NAMEGAME2 POSITION RECALL 2 0.04 2 0.37 2 1.00 2 0.99 2 0.79 2 0.99 2 0.52 2 0.4
ID: 3178083 • Letter: N
Question
NAMEGAME2
POSITION RECALL
2 0.04
2 0.37
2 1.00
2 0.99
2 0.79
2 0.99
2 0.52
2 0.46
2 0.99
2 0.71
2 0.16
2 0.62
2 0.55
2 0.34
2 0.86
2 0.25
2 0.16
2 0.58
3 0.95
3 0.54
3 0.37
3 0.61
3 0.71
3 0.60
3 0.82
3 0.82
3 0.99
3 0.54
3 0.27
3 0.65
3 0.70
3 0.42
3 0.99
3 0.41
3 0.88
3 0.96
4 0.36
4 0.82
4 0.91
4 0.89
4 0.46
4 0.52
4 0.86
4 0.61
4 0.99
4 0.99
4 0.99
4 0.69
4 0.87
4 0.71
4 0.98
4 0.38
4 0.97
4 0.32
5 0.97
5 0.27
5 0.39
5 0.99
5 0.99
5 0.99
5 0.99
5 0.73
5 0.99
5 0.85
5 0.01
5 0.76
5 0.99
5 0.15
5 0.33
5 0.85
5 0.26
5 0.93
6 0.27
6 0.59
6 1.00
6 0.99
6 0.78
6 0.62
6 0.93
6 0.68
6 0.57
6 0.49
6 0.94
6 0.73
6 0.99
6 0.28
6 0.98
6 0.96
6 0.70
6 0.44
7 0.99
7 0.99
7 0.82
7 0.99
7 0.20
7 0.25
7 0.61
7 0.66
7 0.77
7 0.47
7 0.89
7 0.51
7 0.94
7 0.99
7 0.58
7 0.99
7 0.89
7 0.53
8 0.64
8 0.99
8 0.81
8 0.51
8 0.75
8 0.49
8 0.81
8 0.85
8 0.99
8 0.96
8 0.48
8 0.84
8 0.63
8 0.63
8 0.93
8 0.40
8 0.99
8 0.71
9 0.99
9 0.56
9 0.99
9 0.92
9 0.71
9 0.82
9 0.99
9 0.88
9 0.99
9 0.99
9 0.96
9 0.99
9 0.90
9 0.72
9 0.88
9 0.46
9 0.54
9 0.99
(a) Find a 99% confidence interval for the mean recall proportion for students in the fifth position during the "name game". Interpret result.
(b)Find a 99% prediction interval for the recall proportionof a particular student in the fifth position during the "name game". Interpret the result.
(c) Compare the two intervals, parts a and b. Which interval is wider? will this always be the case? Explain.
Explanation / Answer
I am using R software to solve this problem.
First, we need to import this data into R. I have create a csv file with the data given. We can import . csv file in R with below command:
InputData <- read.csv("Input1.txt")
#Printing structure of InputData
> str(InputData)
'data.frame': 144 obs. of 2 variables:
$ POSITION: int 2 2 2 2 2 2 2 2 2 2 ...
$ RECALL : num 0.04 0.37 1 0.99 0.79 0.99 0.52 0.46 0.99 0.71 ...
#Fit a linear model
Model <- lm(RECALL ~ POSITION, data=InputData)
summary(Model)
a) For predicting 99% confidence interval for the mean recall proportion for students in the fifth position, we can use the predict function with interval value as "confidence" as mentioned below:
predict(Model,data.frame(POSITION=5),interval = "confidence")
fit lwr upr
0.7025529 0.6596994 0.7454064
We interpret this to mean that 99% of intervals of this form will contain the true value of RECALL on an average.
b) For predicting 99% prediction interval for the mean recall proportion for students in the fifth position, we can use the predict function with interval value as "prediction" as mentioned below:
predict(Model,data.frame(POSITION=5),interval = "prediction")
fit lwr upr
0.7025529 0.19831 1.206796
We interpret this to mean that 99% of intervals of this form will contain the true value of RECALL for a particular student in 5th position.
c) Prediction interval is wider than confidence interval. Yes, Prediction intervals are always wider than confidence intervals, because they incorporate both the error in the estimate for f(x) (the reducible error) and the uncertainty as to how much an individual point will differ from the population regression plane (the irreducible error)
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