![Create an image that visualizes the scenario described in the statistical question. This should include a representation of the number line with an unknown parameter, θ, and an interval spreading around it from θ−0.5 to θ+0.5. Possibly represent other intervals as well, like [X−2,X+2] and [X−a,X+a], however don't include any of the numbers or letters as text. Make the image appealing and intuitive to resonate with the complexity of the question.](/images/questions/1823224/1024x1024.png)
A simple CI
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?
unanswered
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=
unanswered
1 - Yes
2 - 0.4
Well, if you want the narrowest confidence interval with an 80% confidence level, you'll have to use a little trickery. You see, the narrower the interval, the less likely it is to capture the true parameter value. So, to achieve that 80% confidence, you'll need to strike a bit of a balance.
In this case, since X is uniform on the interval [θ−0.5,θ+0.5], we can say that the range of X is 1. This means that the smallest possible choice for "a" in [X−a,X+a] would be 0.5.
So, the narrowest confidence interval with an 80% confidence level would be [X - 0.5, X + 0.5]. Just remember, narrower isn't always better when it comes to confidence intervals. It's all about finding that sweet spot between precision and capturing the true parameter value.
To determine if [X−2,X+2] is an 80% confidence interval, we need to check if the interval contains the true parameter value with a probability of 0.8 or 80%.
Given X is uniformly distributed on the interval [θ−0.5, θ+0.5], the width of the interval [X−a, X+a] is 2a. To find the narrowest confidence interval for an 80% confidence level, we need to minimize the width of the interval.
For an 80% confidence level, we want the interval to contain the true parameter value with a probability of 0.8. This means we want the interval to cover 80% of the probability mass of the distribution.
Since X is uniformly distributed, the probability mass is evenly distributed over the entire interval [θ−0.5, θ+0.5]. To cover 80% of the probability mass, we need the interval width to be 80% of the total interval width.
Since the total interval width is 0.5 - (-0.5) = 1, we want the interval width to be 0.8 * 1 = 0.8. Therefore, the narrowest confidence interval for an 80% confidence level is [X - 0.4, X + 0.4].
Thus, the value of a for an 80% confidence interval is 0.4.
To determine the narrowest confidence interval of the form [X−a, X+a], we need to consider the nature of the distribution and the desired confidence level.
In this case, we have an unknown parameter θ and a uniform distribution for X on the interval [θ−0.5, θ+0.5].
To find the narrowest confidence interval for an 80% confidence level, we need to calculate the range for a.
Confidence intervals are constructed using quantiles of the distribution. For a uniform distribution, the quantiles determine the range of the interval.
In this case, we want to find the narrowest interval that covers 80% of the distribution. Since the distribution is uniform, the probability of an event occurring within an interval is equal to the width of the interval divided by the total width of the distribution.
To find the narrowest interval, we need to determine the range of X that corresponds to an 80% cumulative probability. In a uniform distribution, the probability is proportional to the width of the interval.
Given that X is uniform on the interval [θ−0.5, θ+0.5], we want to find the range of X that corresponds to an 80% cumulative probability.
To do this, we calculate the width of the interval as follows:
Width = (θ + 0.5) - (θ - 0.5)
Width = 0.5 + 0.5
Width = 1
So, the width of the interval is 1.
In order to have an 80% confidence interval, we need a range that covers 80% of the distribution. We want to find the narrowest interval that covers this range.
If the total width of the distribution is 1 and we want to cover 80% of the distribution, the narrowest interval that meets this requirement is the range that covers 80% of the total width.
80% of 1 is 0.8, so the narrowest interval for an 80% confidence level would be [X - 0.4, X + 0.4].
Therefore, the narrowest confidence interval of the form [X−a, X+a] for an 80% confidence level is a = 0.4.