Low doses of the insecticide toxaphene may cause weight gain in rats. A treatmen
ID: 3206710 • Letter: L
Question
Low doses of the insecticide toxaphene may cause weight gain in rats. A treatment group n = 20 rats were given toxaphene in their diet, while a control group of m = 8 rats were not given toxaphene. The average weight gain among the treatment group was 60 kg and the standard deviation was 30 kg. The average weight gain among the control rats was 10 kg and the standard deviation was 50 kg.
(a) Construct a 95% condence interval for the difference in mean weight gain between the treatment and control groups.
(b) Now let X represent the weight gain of a rat from the treatment group and let Y represent the weight gain of a rat from the control group.
Assume we know that
Using simulation, create a condence interval for the difference in mean weight gain. There are several ways to implement the simulation. Here is an overview/pseudocode of how to use a simple for loop. (The simulation should have 10,000 iterations)
Create a vector to hold 10,000 numeric values
for Loop over an index variable 10,000 times do
Compute the difference in means and store it in the vector
end for
Find the .025 and .975 quantiles of the 10,000 differences
x <- rnorm(20, 60, 30)
Compare the condence interval from the simulation with the condence interval that you com- puted in part 4a. Are the two intervals roughly of the same width? Explain any difference that you observe.
Explanation / Answer
Answer:
Low doses of the insecticide toxaphene may cause weight gain in rats. A treatment group n = 20 rats were given toxaphene in their diet, while a control group of m = 8 rats were not given toxaphene. The average weight gain among the treatment group was 60 kg and the standard deviation was 30 kg. The average weight gain among the control rats was 10 kg and the standard deviation was 50 kg.
95% CI = (ar x1-ar x2) ± z*se
Hypothesis Test: Independent Groups (z-test)
g1
g2
60
10
mean
30
50
std. dev.
20
8
n
50.000
difference (g1 - g2)
18.908
standard error of difference
0
hypothesized difference
2.64
z
.0082
p-value (two-tailed)
12.942
confidence interval 95.% lower
87.058
confidence interval 95.% upper
37.058
margin of error
95% CI = (12.942, 87.058)
(b) Now let X represent the weight gain of a rat from the treatment group and let Y represent the weight gain of a rat from the control group.
Assume we know that
X N ( x = 60; x = 30)
Y N ( y = 10; y = 50)
Using simulation, create a condence interval for the difference in mean weight gain. There are several ways to implement the simulation. Here is an overview/pseudocode of how to use a simple for loop. (The simulation should have 10,000 iterations)
Create a vector to hold 10,000 numeric values
for Loop over an index variable 10,000 times do
Draw 20 samples from N ( x = 60; x = 30)
Draw 8 samples from N ( y = 10; y = 50)
Compute the difference in means and store it in the vector
end for
Find the .025 and .975 quantiles of the 10,000 differences
If you are using R, you can draw 20 random samples from the N ( x = 60; x = 30) distribution and store them in a vector called x like this
x <- rnorm(20, 60, 30)
x <- rnorm(20, 60, 30)
y <- rnorm(8, 10, 50)
t.test(x,y)
x <- rnorm(20, 60, 30)
> y <- rnorm(8, 10, 50)
> t.test(x,y)
Welch Two Sample t-test
data: x and y
t = 5.9075, df = 20.583, p-value = 7.905e-06
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
( 37.28990, 77.88516)
sample estimates:
mean of x mean of y
64.126079 6.538549
Compare the condence interval from the simulation with the condence interval that you com- puted in part 4a. Are the two intervals roughly of the same width? Explain any difference that you observe.
Confidence interval from the simulation is narrow than part 4a. The difference is due to the simulation result is more appropriate.
Hypothesis Test: Independent Groups (z-test)
g1
g2
60
10
mean
30
50
std. dev.
20
8
n
50.000
difference (g1 - g2)
18.908
standard error of difference
0
hypothesized difference
2.64
z
.0082
p-value (two-tailed)
12.942
confidence interval 95.% lower
87.058
confidence interval 95.% upper
37.058
margin of error
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