(a) identify the claim and state H0 and Ha,(b) find the...

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology. The table shows the salaries (in thousands of dollars) for a sample of individuals from the federal, state, and local levels of government. At $\alpha=0.01,$ can you conclude that at least one mean salary is different from the others? (Adapted from Bureau of Labor Statistics) $$\begin{aligned}&-\\&\begin{array}{|c|c|c|}\hline \text { Federal } & \text { State } & \text { Local } \\\hline 70.4 & 52.9 & 48.8 \\63.1 & 37.0 & 38.3 \\74.5 & 54.0 & 42.6 \\82.3 & 54.5 & 41.0 \\81.6 & 56.7 & 51.6 \\85.7 & 61.8 & 45.7 \\56.3 & 39.9 & 60.3 \\71.2 & 50.4 & 40.8 \\80.9 & 53.6 & 37.2 \\64.6 & 47.4 & 33.4 \\\hline\end{array} \end{aligned}$$

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology.
The table shows the salaries (in thousands of dollars) for a sample of individuals from the federal, state, and local levels of government. At $\alpha=0.01,$ can you conclude that at least one mean salary is different from the others? (Adapted from Bureau of Labor Statistics)
$$\begin{aligned}&-\\&\begin{array}{|c|c|c|}\hline \text { Federal } & \text { State } & \text { Local } \\\hline 70.4 & 52.9 & 48.8 \\63.1 & 37.0 & 38.3 \\74.5 & 54.0 & 42.6 \\82.3 & 54.5 & 41.0 \\81.6 & 56.7 & 51.6 \\85.7 & 61.8 & 45.7 \\56.3 & 39.9 & 60.3 \\71.2 & 50.4 & 40.8 \\80.9 & 53.6 & 37.2 \\64.6 & 47.4 & 33.4 \\\hline\end{array}
\end{aligned}$$

Key Concepts

Decision Making in Hypothesis Testing

The decision to reject or fail to reject the null hypothesis is based on comparing the computed F-test statistic with the critical value or by comparing the p-value with the significance level. If the test statistic falls in the rejection region or if the p-value is less than the significance level, the null hypothesis is rejected.

Interpretation in Context

After making the statistical decision, it is important to interpret the result in the context of the original research question or claim. This involves explaining what the rejection or failure to reject the null hypothesis means regarding the population means or group differences, thus providing practical implications and insights based on the analysis.

F-Test Statistic

The F-test statistic in ANOVA is calculated as the ratio of the mean square between groups (variation due to the interaction between the groups) to the mean square within groups (variation due to random error). A higher F value suggests that a significant portion of the total variability is due to differences between group means rather than chance, indicating that at least one group mean is distinct.

Critical Value and Rejection Region

The critical value in an ANOVA F-test is obtained from the F-distribution table based on the chosen significance level, and the degrees of freedom associated with the between-group and within-group variabilities. The rejection region is defined by values of the test statistic that exceed this critical value, meaning that if the computed F statistic falls in this region, the null hypothesis is rejected.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) states that there is no effect or no difference between the groups or populations being compared, while the alternative hypothesis (H?) suggests that there is an effect or a difference. Formulating these hypotheses clearly is crucial as they direct the subsequent analysis and decision-making process.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about population parameters based on sample data. It involves setting up a null hypothesis (typically representing a status quo or equality scenario) and an alternative hypothesis (representing a claim or effect), then using sample data to decide whether there is enough evidence to reject the null hypothesis at a predetermined significance level.

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of three or more independent groups to see if at least one group mean is significantly different from the others. It partitions the overall variability into variability between groups and within groups, and it uses the F-test to determine whether the between-group variability is significantly greater than the within-group variability.

Transcript

00:01 The following is a solution to number eight.

00:02 It's the anova for the difference between federal, state, and local salaries.

00:08 And the null hypothesis and the alternative, they're always the same thing.

00:11 A null is that the means are equal.

00:14 So this would mean that the mean of the federal or the mu 1 is equal to the mean of the state or mu 2, which is equal to the mu of the mean of the local, so mu 3.

00:25 And then the alternative is at least one of those means are different.

00:28 And then the second step is to find the critical value.

00:30 Now you can find the critical value using a table or i programmed it in the calculator, and that's the way i'm going to show you.

00:37 So i'm not going to show you how to program this in the calculator.

00:39 You can google it if you want, but i'm just going to show how to do it in the calculator.

00:45 And i'll call the critical value f star.

00:48 And what you need to find f star is you need the alpha value, and then they give you that generally.

00:52 It's 0 .01.

00:53 And then you also need to know the degrees of freedom for the numerator, which is k .1, which is k minus 1 and then degrees of freedom for the denominator, which is the number of data values minus your number of categories k.

01:04 So there are three categories, the federal state and locals.

01:08 So 3 minus 1 is 2.

01:09 That's your degrees and freedom for the numerator.

01:11 And then for the denominator, there are 30 data values total minus the three columns, the federal, state, and local, so 27 degrees of freedom on the denominator.

01:21 So if you have a tid4, you can program this in your calculator program.

01:26 And i just called it inverse f.

01:29 So inverse f, and it asked me for my area right, so that's my alpha, and then degrees of freedom for the numerator is 2, and then degrees of freedom for the denominator is 27, and then that gives me my critical value of about 5 .488.

01:46 So let's go and write that.

01:48 So f star is 5 .488...

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