Suppose you observe a sample data set consisting of n=64 inter-arrival times X1,…,X64 for the subway, measured in minutes. As before, we assume the statistical model that X1,…,X64∼iidexp(λ) for some unknown parameter λ>0. In this data set, you observe that the sample mean is 164∑64i=1Xi=7.8.
Additional Instructions: For best results, please adhere to the following guidelines and reminders:
For the upcoming calculations, please truncate qα/2 at 2 decimal places, instead of a more exact value. For example, if qα/2=3.84941, use 3.84 instead of 3.85 or 3.849.
Input answers truncated at 4 decimal places. For example, if your calculations yield 11.327458, use 11.3274 instead of 11.3275 or 11.32745.
You will be computing CIs at asymptotic level 90%.
Using the ‘solve method' (refer to the slide ‘Three solutions'), construct a confidence interval Isolve with asymptotic level 90% for the unknown parameter λ.
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Using the ‘plug-in method' Iplug−in (refer to the slide ‘Three solutions'), construct a confidence interval with asymptotic level 90% for the unknown parameter λ.
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Which interval is narrower?
Isolve
Iplug−in
unanswered
Which of these confidence intervals is centered about the sample estimate, λ^n?
Isolve
Iplug−in
Both
Neither
I plugin
I plugin
To construct a confidence interval (CI) using the 'solve method' with asymptotic level 90%, we need to find the critical values.
Step 1: Find the sample standard deviation (sd) using the formula sd = sqrt(1/n * sum((Xi - X-bar)^2)).
Note: X-bar is the sample mean.
Step 2: Calculate the critical value q_alpha/2 for a two-sided confidence interval at alpha = (1 - 0.90)/2 = 0.05 using a statistical table or calculator.
Step 3: Compute the lower bound L_solve = X-bar - q_alpha/2 * sd / sqrt(n).
Step 4: Compute the upper bound U_solve = X-bar + q_alpha/2 * sd / sqrt(n).
The confidence interval (Isolve) for the unknown parameter λ is [L_solve, U_solve].
To construct a confidence interval (CI) using the 'plug-in method' with asymptotic level 90%, we need to follow these steps:
Step 1: Calculate the plug-in estimate (λ-hat) for the unknown parameter λ. In this case, λ-hat is equal to the sample mean X-bar.
Step 2: Calculate the estimated standard error (se) using the formula se = sd / sqrt(n), where sd is the sample standard deviation.
Step 3: Calculate the critical value q_alpha/2 for a two-sided confidence interval at alpha = (1 - 0.90)/2 = 0.05 using a statistical table or calculator.
Step 4: Compute the lower bound L_plugin = λ-hat - q_alpha/2 * se.
Step 5: Compute the upper bound U_plugin = λ-hat + q_alpha/2 * se.
The confidence interval (Iplug-in) for the unknown parameter λ is [L_plugin, U_plugin].
Now, let's compute these confidence intervals step-by-step.
First, we need the value of sd (the sample standard deviation) to compute Isolve.
Second, we compute Isolve step-by-step as follows:
Step 1: Calculate the sample standard deviation (sd) using the formula: sd = sqrt(1/n * sum((Xi - X-bar)^2))
Since you have not provided the values of the inter-arrival times (X1,...,X64), we cannot compute sd and construct Isolve.
Similarly, we cannot compute Iplug-in and determine which interval is narrower or centered around the sample estimate (λ^n) without knowing the values of X1,...,X64.
To construct the confidence intervals, we need to use the solve method and the plug-in method.
1. **Solve Method (Isolve)**:
The solve method involves solving an equation to find the confidence interval. In this case, since X1, ..., X64 are independent and identically distributed exponential random variables with parameter λ, the sample sum of inter-arrival times ∑ Xi follows a gamma distribution with shape parameter n and rate parameter λ.
We know that the sample mean is given as ∑ Xi / n. Given that the sample mean is 7.8, we can solve for λ using the equation:
7.8 = ∑ Xi / n = (∑ Xi) / 64
We need to solve this equation for λ. Let's denote λ as λ^.
2. **Plug-in Method (Iplug-in)**:
The plug-in method involves plugging in the estimated value of λ, denoted as λ^n, into the distribution function to create a confidence interval. In this case, λ^n is given by λ^n = 1 / (∑ Xi / n).
To construct the confidence interval using the plug-in method, we need to find the critical values of the distribution. Since λ^n is the estimated value, we need to estimate the variance of λ^n.
The variance of λ^n can be approximated as σ^2 / n, where σ^2 is the variance of the exponential distribution, given by 1 / λ^2. Therefore, the estimated variance of λ^n is (1 / λ^n^2) / n.
To find the critical values, we use the standard normal distribution with a desired asymptotic level of 90%. We can use the formula: q(α/2) = Φ^(-1)(1 - α/2), where Φ^(-1) represents the inverse of the standard normal cumulative distribution function.
The confidence interval using the plug-in method is given by:
Iplug−in = [λ^n - q(α/2) * sqrt((1 / λ^n^2) / n), λ^n + q(α/2) * sqrt((1 / λ^n^2) / n)]
3. **Comparing the Intervals**:
To determine which interval is narrower, we need to calculate the lengths of both intervals. The length of an interval is given by the difference between its upper and lower bounds.
The length of Isolve is not provided, so it cannot be compared directly to Iplug−in.
4. **Centered about the sample estimate**:
Both intervals, Isolve and Iplug−in, are centered about the sample estimate, λ^n. This means that the midpoint of each interval lies at the same value, which is the estimated value for λ.
5. **Summary**:
To answer the questions:
- The confidence interval using the solve method is denoted as Isolve.
- The confidence interval using the plug-in method is denoted as Iplug−in.
- It is unknown which interval, Isolve or Iplug−in, is narrower.
- Both intervals, Isolve and Iplug−in, are centered about the sample estimate, λ^n.