We are interested in using the pH of the lake water (which is easy to measure) to predict the average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes yielded the following data: Observation (i) pH (x;) Average mercury level (y¡) 0.15 1 3 4 6 7 8 10 8.2 8.4 7.0 7.2 7.3 6.4 9.1 5.8 7.6 8.1 0.04 0.40 0.50 0.27 0.81 0.04 0.83 0.05 0.19 Suppose we fit the data with the following regression model: Y-α+ βα;+ εi, i-1,. . . , 10 , where ɛ; ~ N(0, o²) are independent. We have the following quantities: = J= Σ% = 0.328, ΣΗ= 572.71 , ΣΗ 1.8922 , Σ" T;J; 22.218. Η ΣΗ = 7.51, n n i=1 i=1 Some R output that may help. > p1 <- c(0.01, 0.025, 0.05, 0.1, o.9, 0.95, 0.975, 0.99) > qt (p1, 8) [1] -2.896 -2.306 -1.860 -1.397 1.397 1.860 2.306 2.896 > qt (p1, 9) [1] -2.821 -2.262 -1.833 -1.383 1.383 1.833 2.262 2.821 (a) Find the ordinary least squares (OLS) estimates (denoted as â and B) of the regression coefficients (a and 3).
We are interested in using the pH of the lake water (which is easy to measure) to predict the average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes yielded the following data: Observation (i) pH (x;) Average mercury level (y¡) 0.15 1 3 4 6 7 8 10 8.2 8.4 7.0 7.2 7.3 6.4 9.1 5.8 7.6 8.1 0.04 0.40 0.50 0.27 0.81 0.04 0.83 0.05 0.19 Suppose we fit the data with the following regression model: Y-α+ βα;+ εi, i-1,. . . , 10 , where ɛ; ~ N(0, o²) are independent. We have the following quantities: = J= Σ% = 0.328, ΣΗ= 572.71 , ΣΗ 1.8922 , Σ" T;J; 22.218. Η ΣΗ = 7.51, n n i=1 i=1 Some R output that may help. > p1 <- c(0.01, 0.025, 0.05, 0.1, o.9, 0.95, 0.975, 0.99) > qt (p1, 8) [1] -2.896 -2.306 -1.860 -1.397 1.397 1.860 2.306 2.896 > qt (p1, 9) [1] -2.821 -2.262 -1.833 -1.383 1.383 1.833 2.262 2.821 (a) Find the ordinary least squares (OLS) estimates (denoted as â and B) of the regression coefficients (a and 3).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
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Question
![We are interested in using the pH of the lake water (which is easy to measure) to predict the
average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the
lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes
yielded the following data:
Observation (i)
pH (x;)
Average mercury level (y;) 0.15
1
3
4
6
7
8
9
10
8.2
8.4
7.0
7.2
7.3
6.4
9.1
5.8
7.6
8.1
0.04
0.40
0.50
0.27
0.81
0.04
0.83
0.05
0.19
Suppose we fit the data with the following regression model:
Y; = a + Bx; + Ei, i = 1, ..., 10,
where ɛi ~
N (0, o?) are independent. We have the following quantities: a = E=1 ¤i = 7.51,
j = E1 Yi = 0.328, 1 x = 572.71, 1 Y? = 1.8922, D-1 *iYi = 22.218.
i=1
ri=1
Some R output that may help.
> р1 <- с(0.01, 0.025, 0.05, 0.1, о.9, 0.95, 0.975, 0.99)
> qt (p1, 8)
[1] -2.896 -2.306 -1.860 -1.397
1.397
1.860
2.306
2.896
> qt (p1, 9)
[1] -2.821 -2.262 -1.833 -1.383
1.383
1.833
2.262
2.821
(a) Find the ordinary least squares (OLS) estimates (denoted as â and ß) of the regression
coefficients (a and B).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1076b49d-639e-46e6-94d5-ee66fa9d3246%2F9f0e0c8c-49d8-4e81-ac8f-e5d0a5175a6a%2Fiwxz05b_processed.png&w=3840&q=75)
Transcribed Image Text:We are interested in using the pH of the lake water (which is easy to measure) to predict the
average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the
lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes
yielded the following data:
Observation (i)
pH (x;)
Average mercury level (y;) 0.15
1
3
4
6
7
8
9
10
8.2
8.4
7.0
7.2
7.3
6.4
9.1
5.8
7.6
8.1
0.04
0.40
0.50
0.27
0.81
0.04
0.83
0.05
0.19
Suppose we fit the data with the following regression model:
Y; = a + Bx; + Ei, i = 1, ..., 10,
where ɛi ~
N (0, o?) are independent. We have the following quantities: a = E=1 ¤i = 7.51,
j = E1 Yi = 0.328, 1 x = 572.71, 1 Y? = 1.8922, D-1 *iYi = 22.218.
i=1
ri=1
Some R output that may help.
> р1 <- с(0.01, 0.025, 0.05, 0.1, о.9, 0.95, 0.975, 0.99)
> qt (p1, 8)
[1] -2.896 -2.306 -1.860 -1.397
1.397
1.860
2.306
2.896
> qt (p1, 9)
[1] -2.821 -2.262 -1.833 -1.383
1.383
1.833
2.262
2.821
(a) Find the ordinary least squares (OLS) estimates (denoted as â and ß) of the regression
coefficients (a and B).
Expert Solution
Step 1
The following data is given:
| Observations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| pH (x) | 8.2 | 8.4 | 7.0 | 7.2 | 7.3 | 6.4 | 9.1 | 5.8 | 7.6 | 8.1 |
| Mercury level (y) | 0.15 | 0.04 | 0.40 | 0.50 | 0.27 | 0.81 | 0.04 | 0.83 | 0.05 | 0.19 |
where,
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