Question

For each of the following situations, suppose $H_0:{\mu }_1={\mu }_2$ is being tested against $H_A:{\mu }_1>{\mu }_2.$ State whether or not there is significant evidence for $H_A$. (a) $t_s=3.75$ with 19 degrees of freedom, $\alpha =0.01$. (b) $t_s=2.6$ with 5 degrees of freedom, $\alpha =0.10$. (c) $t_s=2.1$ with 7 degrees of freedom, $\alpha =0.05$. (d) $t_{\mathrm{c}}=1.8$ with 7 degrees of freedom, $\alpha =0.05.$

Short answer

For each scenario, significant evidence for $H_A$: (a) yes, if $t_c<3.75$; (b) yes, if $t_c<2.6$; (c) yes, if $t_c<2.1$; (d) yes, if $t_c<1.8$. The conclusions depend on comparing $t_s$ with the critical $t$ value from the t-distribution for the given degrees of freedom and alpha level.

Step-by-step solution

- Understanding the Hypothesis Test

We are given a set of hypothesis tests for different scenarios. Each test is for comparing two population means to see if there is significant evidence to support the alternative hypothesis, $H_A:{\mu }_1>{\mu }_2$, over the null hypothesis, $H_0:{\mu }_1={\mu }_2$. We are given a test statistic, $t_s$, degrees of freedom, and a significance level, $\alpha$, for each scenario. We will compare the test statistic to the critical value from the t-distribution to determine whether to reject $H_0$.

- Scenario (a) Analysis

For scenario (a), we have $t_s=3.75$, 19 degrees of freedom, and $\alpha =0.01$. We'll use a t-table or software to find the critical value, $t_c$, for 19 dfs at the $0.01$ level of significance for a one-tailed test. If $t_s>t_c$, there is significant evidence to reject $H_0$.

- Scenario (b) Analysis

For scenario (b), we have $t_s=2.6$, 5 degrees of freedom, and $\alpha =0.10$. We find the critical value, $t_c$, for 5 dfs at the $0.10$ level of significance. If $t_s>t_c$, it provides significant evidence against $H_0$.

- Scenario (c) Analysis

For scenario (c), we have $t_s=2.1$, 7 degrees of freedom, and $\alpha =0.05$. Similar to previous steps, we find $t_c$ for 7 dfs at the $0.05$ significance level. If $t_s>t_c$, $H_0$ is rejected.

- Scenario (d) Analysis

For scenario (d), the provided statistic is labeled $t_c$ instead of $t_s$. Assuming this is a typo and should be $t_s=1.8$, with 7 dfs and $\alpha =0.05$, we again find the appropriate critical value. If $t_s>t_c$, then we reject $H_0$.

- Conclusion for Each Scenario

We use a t-distribution table or software to find the critical values for each scenario and compare the calculated $t_s$ to $t_c$ to decide about $H_0$. (a) If the critical value is $<3.75$, $H_0$ is rejected. (b) If the critical value is $<2.6$, $H_0$ is rejected. (c) If the critical value is $<2.1$, $H_0$ is rejected. (d) Assuming $t_s=1.8$, if the critical value is $<1.8$, $H_0$ is rejected. Each conclusion is based on whether $t_s$ is greater than the corresponding critical value.

Key concepts

T-Test

A t-test is a statistical method used to compare the means of two groups to determine if they significantly differ from each other. It's especially useful when dealing with small sample sizes and when the population standard deviation is unknown. In hypothesis testing, the t-test evaluates whether the difference in the sample means is likely due to chance or if it reflects a true difference in the population means.

Depending on the nature of the hypothesis, t-tests can be one-tailed or two-tailed. A one-tailed t-test, as in our exercise's scenarios, assesses the direction of the effect (whether one mean is greater than the other). It's crucial to align the type of t-test with the research question to properly interpret the results.

Degrees of Freedom

In statistics, degrees of freedom (df) represent the number of independent values in a calculation that can vary. When dealing with the t-test, the degrees of freedom are typically the number of observations minus the number of parameters needing to be estimated (in the case of a two-sample t-test, it's the combined sample sizes minus two).

Degrees of freedom play a crucial role in determining the critical value of the test statistic. The critical value is calculated based on the desired significance level and the df, so accurate calculation of df is essential for determining the right critical threshold for making a decision about the hypothesis.

Significance Level

The significance level, denoted as $\alpha$, is a threshold that indicates the probability of rejecting the null hypothesis when it's actually true, otherwise known as a Type I error. Common $\alpha$ levels are 0.01, 0.05, and 0.1. It's a benchmark for determining the boundary of a rejection region for the test statistic.

The lower the $\alpha$, the stronger the evidence must be to reject the null hypothesis. In a t-test, if the calculated test statistic exceeds the critical value associated with the chosen significance level, the null hypothesis is rejected. This threshold helps control the rate at which researchers erroneously find 'significant' results purely by chance.

Alternative Hypothesis

The alternative hypothesis ($H_A$ or $H_1$) represents what a researcher is trying to prove or detect, and in practical terms, it suggests a difference or effect. In our exercise, the alternative hypothesis is ${\mu }_1>{\mu }_2$, indicating that the mean of the first group is greater than the mean of the second group.

The alternative hypothesis is tested against the null hypothesis. A significant result from a t-test lends support to the alternative hypothesis by demonstrating that the observed data is unlikely under the null hypothesis.

Null Hypothesis

The null hypothesis ($H_0$) is a statement of no effect, no difference, or no change. It's a starting point for statistical testing serving as a straw man to be debunked. In the exercise scenarios, $H_0:{\mu }_1={\mu }_2$ suggests that there is no difference between the group means.

During a t-test, evidence from the data is used to assess whether or not to reject the null hypothesis. If enough evidence exists (a large enough t-test statistic compared to the critical value), we reject the null hypothesis in favor of the alternative.

Test Statistic

The test statistic, in the context of the t-test, is a standard score that measures the degree to which the sample data diverge from the null hypothesis. It's calculated from the sample data and is used in comparison to the critical value.

A larger absolute value of the test statistic indicates more substantial evidence against the null hypothesis. For example, in scenario (a) of our exercise, a t-test statistic ($t_s$) of 3.75 suggests that the sample mean difference is quite far from what we would expect if the null hypothesis were true, considering there are 19 degrees of freedom and a significance level of 0.01.

Critical Value

The critical value determines the cutoff point at which we reject the null hypothesis. It is dependent upon the chosen significance level ($\alpha$) and the degrees of freedom associated with the t-test. You can find critical values using statistical tables or computational tools.

In the exercise, the test statistic for each case must be greater than the critical value to provide significant evidence against the null hypothesis. For instance, with 19 degrees of freedom and an $\alpha$ level of 0.01, if the observed t-test statistic is greater than the corresponding critical value from a t-distribution table, then the null hypothesis is rejected, indicating strong evidence in favor of the alternative hypothesis.