The following measurements were recorded for the drying time, in hours, of certa
ID: 3142675 • Letter: T
Question
The following measurements were recorded for the drying time, in hours, of certain brands of latex paint: 3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 3.7 2.8 4.4 4.0 5.2 3.0 4.8 a. Assuming that the measurements represent a random sample from a normal population, find a 99% confidence interval for the mean of the drying time. b. Find a 95% production interval for a new observation of drying time. c. Assuming the latex paint comes from two different brands (both normally distributed) and samples are collected as shown in the following table, calculate a 95% confidence interval for for mu_1 - mu_2 if we know that sigma^2_1 = 0.85 and sigma^2_2 = 0.57. Brand A 3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 3.7 2.8/Brand B 4.4 4.0 5.2 3.0 4.8 d. With the same data collection and distribution assumptions as in part c., calculate a 95% confidence interval for mu_1 - mu_2 if the population variances are unknown but equal.Explanation / Answer
Solution
CONFIDENCE INTERVAL FOR MEAN
Let X = drying time of the paint. We are given X ~ N(µ, 2).
100(1 – ) % confidence interval for µ when 2 is unknown is: {Xbar ± (s/n)(t/2)}, where
Xbar = sample mean,
= population standard deviation,
s = sample standard deviation,
n = sample size and
t/2 = upper (/2) % point of t-Distribution with (n - 1) degrees of freedom..
Given n =
15
=
0.01
Xbar =
3.766667
s =
0.982465
t/2 =
2.976843
99% CI for
3.766667
±
0.75514
Lower Bound =
3.011527
Upper Bound =
4.521807
ANSWER
Details of
Excel
computations
Raw Data
i
xi
n
15
1
3.1
Xbar
3.766667
2
2.5
s
0.982465
3
4.8
s^2
0.965238
4
2.9
5
3.6
6
2.8
7
3.3
8
5.6
9
3.7
10
2.8
11
4.4
12
4
13
5.2
14
3
15
4.8
PREDICTION INTERVAL FOR SINGLE OBSERVATION
100(1 – ) % prediction interval for new observation of drying time = {Xbar ± (s.Z/2)}, where Z/2 is the upper /2 percent point of N(0, 1).
So, 95% prediction interval for new observation of drying time = {3.766667 ± (0.982465 x 1.96)}
= 3.767 ± 1.926 ANSWER
Given n =
15
=
0.01
Xbar =
3.766667
s =
0.982465
t/2 =
2.976843
99% CI for
3.766667
±
0.75514
Lower Bound =
3.011527
Upper Bound =
4.521807
ANSWER
Details of
Excel
computations
Raw Data
i
xi
n
15
1
3.1
Xbar
3.766667
2
2.5
s
0.982465
3
4.8
s^2
0.965238
4
2.9
5
3.6
6
2.8
7
3.3
8
5.6
9
3.7
10
2.8
11
4.4
12
4
13
5.2
14
3
15
4.8
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