Observation (i) pH (x;) Average mercury level (y;) 0.15 1 3 4 5 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-α+ βα + εί, i= 1, . , 10, N(0, o?) are independent. We have the following quantities: = E1 xi = 7.51, 0.328, Σ 572.71, ΣΗ = 1.8922, ΣΗ πυ-22.218. where ɛ; ~ n Li=1 y = ÷Li=1Yi =
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![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;)
1
3
6
7
9
10
8.2
8.4
7.0
7.2
7.3
6.4
9.1
5.8
7.6
8.1
0.15
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 ɛ; ~ N(0, o²) are independent. We have the following quantities: a = E1 ; = 7.51,
j = £i=1 Yi =
0.328, E1 x? = 572.71, 1 y? = 1.8922, -1 Tiyi = 22.218.
n
i=1
Some R output that may help.
> p1 <- c(0.01, 0.025, 0.05, 0.1, 0.9, 0.95, 0.975, 0.99)
> qt (p1, 8)
[1] -2.896 -2.306 -1.860 -1.397
> qt (p1, 9)
[1] -2.821 -2.262 -1.833 -1.383
1.397
1.860
2.306 2.896
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 ß).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F84d7412a-0252-4765-a3f4-d04a4d856ec4%2F41884e7a-04db-48b1-bd81-cc765f3fb782%2Fb09a77e_processed.png&w=3840&q=75)
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- A deficiency of the trace element selenium in the diet can negatively impact growth, immunity, muscle and neuromuscular function, and fertility. The introduction of selenium supplements to dairy cows is justified when pastures have low selenium levels. Authors of a research paper supplied the following data on milk selenium concentration (mg/L) for a sample of cows given a selenium supplement (the treatment group) and a control sample given no supplement, both initially and after a 9-day period. Initial Measurement Treatment Control 11.3 9.1 9.7 8.7 10.1 9.7 8.5 10.8 10.4 10.9 10.7 10.6 11.8 10.1 9.8 12.3 10.6 8.8 10.4 10.4 10.2 10.9 11.3 10.4 9.2 11.6 10.7 10.9 10.8 8.2 After 9 Days Treatment Control 138.3 9.4 104 8.8 96.4 8.8 89 10.1 88 9.7 103.8 8.7 147.3 10.3 97.1 12.3 172.6 9.4 146.3 9.5 99 8.3 122.3 8.9 103 12.5 117.8 9.1 121.5 93 (a) Use the given data for the treatment group to determine if…A deficiency of the trace element selenium in the diet can negatively impact growth, immunity, muscle and neuromuscular function, and fertility. The introduction of selenium supplements to dairy cows is justified when pastures have low selenium levels. Authors of a research paper supplied the following data on milk selenium concentration (mg/L) for a sample of cows given a selenium supplement (the treatment group) and a control sample given no supplement, both initially and after a 9-day period. Initial Measurement Treatment Control 11.4 9.1 9.6 8.7 10.1 9.7 8.5 10.8 10.2 10.9 10.6 10.6 11.9 10.1 9.9 12.3 10.7 8.8 10.2 10.4 10.3 10.9 11.4 10.4 9.3 11.6 10.6 10.9 10.9 8.3 After 9 Days Treatment Control 138.3 9.2 104 8.9 96.4 8.9 89 10.1 88 9.6 103.8 8.6 147.3 10.4 97.1 12.4 172.6 9.2 146.3 9.5 99 8.4 122.3 8.8 103 12.5 117.8 9.1 121.5 93 (a) Use the given data for the treatment group to determine if…Researchers interested in lead exposure due to car exhaust sampled the blood of 52 police officers subjected to constant inhalation of automobile exhaust fumes while working traffic enforcement in a primarily urban environment. The blood samples of these officers had an average lead concentration of 124.32 µg/l and a SD of 37.74 µg/l; a previous study of individuals from a nearby suburb, with no history of exposure, found an average blood level concentration of 35 µg/l. Based on your preceding result, without performing a calculation, would a 99% confidence interval for the average blood concentration level of police officers contain 35 µg/l? Based on your preceding result, without performing a calculation, would a 99% confidence interval for this difference contain 0? Explain why or why not.
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