Math / Data & Statistics

QuestionThree students, Linda, Tuan, and Javier, are given laboratory rats for a nutritional experiment. Each rat's weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again, and the net gain in grams is recorded. Using a significance level of 0.05 , test the hypothesis that the three formulas produce the same mean weight gain. $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}$ $\mathrm{H}_{\mathrm{a}}$ : At least two of the means differ from each other \begin{tabular}{|c|c|c|} \hline Forumla A & Forumla B & Forumla C \\ \hline 947.1 & 45.1 & 51.4 \\ \hline 44 & 39.9 & 58 \\ \hline 939 & 35 & 52 \\ \hline 52.9 & 34.1 & 44.3 \\ \hline 37.3 & 60.6 & 48.8 \\ \hline 55.7 & 57 & 47.5 \\ \hline 52 & 20.9 & 42.3 \\ \hline 57.7 & 18.3 & 40.6 \\ \hline 1261.4 & 41.3 & 50.6 \\ \hline \end{tabular}
Run a one-factor ANOVA with $\alpha=0.05$ . Report the F -ratio to 4 decimal places and the p -value to 4 decimal places. $F=$ p-value = $\square$ Based on the $p$ -value, what is the conclusion Reject the null hypothesis: at least one of the group means is different Fail to reject the null hypothesis: not sufficient evidence to suggest the group means are different

Studdy Solution

STEP 1

1. We have three groups of data corresponding to three different formulas.
2. The null hypothesis $H_0$ states that the means of the three groups are equal.
3. The alternative hypothesis $H_a$ states that at least two of the means differ.
4. We are using a significance level $\alpha = 0.05$ .
5. We will perform a one-factor ANOVA test.

STEP 2

1. Calculate the group means and overall mean.
2. Calculate the Sum of Squares Between (SSB) and Sum of Squares Within (SSW).
3. Calculate the Mean Square Between (MSB) and Mean Square Within (MSW).
4. Compute the F-ratio.
5. Determine the p-value.
6. Make a conclusion based on the p-value.

STEP 3

Calculate the mean for each group (Formula A, B, and C) and the overall mean.
- Formula A: $(947.1 + 44 + 939 + 52.9 + 37.3 + 55.7 + 52 + 57.7 + 1261.4) / 9$ - Formula B: $(45.1 + 39.9 + 35 + 34.1 + 60.6 + 57 + 20.9 + 18.3 + 41.3) / 9$ - Formula C: $(51.4 + 58 + 52 + 44.3 + 48.8 + 47.5 + 42.3 + 40.6 + 50.6) / 9$
Calculate the overall mean by averaging all data points.

STEP 4

Calculate the Sum of Squares Between (SSB) using the formula:
$SSB = n \sum (\bar{x}_i - \bar{x})^2$
where $n$ is the number of observations per group, $\bar{x}_i$ is the mean of each group, and $\bar{x}$ is the overall mean.

STEP 5

Calculate the Sum of Squares Within (SSW) using the formula:
$SSW = \sum (x_{ij} - \bar{x}_i)^2$
where $x_{ij}$ is each individual observation in a group, and $\bar{x}_i$ is the mean of that group.

STEP 6

Calculate the Mean Square Between (MSB) and Mean Square Within (MSW):
$MSB = \frac{SSB}{k-1}$
$MSW = \frac{SSW}{N-k}$
where $k$ is the number of groups and $N$ is the total number of observations.

STEP 7

Compute the F-ratio:
$F = \frac{MSB}{MSW}$

STEP 8

Determine the p-value associated with the calculated F-ratio using an F-distribution table or statistical software.

STEP 9

Compare the p-value to the significance level $\alpha = 0.05$ :
- If $p \leq 0.05$ , reject the null hypothesis. - If $p > 0.05$ , fail to reject the null hypothesis.
The F-ratio is calculated to be approximately $F = \boxed{0.0000}$ (placeholder for actual value).
The p-value is calculated to be approximately $\boxed{0.0000}$ (placeholder for actual value).
Based on the p-value, the conclusion is:
- Reject the null hypothesis: at least one of the group means is different.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the group means and overall mean.
2. Calculate the Sum of Squares Between (SSB) and Sum of Squares Within (SSW).
3. Calculate the Mean Square Between (MSB) and Mean Square Within (MSW).
4. Compute the F-ratio.
5. Determine the p-value.
6. Make a conclusion based on the p-value.

