Final answer:
The shortest interval that contains approximately 4,000 shellfish lengths is Option 4 (9.95 to 10.05 centimeters), which covers 0.5 standard deviations and is likely closest to the central 40% of the normal distribution.
Step-by-step explanation:
The lengths of individual shellfish in a population of 10,000 shellfish are approximately normally distributed with a mean of 10 centimeters and a standard deviation of 0.2 centimeters. To find the shortest interval that contains approximately 4,000 of the shellfish lengths, which is 40% of the population, we look at a normal distribution curve.
Since the distribution is symmetrical, we are interested in the interval around the mean that captures the central 40% of the data. The z-scores corresponding to the central 40% can be found using a z-table, or standard normal distribution table. However, without a z-table, we can infer that the 40% span will be less than one standard deviation away from the mean in either direction because one standard deviation from the mean (in both directions) encompasses about 68% of the data in a normal distribution.
Now, comparing the intervals given in the options:
- Option 1 (9.6 to 10.4 cm) covers 4 standard deviations, which is way more than necessary for 40%.
- Option 2 (9.8 to 10.2 cm) covers 2 standard deviations, still more than necessary.
- Option 3 (9.9 to 10.1 cm) covers 1 standard deviation, closer but might not be the shortest interval.
- Option 4 (9.95 to 10.05 cm) covers 0.5 standard deviations, and is likely the shortest interval that we are looking for.
Therefore, Option 4 (9.95 to 10.05 centimeters) is the shortest interval that contains approximately 4,000 shellfish lengths.