Continue from the question above. Find a 95% confidence interval for β.

Report the lower limit to two decimal places.

Question

Continue from the question above.  Find a 95% confidence interval for β.

Report the lower limit to two decimal places.

BlackTom AI · Accepted Answer

The prompt asks to continue from a prior question and find a 95% confidence interval for β, reporting the lower limit to two decimal places. Since no answer options are provided, we cannot evaluate multiple choices or rank them.

To construct a 95% confidence interval for β in a typical regression or parameter-estimation context, you generally need two key pieces of information: the point estimate β̂ and its standard error SE(β̂).

The standard approach is as follows:
- Step 1: Obtain the point estimate β̂ for the parameter β from your model or estimate method.
- Step 2: Compute or obtain the standard error SE(β̂) associated with β̂. This SE reflects the variability of the estimator given the data and model assumptions.
- Step 3: Determine the critical value for a 95% interval. If the sample size is large or the underlying distribution is well-approximated by a normal distribution, you typically use the z critical value z* = 1.96. If the sample size is small or the t-distribution is appropriate, you use t_{α/2, df} with the relevant degrees of freedom df.
- Step 4: Form the confidence interval using:
  • Lower limit = β̂ − z*(or t*) × SE(β̂)
  • Upper limit = β̂ + z*(or t*) × SE(β̂)

- Step 5: Report the lower limit to two decimals, as requested.

If the given answer 3.82 corresponds to the lower limit, you could back-calculate what combinations of β̂ and SE(β̂) would yield that value. For instance, if you know β̂ and SE(β̂), you would compute 3.82 = β̂ − 1.96 × SE(β̂) (assuming a normal approximation). Solving for SE(β̂) gives SE(β̂) = (β̂ − 3.82) / 1.96. However, without β̂, SE(β̂), or the df context, this back-calculation remains speculative.

If you can provide the preceding question details, the estimated β̂, and the standard error (or the regression output from which they derive), I can walk through the numerical steps and verify the exact lower limit to two decimals.

In summary, to verify or compute the requested 95% CI lower limit, you must know the point estimate β̂ and its standard error, choose the appropriate critical value (1.96 for large samples, or a t-quantile for smaller samples), and apply the formula for the lower bound. Without those inputs or any answer choices, the analysis stops at outlining the method and illustrating how a 3.82 lower limit would relate to β̂ and SE(β̂).

MAST90007_2025_WIN 2025 Quiz 3

Continue from the question above. Find a 95% confidence interval for β. Report the lower limit to two decimal places.

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