(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 sale prices (in thousands of dollars) of a sample of one-family houses in three cities. At $\alpha=0.10$ can you conclude that at least one mean sale price is different from the others? (Adapted from National Association of Realtors) $$\begin{array}{|c|c|c|}\hline \text { Gainesville } & \text { Orlando } & \text { Tampa } \\\hline 139.0 & 169.9 & 184.7 \\111.5 & 127.1 & 69.7 \\156.6 & 111.3 & 165.0 \\152.3 & 113.5 & 157.5 \\214.7 & 133.9 & 103.9 \\172.4 & 160.8 & 120.8 \\52.8 & 179.2 & 88.1 \\170.6 & 70.7 & 168.2 \\140.5 & 89.9 & 59.5 \\186.0 & 99.3 & 170.2 \\139.0 & & \\ \hline\end{array}$$

(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 sale prices (in thousands of dollars) of a sample of one-family houses in three cities. At $\alpha=0.10$ can you conclude that at least one mean sale price is different from the others? (Adapted from National Association of Realtors)
$$\begin{array}{|c|c|c|}\hline \text { Gainesville } & \text { Orlando } & \text { Tampa } \\\hline 139.0 & 169.9 & 184.7 \\111.5 & 127.1 & 69.7 \\156.6 & 111.3 & 165.0 \\152.3 & 113.5 & 157.5 \\214.7 & 133.9 & 103.9 \\172.4 & 160.8 & 120.8 \\52.8 & 179.2 & 88.1 \\170.6 & 70.7 & 168.2 \\140.5 & 89.9 & 59.5 \\186.0 & 99.3 & 170.2 \\139.0 & & \\
\hline\end{array}$$

Key Concepts

F-Statistic

The F-statistic is the test statistic used in ANOVA. It is calculated as a ratio of the variance between group means to the variance within the groups. A larger F value indicates a greater likelihood that there are significant differences among the group means, which could lead to rejecting the null hypothesis.

Null and Alternative Hypotheses

These are the foundation of any hypothesis test. In comparing more than two means, the null hypothesis (H0) typically states that all group means are equal, while the alternative hypothesis (Ha) asserts that at least one group mean is different. Clearly defining these hypotheses is essential to determining whether observed differences are statistically significant.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more independent groups to see if at least one of them significantly differs from the others. It works by analyzing the variation within groups versus the variation between groups, and it assumes normally distributed populations with equal variances.

Critical Value and Rejection Region

The critical value is determined from the F-distribution table based on the chosen significance level (alpha) and the degrees of freedom for the numerator (between groups) and the denominator (within groups). The rejection region consists of F-statistic values that, if exceeded, indicate that the null hypothesis should be rejected.

Assumptions Underlying ANOVA

ANOVA relies on several critical assumptions: the samples must be random and independent, the populations should be normally distributed, and the variances across the groups must be equal. Verifying these assumptions is important to ensure that the results of the analysis are valid and reliable.

Transcript

00:01 Following is the solution to number 14 at one -way a nova test.

00:04 And this is about the mean sales prices for three cities.

00:09 And the null hypothesis here is that the mean sale prices are the same for these certain houses.

00:13 And then the alternative is that at least one of them is different.

00:16 The second step is to find the critical value.

00:19 And you need three pieces of information to find the critical value.

00:22 One is your alpha, your significance level.

00:24 And that significance level in this case, and that's usually given to you is 0 .10.

00:27 You also need the degrees of freedom for the numerator, which is, the number of categories, in this case the number of cities minus one.

00:33 So there were three cities that we looked at minus one is two.

00:36 So degrees of freedom for the numerator is two.

00:38 Degrees of freedom for the denominator is the total number of data values minus the number of categories.

00:43 So in this case, if you counted up those data values, there were 31 minus the three cities that we looked at.

00:48 So 31 minus three is 28.

00:50 So that's what we need.

00:51 So from there, you can use a table or you can use software.

00:54 I'm going to use software.

00:55 So i wrote a program and i called it inverse i'm not going to show you how to write this program.

00:59 You can youtube it if you wish, but it makes it easier for me.

01:05 So the area is the alpha value, so we'll put in point 10 for that.

01:09 Degrees of freedom for the numerator was 2, and then degrees of freedom for the denominator was 28.

01:14 And that's going to spit out my f star, my alpha value, or my critical value, which is about 2 .503, let's call it.

01:21 So 2 .503 is my f star.

01:24 So 2 .503.

01:29 So anything greater than 2 .503, we're going to reject h0.

01:34 Anything less than 2 .503, then we're going to fail to reject the null...

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