The following table lists the temperature (in Fahrenheit, ${ }^{\circ} F$ ) and the number of chirps per second for the striped ground cricket.
| Temperature ( x ) | Chirps per second (y) |
|---|---|
| 88.6 | 20 |
| 93.3 | 19.8 |
| 80.6 | 17.1 |
| 69.7 | 14.7 |
| 69.4 | 15.4 |
| 79.6 | 15 |
What's the appropriate $t$-table value that needs to be used in calculating the $95 \%$ c.i. for the true slope?
A. 2.571 (df=5)
B. 2.776 (df-4)
C. 2.015 (df=5)
D. $2.132( df =4)$
Answer (1)
2.776 ( df = 4 )
Explanation
Understanding the Problem We are given a table of temperature (x) and chirps per second (y) data for striped ground crickets. We want to find the appropriate t-table value for calculating the 95% confidence interval for the true slope. The degrees of freedom (df) is needed to find the t-table value. The degrees of freedom for the slope is n-2, where n is the number of data points.
Calculating Degrees of Freedom The degrees of freedom (df) is calculated as df = n − 2 , where n is the number of data points. In this case, we have n = 6 data points.
Determining df Substituting n = 6 into the formula, we get df = 6 − 2 = 4 .
Finding the t-table value We need to find the t-table value for a 95% confidence interval with df = 4 . This value corresponds to α /2 = 0.025 with df = 4 . Looking at the answer choices, we need to select the one that has df = 4 .
Final Answer The correct t-table value is 2.776 with df = 4 .
Examples
In statistical analysis, determining the appropriate t-table value is crucial for constructing confidence intervals, which estimate population parameters like the true slope in linear regression. For instance, in environmental science, you might want to understand the relationship between pollutant levels and plant growth. By calculating a confidence interval for the slope of the regression line, you can estimate the impact of pollutants on plant growth with a certain level of confidence. This helps in making informed decisions about environmental regulations and conservation efforts. Similarly, in medical research, confidence intervals are used to estimate the effect of a drug on patient outcomes, providing a range within which the true effect is likely to lie.
