The average length of a dog pregnancy is a normal random variable with mean
𝜇
=
63
days and standard deviation
𝜎
=
5
days. Use the empirical rule to determine what percent of dogs are pregnant longer than 68 days.

Question

The average length of a dog pregnancy is a normal random variable with mean 
𝜇
=
63
 days and standard deviation 
𝜎
=
5
 days. Use the empirical rule to determine what percent of dogs are pregnant longer than 68 days.

BlackTom AI · Accepted Answer

Since 68 days is one standard deviation above the mean (63 days, σ = 5 days), apply the empirical (68-95-99.7) rule.

1) Relevant knowledge points and concepts
- The empirical rule for a normal distribution: about 68% of values lie within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ.
- Symmetry of the normal distribution implies tails beyond ±1σ share the remaining probability equally.

2) Step-by-step reasoning
- Compute the z-score for 68 days: z = (x − μ) / σ.
- Using the empirical rule, determine the proportion above z = +1.

3) Calculations (all intermediate steps)
- z = (68 − 63) / 5 = 5 / 5 = 1.
- About 68% of dogs have pregnancy lengths between 63 − 5 = 58 days and 63 + 5 = 68 days.
- The middle 68% is symmetric around the mean, so the portion from the mean up to +1σ is half of 68%: 68% / 2 = 34%.
- Since 50% of values lie above the mean, the proportion above +1σ = 50% − 34% = 16%.
- Or equivalently, the two tails outside ±1σ sum to 100% − 68% = 32%; each tail is 32% / 2 = 16%.

4) Why the correct answer is correct
- 68 days equals μ + 1σ, and the empirical rule directly gives the area to the right of +1σ as 16%. Therefore 16% of dogs are pregnant longer than 68 days.

5) Why the other options are incorrect (brief)
- 0.15%: corresponds to being beyond +3σ (one-tail), not +1σ.
- 84%: equals the cumulative proportion ≤ +1σ (50% + 34%), i.e., proportion shorter than or equal to 68 days, not longer.
- 13.5%: is the proportion between +1σ and +2σ (one-sided), not the tail beyond +1σ.
- 97.5%: equals cumulative proportion ≤ +2σ, not the tail beyond +1σ.
- 2.5%: is the one-tail beyond +2σ, not beyond +1σ.

6) Conclusion
Therefore, by the empirical rule, 16% of dogs are pregnant longer than 68 days, which matches the provided answer: 16%.

The average length of a dog pregnancy is a normal random variable with mean 𝜇 = 63 days and standard deviation 𝜎 = 5 days. Use the empirical rule to determine what percent of dogs are pregnant longer than 68 days.单项选择题

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