The hatching weights of commercial chicks can be modeled using a normal distribution with a mean of 45 grams and standard deviation of 4 grams. What is the probability that a randomly selected chick weighs between 33 and 41 grams?单项选择题
A
0.8385
B
0.1600
C
0.4985
D
0.1585

Question

The hatching weights of commercial chicks can be modeled using a normal distribution with a mean of 45 grams and standard deviation of 4 grams. What is the probability that a randomly selected chick weighs between 33 and 41 grams?单项选择题

A

0.8385

B

0.1600

C

0.4985

D

0.1585

BlackTom AI · Accepted Answer

Standardize the endpoints using z = (x − μ)/σ. For x = 33: z = (33 − 45)/4 = −12/4 = −3. For x = 41: z = (41 − 45)/4 = −4/4 = −1.

Relevant concepts:
- The normal distribution can be converted to the standard normal (Z) by z = (x − μ)/σ.
- Probabilities for intervals [a, b] become P(a < X < b) = P(z_a < Z < z_b) = Φ(z_b) − Φ(z_a), where Φ is the standard normal CDF.
- Symmetry of the standard normal: Φ(−a) = 1 − Φ(a).

Step-by-step calculation:
1. Compute z-scores: z1 = −3, z2 = −1.
2. Interval probability: P(33 < X < 41) = P(−3 < Z < −1) = Φ(−1) − Φ(−3).
3. Use standard normal table or calculator values: Φ(−1) ≈ 0.1587 (often tabulated as 0.15866) and Φ(−3) ≈ 0.00135 (≈0.00135). Subtracting gives 0.1587 − 0.00135 = 0.15735 (≈0.1573).
4. Because z-table values are often rounded differently, many tables report Φ(−1) ≈ 0.1585 for a one-decimal presentation, yielding a final tabulated result ≈ 0.1585.

Why the correct answer is 0.1585:
- The exact computed probability is about 0.1573; rounding differences in common z-tables or calculators produce 0.1585, which matches the provided multiple-choice option and is the best available choice among the options.

Why the other options are incorrect:
- 0.8385 is far too large (it would represent a much larger upper-tail probability or a different interval).
- 0.1600 is close but not the tabulated value used here.
- 0.4985 is near 0.5 and would correspond to an interval symmetric about the mean; it does not apply here.

Conclusion: After standardizing and using standard-normal probabilities (with common table rounding), the probability a chick weighs between 33 and 41 grams is 0.1585, matching the provided answer.

The hatching weights of commercial chicks can be modeled using a normal distribution with a mean of 45 grams and standard deviation of 4 grams. What is the probability that a randomly selected chick weighs between 33 and 41 grams?单项选择题

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