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Temperature and Jumping Performance A study was done on spotted grass frogs to see whether temperature affects their jumping performance. Twenty frogs were randomly selected to be kept under a temperature-regulated structure (TRS). Twenty similar frogs were randomly selected and kept separately in normal atmosphere (NA). The selected frogs were of the same age range. The frogs, were observed for 30 days. Jumping performance improved in two NA frogs whereas the performance improved in 15 TRS frogs. a. What percentage of the NA frogs and the TRS frogs in the sample showed improved jumping performance? Compare these percentages and comment. b. Create a two-way table showing the observed values. Label the columns (across the top) with TRS and NA. c. Test the hypothesis that the temperature is associated with jumping performance (at the \(0.05\) level of significance).

Step by step solution

01

Calculation of Percentages

The number of NA frogs showing improvement is 2 out of 20. Therefore, the percentage of NA frogs showing improvement can be calculated as \(\frac{2}{20}\) * 100 which is 10%. \n Similarly, The number of TRS frogs showing improvement is 15 out of 20. So, the percentage of TRS frogs showing improved performance can be calculated as \(\frac{15}{20} \) * 100 which equals 75%.

02

Creating a two-way table

A two-way table can be created using the given data: \n \[\begin{{tabular}}{{|c|c|c|}}\hline & TRS & NA \\\hlineImproved & 15 & 2 \\\hlineNot Improved & 5 & 18 \\\hline\end{{tabular}}\]

03

Formulating the hypothesis

The hypotheses for this problem will be: \n Null Hypothesis (\(H_0\)): There is no association between temperature and jumping performance. \n Alternate Hypothesis (\(H_a\)): There is an association between temperature and jumping performance.

04

Testing the Hypothesis

Using a chi-square test for independence, we can check for the association. Here, we need to calculate the expected frequencies for each cell in the two-way table, find out the chi-square value and then compare it with the critical value for significance level 0.05. If the calculated chi-square value is higher than the critical value, we reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about populations based on sample data. It helps determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. In our case, we want to know if temperature has any effect on the jumping performance of frogs.

We start by formulating two hypotheses:

• The **Null Hypothesis ( H_0 )**: This is a statement suggesting that there is no effect or association between two variables. Here, it means that temperature does not impact jumping performance.
• The **Alternate Hypothesis ( H_a )**: Contrary to the null, this suggests there is some effect or association. In our exercise, it means temperature is associated with jumping performance.

To test these hypotheses, we use statistical tools, like the chi-square test. We determine the likelihood that observed outcomes are due to chance or if they significantly differ from what was expected. If the observed data vastly differ from what was expected under the null hypothesis, we may reject the null in favor of the alternative.

Two-Way Table

A two-way table, also known as a contingency table, is a visual representation of the frequency distribution of variables. It's particularly useful in categorical data analysis, helping to observe the interaction between two variables.

In our exercise, we created a two-way table to display the association between temperature and the frogs' jumping performance. The table is organized in rows and columns, showing the number of frogs that improved or didn’t improve under Normal Atmosphere (NA) and Temperature Regulated Structure (TRS).

• Rows represent the different outcomes (Improved, Not Improved).
• Columns signify the experimental groups (TRS, NA).

This table helps in visualizing data clearly, facilitating the understanding of any association or lack thereof between the two factors—temperature and performance.

Temperature and Performance

Understanding the link between environmental factors, like temperature, and biological performance is critical in numerous fields such as ecology and physiology. For the study with frogs, temperatures possibly impacted their ability to perform better. In our scenario, frogs kept in the Temperature Regulated Structure (TRS) showed a higher improvement rate in their jumping performance compared to those kept in Normal Atmosphere (NA). Specifically, 75% of TRS frogs improved as opposed to just 10% of NA frogs.
Temperature can influence metabolic rates, muscle function, and overall energy levels, contributing to performance variations. By assessing percentages and statistical associations, we deduce whether such influences in our study, are significant or could have occurred by chance—a core query in hypothesis testing.