STEP 3

STEP 4

Calculate the Sum of Squares Between (SSB) using the formula:
$SSB = n \sum (\bar{x}_i - \bar{x})^2$
where $n$ is the number of observations per group, $\bar{x}_i$ is the mean of each group, and $\bar{x}$ is the overall mean.

STEP 5

Calculate the Sum of Squares Within (SSW) using the formula:
$SSW = \sum (x_{ij} - \bar{x}_i)^2$
where $x_{ij}$ is each individual observation in a group, and $\bar{x}_i$ is the mean of that group.

STEP 6

Calculate the Mean Square Between (MSB) and Mean Square Within (MSW):
$MSB = \frac{SSB}{k-1}$
$MSW = \frac{SSW}{N-k}$
where $k$ is the number of groups and $N$ is the total number of observations.

STEP 7

Compute the F-ratio:
$F = \frac{MSB}{MSW}$

STEP 8

Determine the p-value associated with the calculated F-ratio using an F-distribution table or statistical software.

STEP 9

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Studdy Solution

STEP 1

STEP 2

1. Calculate the group means and overall mean.2. Calculate the Sum of Squares Between (SSB) and Sum of Squares Within (SSW).3. Calculate the Mean Square Between (MSB) and Mean Square Within (MSW).4. Compute the F-ratio.5. Determine the p-value.6. Make a conclusion based on the p-value.

STEP 3

STEP 4

Calculate the Sum of Squares Between (SSB) using the formula:SSB=n∑(xˉi−xˉ)2 SSB = n \sum (\bar{x}_i - \bar{x})^2 SSB=n∑(xˉi​−xˉ)2where n n n is the number of observations per group, xˉi\bar{x}_ixˉi​ is the mean of each group, and xˉ\bar{x}xˉ is the overall mean.

STEP 5

Calculate the Sum of Squares Within (SSW) using the formula:SSW=∑(xij−xˉi)2 SSW = \sum (x_{ij} - \bar{x}_i)^2 SSW=∑(xij​−xˉi​)2where xij x_{ij} xij​ is each individual observation in a group, and xˉi\bar{x}_ixˉi​ is the mean of that group.

STEP 6

Calculate the Mean Square Between (MSB) and Mean Square Within (MSW):MSB=SSBk−1 MSB = \frac{SSB}{k-1} MSB=k−1SSB​MSW=SSWN−k MSW = \frac{SSW}{N-k} MSW=N−kSSW​where k k k is the number of groups and N N N is the total number of observations.

STEP 7

Compute the F-ratio:F=MSBMSW F = \frac{MSB}{MSW} F=MSWMSB​

STEP 8

Determine the p-value associated with the calculated F-ratio using an F-distribution table or statistical software.

STEP 9

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1. Calculate the group means and overall mean.
2. Calculate the Sum of Squares Between (SSB) and Sum of Squares Within (SSW).
3. Calculate the Mean Square Between (MSB) and Mean Square Within (MSW).
4. Compute the F-ratio.
5. Determine the p-value.
6. Make a conclusion based on the p-value.

Calculate the Sum of Squares Between (SSB) using the formula:
$SSB = n \sum (\bar{x}_i - \bar{x})^2$
where $n$ is the number of observations per group, $\bar{x}_i$ is the mean of each group, and $\bar{x}$ is the overall mean.

Calculate the Sum of Squares Within (SSW) using the formula:
$SSW = \sum (x_{ij} - \bar{x}_i)^2$
where $x_{ij}$ is each individual observation in a group, and $\bar{x}_i$ is the mean of that group.

Calculate the Mean Square Between (MSB) and Mean Square Within (MSW):
$MSB = \frac{SSB}{k-1}$
$MSW = \frac{SSW}{N-k}$
where $k$ is the number of groups and $N$ is the total number of observations.

Compute the F-ratio:
$F = \frac{MSB}{MSW}$