Questions: A study gave data on body mass (in grams) from representative samples of lizards from disturbed habitats on each of four islands in the Bahamas (Bimini, Great Exuma, Harbor, and New Providence). Summary statistics are given in the accompanying table. Island Sample Mean Weight Sample Standard Deviation Sample Size ------------ Bimini 4.32 0.87 20 Great Exuma 7.21 1.31 18 Harbor 5.85 1.00 20 New Providence 5.23 1.12 20 N=78, grand total T=437.78, and x̄=5.61 Does this data provide convincing evidence that the mean mass for this species of lizards is not the same for all four island lizard populations? You can assume that the mass distribution is approximately normal for each island. (Let μ1, μ2, μ3, and μ4 be true mean mass for each of four island lizard populations. Use α=0.05.)

A study gave data on body mass (in grams) from representative samples of lizards from disturbed habitats on each of four islands in the Bahamas (Bimini, Great Exuma, Harbor, and New Providence). Summary statistics are given in the accompanying table.

Island Sample Mean Weight Sample Standard Deviation Sample Size
------------
Bimini 4.32 0.87 20
Great Exuma 7.21 1.31 18
Harbor 5.85 1.00 20
New Providence 5.23 1.12 20

N=78, grand total T=437.78, and x̄=5.61

Does this data provide convincing evidence that the mean mass for this species of lizards is not the same for all four island lizard populations? You can assume that the mass distribution is approximately normal for each island. (Let μ1, μ2, μ3, and μ4 be true mean mass for each of four island lizard populations. Use α=0.05.)

Transcript text: A study gave data on body mass (in grams) from representative samples of lizards from disturbed habitats on each of four islands in the Bahamas (Bimini, Great Exuma, Harbor, and New Providence). Summary statistics are given in the accompanying table. \begin{tabular}{|c|c|c|c|} \hline Island & Sample Mean Weight & Sample Standard Deviation & Sample Size \\ \hline Bimini & 4.32 & 0.87 & 20 \\ \hline Great Exuma & 7.21 & 1.31 & 18 \\ \hline Harbor & 5.85 & 1.00 & 20 \\ \hline New Providence & 5.23 & 1.12 & 20 \\ \hline \end{tabular} $N=78$, grand total $T=437.78$, and $\overline{\bar{x}}=5.61$ Does this data provide convincing evidence that the mean mass for this species of lizards is not the same for all four island lizard populations? You can assume that the mass distribution is approximately normal for each island. (Let $\mu_{1}, \mu_{2}, \mu_{3}$, and $\mu_{4}$ be true mean mass for each of four island lizard populations. Use $\alpha=0.05$.)

Solution

Solution Steps

Step 1: Calculate Sum of Squares Between Groups

The sum of squares between groups ($SS_{between}$) is calculated as follows:

\[ SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 83.4015 \]

Step 2: Calculate Sum of Squares Within Groups

The sum of squares within groups ($SS_{within}$) is calculated as:

\[ SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 = 0.0 \]

Step 3: Calculate Mean Squares

The mean square between groups ($MS_{between}$) and mean square within groups ($MS_{within}$) are calculated as follows:

\[ MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{83.4015}{3} = 27.8005 \]

\[ MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{0.0}{74} = 0.0 \]

Step 4: Calculate F-statistic

The F-statistic is calculated using the mean squares:

\[ F = \frac{MS_{between}}{MS_{within}} = \frac{27.8005}{0.0} = 2.1732144308173038 \times 10^{31} \]

Step 5: Calculate P-value

The p-value is calculated as:

\[ P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(2.1732144308173038 \times 10^{31}; 3, 74) = 0.0 \]

Step 6: Conclusion

The degrees of freedom between groups is $df_{between} = 3$ and the degrees of freedom within groups is $df_{within} = 74$. The calculated F-statistic is $F = 2.1732144308173038 \times 10^{31}$ and the p-value is $P = 0.0$.

Since the p-value $P < \alpha = 0.05$, we reject the null hypothesis. This provides convincing evidence that the mean mass for this species of lizards is not the same for all four island lizard populations.

Final Answer

$\boxed{\text{Reject the null hypothesis: There is convincing evidence that the mean mass for this species of lizards is not the same for all four island lizard populations.}}$

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