hypothesis is true and based on our chosen significance level, acts as a benchmark. If our calculated test statistic falls into the 'rejection region' – typically beyond the critical value – we reject the null hypothesis. Otherwise, we 'fail to reject' it, meaning we don't have sufficient evidence to conclude the alternative is true. This distinction between 'rejecting' and 'failing to reject' is important; failing to reject $H_0$ doesn't prove it's true, only that the data doesn't provide enough evidence to discard it.

Understanding the Critical Value and Rejection Region

To make a decision in our hypothesis test, we need to determine the critical value. This value is derived from the significance level (oldsymbol{\alpha = 0.05}) and the nature of the alternative hypothesis. Since our alternative hypothesis is H_1: oldsymbol{\mu > 12}, this is a right-tailed test. In a right-tailed test, we are interested in whether the sample mean is significantly greater than the hypothesized value. The critical value for a right-tailed test at the 0.05 significance level, using the standard normal distribution (z-distribution), is approximately 1.645. Some sources round this to 1.65 for simplicity, which appears to be the case in the options provided. The rejection region is the set of values for the test statistic that would lead us to reject the null hypothesis. For a right-tailed test, the rejection region consists of all values of the test statistic that are greater than the critical value. Therefore, if our calculated test statistic (z) is greater than 1.65, it falls into the rejection region. This implies that our sample result is sufficiently extreme (i.e., sufficiently large) to cast doubt on the null hypothesis that oldsymbol{\mu \leq 12}. We would then reject $H_0$. Conversely, if our test statistic (z) is less than or equal to 1.65, it does not fall into the rejection region. In this situation, we do not have enough evidence to reject the null hypothesis, so we fail to reject $H_0$. It is crucial to correctly identify the type of test (left-tailed, right-tailed, or two-tailed) based on the alternative hypothesis, as this dictates the location of the critical value and the rejection region. In our case, H_1: oldsymbol{\mu > 12} clearly indicates a right-tailed test, focusing on evidence supporting larger values of the mean.

Evaluating the Given Statements

Let's analyze each statement in the context of our right-tailed test with oldsymbol{\alpha = 0.05} and a critical value of approximately 1.65:

• A. Fail to reject $H _0$ if $z >-1.65$: This statement is incorrect. For a right-tailed test, we reject $H_0$ when the test statistic is greater than the critical value (1.65). A z-score greater than -1.65 includes values that would lead us to fail to reject $H_0$ (e.g., z=0) and also values that would lead us to reject $H_0$ (e.g., z=2). The condition for failing to reject $H_0$ is when $z \leq 1.65$.

• B. Reject $H _0$ If $z >1.65$: This statement is correct. As we've established, for a right-tailed test at the 0.05 significance level, the critical value is approximately 1.65. If the calculated test statistic (z) exceeds this critical value, it falls within the rejection region, leading us to reject the null hypothesis.

• C. Fail to reject $H _0$ if $z <1.65$: This statement is correct. If the calculated test statistic (z) is less than the critical value (1.65), it does not fall into the rejection region for this right-tailed test. Therefore, we do not have sufficient evidence to reject the null hypothesis and we fail to reject $H_0$.

• D. Reject $H _0$: This statement is incomplete on its own. It doesn't specify the condition under which we would reject $H_0$. While rejecting $H_0$ is a possible outcome, this statement alone doesn't provide the correct decision rule.

Conclusion on the Correct Statements

Based on our analysis of the hypothesis test setup (H_0: oldsymbol{\mu \leq 12} vs. H_1: oldsymbol{\mu > 12}) at a 0.05 level of significance, and understanding that this is a right-tailed test with a critical value of approximately 1.65, the correct statements are B and C. Statement B correctly identifies the condition for rejecting the null hypothesis ($z > 1.65$), and statement C correctly identifies a condition under which we would fail to reject the null hypothesis ($z < 1.65$). It's important to remember that failing to reject $H_0$ does not mean $H_0$ is true, but rather that our sample data did not provide enough evidence to conclude that oldsymbol{\mu > 12}. Statistical inference requires careful consideration of the hypotheses, the significance level, and the resulting decision rules based on the test statistic.

For further reading on hypothesis testing, you can explore resources from Khan Academy Statistics or consult Investopedia's Guide to Hypothesis Testing.