Finding Formulas for Approximately Linear Data

In the previous labs you have worked with linear equations. You are now going to use what you learned in those labs to find equations that describe the relationship between points that appear to be roughly in a straight line.

The data that you will use in this lab shows the relationship between the number of cigarettes bought and the death rates from cancer in different states during 1960. This data was taken from http://lib.stat.cmu.edu/DASL/Datafiles/cigcancerdat.html.

Bladder cancer

$x$	$y$	State
30.34	3.46	AK
18.20	2.90	AL
18.24	2.99	AR
25.82	3.52	AZ
28.60	4.46	CA
31.10	5.11	CT
40.46	5.60	DC
33.60	4.78	DE
28.27	4.46	FL
22.12	4.23	IA
20.10	3.08	ID
27.91	4.75	IL
26.18	4.09	IN
21.84	2.91	KS
23.44	2.86	KY
21.58	4.65	LA
26.92	4.69	MA
25.91	5.21	MD
28.92	4.79	ME
24.96	5.27	MI
22.06	3.72	MN
27.56	4.04	MO
16.08	3.06	MS
23.75	3.95	MT
19.96	2.89	ND
23.32	3.72	NE
28.64	5.98	NJ
21.16	2.90	NM
42.40	6.54	NV
29.14	5.30	NY
26.38	4.47	OH
23.44	2.93	OK
23.78	4.89	PA
29.18	4.99	RI
18.06	3.25	SC
20.94	3.64	SD
20.08	2.94	TN
22.57	3.21	TX
14.00	3.31	UT
25.89	4.63	VT
21.17	4.04	WA
21.25	5.14	WI
22.86	4.78	WV
28.04	3.20	WY

Question 1. The numbers in the $x$ column of this table tell us how many hundreds of cigarettes were sold per person in each state during 1960. For example, in Alabama (AL), the number in the $x$ column is 18.2. This means that for each person living in Alabama in 1960, 1,820 cigarettes were sold (that is, 18.2 hundreds).

In 1960, how many hundreds of cigarettes were sold per person in Arizona (AZ)?

Which state sold the most cigarettes per person?

Which state sold the fewest cigarettes per person?

Question 2. The numbers in the $y$ column tell us how many people died of bladder cancer for each 100,000 people living in the state. For example, in Alabama (AL), for each 100,000 people, 2.9 people died of bladder cancer.

In 1960, for every 100,000 people living in Idaho (ID), how many died of bladder cancer?

Which state had the highest death rate due to bladder cancer?

Which state had the lowest death rate due to bladder cancer?

Finding the "best fit" line

Just by looking at the table, it's hard to tell whether there is a relationship between the number of cigarettes sold in each state and the number of deaths due to bladder cancer. Another way to look at the data in the table is to graph it. You can see a graph showing the points that represent the data to the left.

When the points on a grid are not all on a straight line, but seem to have a somewhat linear pattern, you can find a line that is "best fit" (closest) to the pattern.

The number 2.981, shown to the right of the graph, is called the "root-mean-square error" for the graphed line ($y=0.05x$). This number tells you how far away the line is from the points (a smaller number means the line is a better “fit” to the points).

Question 3. Use both sliders in the lower left portion of your screen to change the values of $m$ and $b$ in the equation $y=mx+b$, and find an equation for a line that is a better approximation of the points on the grid (i.e. has a smaller root-mean-square error). The equation for the line you are looking at is written below the grid. Try to make the root-mean-square error smaller than 0.69.

Lung cancer

$x$	$y$	State
30.34	25.88	AK
18.20	17.05	AL
18.24	15.98	AR
25.82	19.80	AZ
28.60	22.07	CA
31.10	22.83	CT
40.46	27.27	DC
33.60	24.55	DE
28.27	23.57	FL
22.12	16.59	IA
20.10	13.58	ID
27.91	22.80	IL
26.18	20.30	IN
21.84	16.84	KS
23.44	17.71	KY
21.58	25.45	LA
26.92	22.04	MA
25.91	26.48	MD
28.92	20.94	ME
24.96	22.72	MI
22.06	14.20	MN
27.56	20.98	MO
16.08	15.60	MS
23.75	19.50	MT
19.96	12.12	ND
23.32	16.70	NE
28.64	25.95	NJ
21.16	14.59	NM
42.40	23.03	NV
29.14	25.02	NY
26.38	21.89	OH
23.44	19.45	OK
23.78	12.11	PA
29.18	23.68	RI
18.06	17.45	SC
20.94	14.11	SD
20.08	17.60	TN
22.57	20.74	TX
14.00	12.01	UT
25.89	21.22	VT
21.17	20.34	WA
21.25	20.55	WI
22.86	15.53	WV
28.04	15.92	WY

Question 4. Let's look at deaths due to lung cancer. The numbers in the $x$ column of this table are the same as before, but the numbers in the $y$ column tell us how many people died of lung cancer for each 100,000 people living in the state.

