Empirical Rule

Q3 d) Find P(LB

Question

Empirical Rule

Q3 d) Find P(LB < Z< UB). Enter your answer to 4 decimal places.

BlackTom AI · Accepted Answer

The problem mentions the Empirical Rule and asks for P(LB < Z < UB) for a standard normal Z, with the answer provided as 0.9973.

First, recall what the Empirical Rule (68-95-99.7 rule) tells us about a standard normal distribution: about 68% of values fall within ±1 standard deviation, about 95% within ±2 standard deviations, and about 99.7% within ±3 standard deviations from the mean.

To connect this to P(LB < Z < UB), you would typically identify LB and UB as specific z-scores. If the interval is symmetric around the mean and spans from -3 to 3, then P(-3 < Z < 3) ≈ 0.9973, which matches the given numerical value.

However, the question as written does not display explicit LB and UB values. Without those bounds, you cannot compute a probability; you would need to know the exact z-scores that define LB and UB.

Assuming the target interval is the ±3σ interval, the logic is: 
- The Empirical Rule states that roughly 99.7% of the distribution lies within 3 standard deviations of the mean. 
- For a standard normal Z, this corresponds to P(-3 < Z < 3) ≈ 0.9973.

If instead LB and UB were different symmetric bounds (for example ±1 or ±2), you would apply the corresponding percentages: ±1 would give about 0.6827, ±2 about 0.9545, and ±3 about 0.9973.

In summary, to evaluate P(LB < Z < UB) you must know LB and UB. If the interval is indeed [-3, 3], the result 0.9973 aligns with the Empirical Rule for a standard normal distribution. If the interval is some other pair like [-a, a], you would look up or calculate P(-a < Z < a) accordingly and compare with the stated answer.

Potential pitfalls to avoid: assuming a symmetric interval without verification, or using the wrong tail probabilities for non-standard intervals. Always confirm the exact z-score bounds before selecting the probability.

STAT_008_A01_25U Lab #10

Empirical Rule Q3 d) Find P(LB < Z< UB). Enter your answer to 4 decimal places.

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