Nadia collected the data in the table on the left using a CBR after a ball was
dropped.
(a) Enter the data into L1 and L2 using a TI-83 Plus. Graph the relation.
(b) Without looking at the graph, how do you know that the relation is not
linear?
(c) Use regression to find the equation of the quadratic of best fit.
(d) Is this equation a good model? Justify your answer.
You can use the TI-83 Plus to estimate different regression models, but
often the model does not perfectly fit the data. A common method in data
analysis is to transform one variable until the graph becomes linear. In this
case, observe that time increases by a constant, but height decreases by a
larger amount in each successive observation. This suggests a
transformation that compresses height or expands time.
(e) Calculate the square root of each entry in L2 (height) and store the
values in L3.
(f) Graph the square root of height versus time. Does this relation appear
to be linear? Calculate the correlation coefficient for the relation.
What can you conclude?
(g) Calculate the square of each time-value in L1 and store the results in
L3.
(h) Graph height versus time2. Does this relation appear to be linear?
Calculate the correlation coefficient. What can you conclude?
(i) Let h represent height and t represents
TIME(S) HEIGHT(M)
0.0 3.0
0.2 2.8
0.4 2.2
0.6 1.2
0.8 0.0
(a) To enter the data into L1 and L2 using a TI-83 Plus:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Edit" and press [ENTER].
3. Enter the values for time in L1 and the values for height in L2. The data should be entered as follows:
L1: {0.0, 0.2, 0.4, 0.6, 0.8}
L2: {3.0, 2.8, 2.2, 1.2, 0.0}
4. Once you have entered the data, press [STAT] again.
5. Use the arrow keys to highlight "Plot1" and press [ENTER].
6. Choose an appropriate graph type, such as a scatter plot, and select the relevant data columns (L1 and L2).
7. The TI-83 Plus will graph the relation.
(b) Without looking at the graph, you can analyze the data to determine if the relation is linear. In this case, the relation is not linear because the difference in height (dependent variable) is not constant for each unit increase in time (independent variable). The height decreases by a larger amount in each successive observation.
(c) To use regression to find the equation of the quadratic of best fit:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Calc" and press [ENTER].
3. Choose "QuadReg" from the list of regression models and press [ENTER].
4. Enter the relevant data columns (L1 and L2) and press [ENTER].
5. The TI-83 Plus will display the equation of the quadratic of best fit.
(d) To determine if the equation of the quadratic of best fit is a good model, you can analyze the residual plot and the coefficient of determination (R²). A residual plot should show no obvious patterns or trends, and a higher R² value (closer to 1) indicates a better fit of the model to the data.
(e) To calculate the square root of each entry in L2 (height) and store the values in L3:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Edit" and press [ENTER].
3. Enter the following command in L3: L3 = √(L2)
4. The TI-83 Plus will calculate the square root of each entry in L2 and store the resulting values in L3.
(f) To graph the square root of height versus time and calculate the correlation coefficient:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Plot1" and press [ENTER].
3. Choose an appropriate graph type, such as a scatter plot, and select the relevant data columns (L1 and L3).
4. The TI-83 Plus will graph the relation.
5. Press [STAT], use the right arrow key to highlight "Calc", and press [ENTER].
6. Select "LinReg(ax+b)" from the list of regression models and press [ENTER].
7. Enter the relevant data columns (L1 and L3) and press [ENTER].
8. The TI-83 Plus will display the correlation coefficient for the relation.
(g) To calculate the square of each time-value in L1 and store the results in L3:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Edit" and press [ENTER].
3. Enter the following command in L3: L3 = L1²
4. The TI-83 Plus will calculate the square of each entry in L1 and store the resulting values in L3.
(h) To graph height versus time² and calculate the correlation coefficient:
1. Press the [STAT] button.
2. Use the arrow keys to highlight "Plot1" and press [ENTER].
3. Choose an appropriate graph type, such as a scatter plot, and select the relevant data columns (L3 and L2).
4. The TI-83 Plus will graph the relation.
5. Press [STAT], use the right arrow key to highlight "Calc", and press [ENTER].
6. Select "LinReg(ax+b)" from the list of regression models and press [ENTER].
7. Enter the relevant data columns (L3 and L2) and press [ENTER].
8. The TI-83 Plus will display the correlation coefficient for the relation.
(i) It appears that the question was cut off. Please provide more information for the remaining part of the question.
look at the differences.
1st: -0.2, -0.6, -1.0, -1.2, ...
2nd: -0.4, -0.4, -0.4, ...
so, look for a linear relationship between y and x^2:
y = 3-5x^2
fits pretty well, at least at first