Let
𝑋
equal the number of tomato seeds out of
𝑛
=
48
seeds that will germinate when
𝑝
=
0.75
is the probability that a particular seed germinates. Approximate
𝑃
(
35
≤
𝑋
≤
40
)
.

Choose the correct answer below.

Question

Let 
𝑋
 equal the number of tomato seeds out of 
𝑛
=
48
 seeds that will germinate when 
𝑝
=
0.75
 is the probability that a particular seed germinates. Approximate 
𝑃
(
35
≤
𝑋
≤
40
)
.

Choose the correct answer below.

BlackTom AI · Accepted Answer

Using the normal approximation to the binomial: μ = np = 48(0.75) = 36 and σ = sqrt(np(1 − p)) = sqrt(36·0.25) = sqrt(9) = 3. The normal approximation is appropriate because np = 36 and n(1 − p) = 12 are both ≥ 10.

Apply the continuity correction for the discrete binomial probability P(35 ≤ X ≤ 40). Replace the integer bounds with half-integers: P(35 ≤ X ≤ 40) ≈ P(34.5 < Y < 40.5) where Y ~ N(μ = 36, σ = 3).

Compute z-scores for the endpoints:
z1 = (34.5 − 36)/3 = −1.5/3 = −0.5.
z2 = (40.5 − 36)/3 = 4.5/3 = 1.5.

Use standard normal cumulative distribution values (Φ):
Φ(1.5) ≈ 0.93319,
Φ(−0.5) ≈ 0.30854.

The desired probability is Φ(1.5) − Φ(−0.5) = 0.93319 − 0.30854 = 0.62465 ≈ 0.6247.

Why this is correct: the calculation follows the standard normal approximation with the continuity correction (necessary to approximate a discrete binomial interval), uses correctly computed mean and standard deviation, and evaluates the correct z-scores and table values. The numerical subtraction yields 0.62465, which rounds to 0.6247.

Why the other choices are incorrect:
- 0.6983 and 0.5802 likely result from miscomputed z-scores or neglecting/incorrectly applying the continuity correction; they do not match the correct z = −0.5 and z = 1.5 interval.
- 0.5541 is also inconsistent with the correct normal approximation and arises from arithmetic or tabulation errors.
- “none of these” is wrong because one option does match the correct computed value.

Final confirmation: the correct approximate probability is 0.6247, matching the provided answer.

Let 𝑋 equal the number of tomato seeds out of 𝑛 = 48 seeds that will germinate when 𝑝 = 0.75 is the probability that a particular seed germinates. Approximate 𝑃 ( 35 ≤ 𝑋 ≤ 40 ) . Choose the correct answer below.单项选择题

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