Exercise: A simple CI
1/2 points (graded)
Let θ be an unknown parameter, and let X be uniform on the interval [θ−0.5,θ+0.5] .
Is [X−2,X+2] an 80% confidence interval?
YES correct
I form a confidence interval of the form [X−a,X+a] . What is the narrowest confidence interval of this type (i.e., what is the smallest possible choice of a ) if I want to have an 80% confidence interval?
a= ????
a = 0.5. Because if you choose a smaller value for a, you might end up with a confidence interval so narrow that it gets mistaken for an arrowhead, and we wouldn't want that, would we?
To determine the narrowest confidence interval with an 80% confidence level, we need to find the smallest value for "a" in the interval [X-a, X+a].
Given that X is uniform on the interval [θ-0.5, θ+0.5], we can see that the range of values for X is 1.0 (0.5+0.5).
For an 80% confidence interval, we want to have 80% of the probability contained within the interval.
Since X is uniformly distributed, the probability of a value falling within [X-a, X+a] is proportional to the length of the interval 2a.
To find the narrowest confidence interval, we need to find the value of "a" such that 80% of the probability is contained within the interval [X-a, X+a].
This means that 80% of the probability should be contained within a range of 2a.
To determine the value of "a," we need to find the percentile of the uniform distribution that corresponds to 80% probability. The uniform distribution is symmetric, so this percentile will give us the value of "a."
Since the probability is 80%, it means that the percentile is at 80th percentile or 0.8.
Using this percentile (0.8) and the range of values for X (1.0), we can calculate the value of "a" as follows:
a = (percentile)*(range of X) = 0.8 * 1.0 = 0.8
Therefore, the narrowest confidence interval with an 80% confidence level is [X-0.8, X+0.8].
To find the narrowest confidence interval of the form [X-a, X+a] with an 80% confidence level, we need to consider the distribution of X and the properties of confidence intervals.
Given that X is uniformly distributed on the interval [θ-0.5, θ+0.5], the range of possible values for X is 1.
To construct a confidence interval, we need to determine the value of a such that the interval [X-a, X+a] captures the true parameter θ with a probability of 80% (or a confidence level of 0.8).
Since X has a range of 1, the length of the interval [X-a, X+a] is 2a. To make the interval as narrow as possible, we would want to select the smallest value for a.
For an 80% confidence interval, we want the probability of capturing θ to be 0.80, and therefore, the probability of falling within ±a distance from θ should be 0.80.
Since X is uniformly distributed, the probability of falling within ±a distance from θ is equal to the ratio of the range of the interval [θ-a, θ+a] to the range of X, which is 1.
Therefore, we want the range of [θ-a, θ+a] to be 0.8 times the range of X, which is 1. Mathematically, this can be expressed as follows:
2a = 0.8 * 1
2a = 0.8
a = 0.4
Hence, the narrowest confidence interval of the form [X-a, X+a] will be achieved by selecting a = 0.4.