Simple Linear Regression

Biologists have speculated that the frequency of insects chirping is related to the temperature in their environment at the time. To test this relationship, they identified 15 crickets and recorded their chirps over a period of one hour, and calculated their average chirps per second. They simultaneously recorded the temperature over that same hour. The data from this study are in the table below. The researchers want to know if they can predict with statistical accuracy the temperature if they know the number of cricket chirps per second.

Cricket	Chirps per second	Temperature
1	20.0	88.6
2	16.0	71.6
3	19.8	93.3
4	18.4	84.3
5	17.1	80.6
6	15.5	75.2
7	14.7	69.4
8	17.1	82.0
9	15.4	69.4
10	16.2	83.3
11	15.0	79.6
12	17.2	82.6
13	16.0	80.6
14	17.0	83.5
15	14.4	76.3

1. What is the independent variable?
2. What is the dependent variable?
3. State the null and the alternative hypotheses.
4. Open SPSS and create three variables. The first is the id code for each cricket so that is a Nominal Scale. Then enter the two variables of interest in the study: Chirps per second and Temperature. Make sure that the Chirps variable and the Temperature variable are both marked Scale under Measure.
5. Enter the data above, and save when finished.
6. First, create a graph to confirm that the variables are linearly related. Click on Graphs in the top menu, and then on Chart Builder. You may get a warning to make sure your variables are properly labeled. If you did as I instructed in Step 4, then you're good to go, and click OK.
7. Your variables will appear in a box on the upper left, and a blank box will be on the right of that. Under those boxes on the left is a "Choose from:" menu. Click on "Scatter/Dot"; then click on the example of a scatter plot and drag it to the empty box above it. A straight line of dots will appear because it doesn't yet know your variables.
8. Click on your chirps variable and drag it to the x-axis (it's the independent variable). Click on your temperature variable and drag it to the y-axis (it's the dependent variable). Click OK, and a scatterplot of the data will appear in your output.
9. Do the variables appear to have a linear relationship (it will be extremely rare to see a straight line. That's not the question)?
10. To run a regression, click on Analyze/Regression/Linear
11. Move the independent and the dependent variables into the boxes marked independent and dependent.
12. Click on Statistics, and make sure Model fit box is checked and under Regression Coefficients, Estimates is checked. Click OK.
13. Several tables of output appear. Supply the following information:

1. Under Model Summary, what is R Square?
2. What percentage of the variance in the data is explained by the regression line? (Hint: multiply R squared by 100 to turn it into a percentage.)
3. Under Coefficients, what is the Unstandardized Beta for your chirps variable? (This is the slope of the line; the Constant is the y-intercept of the line.)?
4. What is the t score for your chirps variable?
5. What is the p-value for the chirps variable? (it's under Sig. column)
6. What is your conclusion (based on your p-value compared to alpha = .01) about the researchers' question about being able to predict the temperature if we know the chirps per second of crickets?
7. Save the scatterplot and your output and submit with your paragraph explained below in h.

1. Write up the results. Here is a model. You can't just fill in the blanks. Type up a paragraph with your results. I have bolded what you need to insert and put them into parentheses. Do not put those in your written paper with the exception of ( = XXX, p = XXX). That does go in parentheses but don't bold.

To investigate (STATE THE RESEARCH QUESTION), a simple linear regression was conducted. The predictor variable was (THE INDEPENDENT VARIABLE), and the outcome was (THE DEPENDENT VARIABLE). The predictor variable was found to be statistically significant ( = XXX, p = XXX), indicating that for every one unit increase in (INDEPENDENT VARIABLE) the (DEPENDENT VARIABLE) changed by (+/- BETA FOR INDEPENDENT VARIABLE) units. The model explained approximately (ANSWER TO 13b ABOVE) of the variability in (DEPENDENT VARIABLE). Therefore, the null hypothesis is rejected, and the alternative is retained. / OR Therefore, the null hypothesis failed to be rejected, and it is retained. (CHOOSE OF THE TWO PREVIOUS SENTENCES, WHICHEVER IS CORRECT.)

Model results paragraph using different data.

To investigate if depression symptoms could be predicted from the level of domestic violence a woman experienced, a simple linear regression was conducted. The predictor variable was domestic violence experiences, and the outcome was depression. The predictor variable was found to be statistically significant ( = 5.493, p = .004), indicating that for every one unit increase in domestic violence experiences the level of depression changed by +5.403 units. The model explained approximately 6.0% of the variability in the depression measure. Therefore, the null hypothesis is rejected, and the alternative is retained.