(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 prices (in dollars) for a sample of automobile batteries. The prices are classified according to battery type. At $\alpha=0.05,$ is there enough evidence to conclude that at least one mean battery price is different from the others? (Source: Consumer Reports) $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Group size 35 } & 90 & 90 & 75 & 105 & 65 & \\\hline \text { Group size 65 } & 100 & 75 & 105 & 110 & 90 & 90 \\\hline \text { Group size 24/24F } & 115 & 75 & 75 & 90 & 110 & 90 \\ \hline\end{array}$$

(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 prices (in dollars) for a sample of automobile batteries. The prices are classified according to battery type. At $\alpha=0.05,$ is there enough evidence to conclude that at least one mean battery price is different from the others? (Source: Consumer Reports)
$$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Group size 35 } & 90 & 90 & 75 & 105 & 65 & \\\hline \text { Group size 65 } & 100 & 75 & 105 & 110 & 90 & 90 \\\hline \text { Group size 24/24F } & 115 & 75 & 75 & 90 & 110 & 90 \\
\hline\end{array}$$

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more independent groups to determine whether there is any statistically significant difference among them. It partitions the total variation observed in the data into variation between groups and within groups, using the assumption of normally distributed populations with equal variances.

Null and Alternative Hypotheses

These are statements about the parameters of the population. The null hypothesis (H0) typically asserts that there is no difference among the group means (i.e., all population means are equal), while the alternative hypothesis (Ha) states that at least one group mean is different. This framework forms the basis for testing claims about group differences in many statistical procedures.

F-Test Statistic

The F-test statistic in ANOVA is the ratio of the variance estimated between the group means to the variance within the groups. A larger F-value suggests that the between-group variability exceeds the variability expected due to random sampling error, thereby indicating evidence against the null hypothesis.

Critical Value and Rejection Region

In hypothesis testing, the critical value is a threshold value derived from the F-distribution table (using the chosen significance level and the degrees of freedom). The rejection region defines the values of the F-test statistic for which the null hypothesis will be rejected. When the calculated F exceeds this critical value, the test indicates significant differences among the group means.

Decision and Interpretation

Based on the comparison between the calculated F-statistic and the critical value, a decision is made to either reject or fail to reject the null hypothesis. The decision is then interpreted in the context of the original research claim, explaining whether there is sufficient evidence to conclude that not all group means are equal.

Transcript

00:01 Here's the solution for the anova test for the mean cost of batteries given on, based on a variation of sizes.

00:12 So the null hypothesis is that the means are equal.

00:16 The mean prices are equal.

00:17 So mu1 equals mu2 equals mu3.

00:19 And then the alternative is that not all of them are equal.

00:23 And actually this is, this should be mu one, just to keep the same thing going.

00:29 On some mu1, mu2, and you three.

00:32 Okay, and the next step is to find the critical value, f star.

00:35 Now you can find that using a table, or you can, i build a program in the calculator, so that's the way i'm gonna show you, but you can certainly use a table.

00:42 It's probably either in your book, or you can probably just find it online, but i call it f star, and you need an alpha value, and that's given in your problem, the alpha's 0 .05.

00:52 The degrees of freedom in this case is two, because it's separated into three different categories, and it's always the number of categories minus one.

00:58 So three minus one is two.

00:59 And then the degrees of freedom for the denominator is 14.

01:02 That's the total number of data values minus the number of categories.

01:05 So there are 17 data values minus the three categories of 14.

01:09 Okay, so then that's the information and that you can use the table or you can build this...

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