For each 100,000 people living in Kansas (KS) in 1960, how many died of lung cancer in 1960?

Which state (or district) had the highest death rate due to lung cancer?

Which state had the lowest death rate due to lung cancer?

Question 5. Find the point on the grid that represents the cigarette sales and lung cancer death rates for Nevada (NV) in 1960. Where is this point compared to the other points on the grid? What does that mean?

Does the graph indicate that there is a connection between the number of cigarettes sold and the number of deaths from lung cancer? Explain.

Question 6. Slide $m$ and $b$ to find an equation for a line that is a better approximation of the points on the graph. Your root-mean-square error should be smaller than 3.2.

What value for $m$ did you find?

What value for $b$ did you find?

What is your equation for $y$?

What is the root-mean-square error for your line?

Leukemia

$x$	$y$	State
30.34	4.90	AK
18.20	6.15	AL
18.24	6.94	AR
25.82	6.61	AZ
28.60	7.06	CA
31.10	7.20	CT
40.46	7.08	DC
33.60	6.45	DE
28.27	6.07	FL
22.12	7.69	IA
20.10	6.62	ID
27.91	7.27	IL
26.18	7.00	IN
21.84	7.42	KS
23.44	6.41	KY
21.58	6.71	LA
26.92	6.89	MA
25.91	6.81	MD
28.92	6.24	ME
24.96	6.91	MI
22.06	8.28	MN
27.56	6.82	MO
16.08	6.08	MS
23.75	6.90	MT
19.96	6.99	ND
23.32	7.80	NE
28.64	7.12	NJ
21.16	5.95	NM
42.40	6.67	NV
29.14	7.23	NY
26.38	7.38	OH
23.44	7.46	OK
23.78	6.83	PA
29.18	6.35	RI
18.06	5.82	SC
20.94	8.15	SD
20.08	6.59	TN
22.57	7.02	TX
14.00	6.71	UT
25.89	6.56	VT
21.17	7.48	WA
21.25	6.73	WI
22.86	7.38	WV
28.04	5.78	WY

Question 7. This time, let's analyze the relationship between the number of cigarettes sold and the number of people who died from leukemia.

Which state is represented by the highest dot on this graph?

What does it mean (about cigarettes or leukemia deaths) to be the highest dot?

Does the graph indicate that there is a connection between cigarettes sold and deaths from leukemia? Explain.

Question 8. Use the sliders for $m$ and $b$ to find a line with the smallest root-mean-square error you can (you should get below 0.66).

What value for $m$ did you find?

What value for $b$ did you find?

What is your equation for $y$?

What is the root-mean-square error for your line?

Kidney cancer

$x$	$y$	State
30.34	4.32	AK
18.20	1.59	AL
18.24	2.02	AR
25.82	2.75	AZ
28.60	2.66	CA
31.10	3.35	CT
40.46	3.13	DC
33.60	3.36	DE
28.27	2.41	FL
22.12	2.90	IA
20.10	2.46	ID
27.91	2.95	IL
26.18	2.81	IN
21.84	2.88	KS
23.44	2.13	KY
21.58	2.30	LA
26.92	3.03	MA
25.91	2.85	MD
28.92	3.22	ME
24.96	2.97	MI
22.06	3.54	MN
27.56	2.55	MO
16.08	1.77	MS
23.75	3.43	MT
19.96	3.62	ND
23.32	2.92	NE
28.64	3.12	NJ
21.16	2.52	NM
42.40	2.85	NV
29.14	3.10	NY
26.38	2.95	OH
23.44	2.45	OK
23.78	2.75	PA
29.18	2.84	RI
18.06	2.05	SC
20.94	3.11	SD
20.08	2.18	TN
22.57	2.69	TX
14.00	2.20	UT
25.89	3.17	VT
21.17	2.78	WA
21.25	2.34	WI
22.86	3.28	WV
28.04	2.66	WY

Question 9. You are looking at the data for deaths from kidney cancer. Use the sliders to find the line with the smallest root-mean-square error that you can (you should get below 0.49).

What value for $m$ did you find?

What value for $b$ did you find?

What is your equation for $y$?

What is the root-mean-square error for your line?

Question 10. Does the graph indicate that there is a connection between cigarettes sold and deaths from kidney cancer? Explain.

The meaning of the slope

Question 11. From the table below, you can see that in the case of bladder cancer, the slope of the line is 0.13. This means that for every 100 cigarettes sold per person in 1960, the death rate by bladder cancer increased by about 0.13 people per 100,000 population.

Disease	slope ($m$)	$y$-intercept ($b$)	Equation	Error
Bladder Cancer	0.13	0.88	$y=0.13x+0.88$	0.6793
Lung Cancer
Leukemia
Kidney Cancer

What does $m$ tell you in the case of leukemia?

What does $m$ tell you in the case of lung cancer?

What does $m$ tell you in the case of kidney cancer?

Which of these four death rates appears to be most affected by cigarette sales?

Which of these four death rates appears to be least affected by cigarette sales?