Question

For his term project in biology, Robert believed he could increase the weight of mice by feeding them a hormone. Do his results, in Table 17.21 , support the claim that the hormone increases weight? $$ \begin{array}{c|c|c|c} \hline & \text { Weight increase } & \text { No weight increase } & \text { Total } \\ \hline \text { Fed hormone } & 120 & 30 & 150 \\ \hline \text { Not fed hormone } & 25 & 25 & 50 \\ \hline \text { Total } & 145 & 55 & 200 \\ \hline \end{array}$$

Short answer

Answer: Yes, Robert's results support the claim that the hormone increases weight in mice, as the Chi-Square Test Statistic (16.98) is greater than the critical value (3.84) at a 0.05 significance level and 1 degree of freedom.

Step-by-step solution

Identify Observed Frequencies

First, identify the observed frequencies from the given contingency table. The table is already provided with the observed frequencies: Weight increase (Fed hormone) = 120 No weight increase (Fed hormone) = 30 Weight increase (Not fed hormone) = 25 No weight increase (Not fed hormone) = 25

Calculate Expected Frequencies

Next, calculate the expected frequencies for each cell in the contingency table. To do this, multiply the row total by the column total and divide by the grand total. For example, to calculate the expected frequency for the cell with weight increase and fed hormone: Expected Frequency = (Row Total * Column Total) / Grand Total Weight increase (Fed hormone): (150 * 145) / 200 = 21750/200 = 108.75 No weight increase (Fed hormone): (150 * 55) / 200 = 8250/200 = 41.25 Weight increase (Not fed hormone): (50 * 145) / 200 = 7250/200 = 36.25 No weight increase (Not fed hormone): (50 * 55) / 200 = 2750/200 = 13.75

Calculate Chi-Square Test statistic

Now, calculate the Chi-Square Test statistic using the formula: Chi-Square = Σ [ (Observed Frequency - Expected Frequency)^2 / Expected Frequency] Chi-Square = [ (120 - 108.75)^2 / 108.75 ] + [ (30 - 41.25)^2 / 41.25 ] + [ (25 - 36.25)^2 / 36.25 ] + [ (25 - 13.75)^2 / 13.75 ] Chi-Square = [ 126.56 /108.75] + [ 127.69 /41.25] + [ 127.69 /36.25] + [126.56 /13.75] Chi-Square ≈ 1.16 + 3.09 + 3.53 + 9.20 = 16.98

Determine Degrees of Freedom

Calculate the degrees of freedom (df) for this test statistic: Degrees of Freedom = (number of rows - 1) * (number of columns - 1) Degrees of Freedom = (2 - 1) * (2 - 1) = 1

Compare Chi-Square Test Statistic to Critical Value

Now, compare the calculated Chi-Square Test Statistic (16.98) to the critical value from the Chi-Square distribution table at the desired significance level (usually 0.05) with 1 degree of freedom. The critical value at a 0.05 significance level and 1 degree of freedom is 3.84. Since 16.98 > 3.84, we can reject the null hypothesis (no relationship between hormone and weight increase) and conclude that there is a statistically significant relationship between feeding the hormone to mice and their weight increase. Therefore, Robert's results support the claim that the hormone increases weight in mice.

Key concepts

Expected Frequencies

Expected frequencies provide a theoretical count for what we would expect if there were no relationship between groups being tested. In the context of Robert's biology project, this means if the hormone had no effect on the mice's weight, what would the weight increase or no weight increase distributions look like? We calculate expected frequencies to use them later in the Chi-Square Test.

To calculate expected frequency, you can use the formula: \[\text{Expected Frequency} = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}} \]This formula is applied to each cell of the table.

For example, to calculate the expected frequency for the cell where mice were fed the hormone and showed a weight increase, you would determine:

• Row Total for mice fed hormone: 150
• Column Total for weight increase: 145
• Grand Total of all mice: 200
• Expected Frequency = (150 * 145) / 200 = 108.75

This helps in comparing what was observed in reality against what was expected under the assumption of no effect.

Observed Frequencies

Observed frequencies are the actual data collected from an experiment or study—what really happened. In the Chi-Square Test, these values are critical as they are compared against expected frequencies.

In Robert's project, the observed frequencies are obtained directly from the study of mice. They show how many mice gained weight when fed a hormone or not fed a hormone. For example:

• Mice Fed Hormone with Weight Increase: 120
• Mice Fed Hormone without Weight Increase: 30
• Mice Not Fed Hormone with Weight Increase: 25
• Mice Not Fed Hormone without Weight Increase: 25

These observed frequencies become essential data points in determining whether there is a statistically significant effect from feeding the hormone, helping to drive the conclusions of Robert's study forward.

Degrees of Freedom

In statistical analyses like the Chi-Square Test, degrees of freedom (df) refer to the number of categories that are independent and free to vary in the analysis. It is crucial because it influences the shape of the Chi-Square distribution and affects the critical value needed to determine significance.

In Robert's study, the degrees of freedom are calculated based on the number of rows and columns in the contingency table minus 1 for each.

Here’s how to calculate it:\[\text{Degrees of Freedom} = (\text{number of rows} - 1) \times (\text{number of columns} - 1)\]For his table, this becomes (2 - 1) * (2 - 1) = 1.

Why does this matter? Because with 1 degree of freedom, a critical value can be determined from the Chi-Square distribution table, allowing Robert to assess if his Chi-Square Test Statistic indicates a significant difference. This helps to validate his hypothesis about the hormone's effectiveness.