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

类似问题

Quackers decides to measure the number of quacks he makes in one hour (60 minutes). It turns out that it is 34. He wants to calculate the probability that he would have made that many quacks if the underlying true probability of a quack in any minute was 50%. How should he do this?

Quackers decides to measure the number of quacks he makes in half an hour. It turns out that it is 18. He wants to calculate the probability that he would have made that many quacks if the underlying true probability of a quack in any minute was 55%. How should he do this?

A basketball player takes 20 free throws in a practice session. Each shot is made independently with probability [math: p=0.7]. Which probability distribution is most appropriate for modelling the number of successful shots?[Fill in the blank]

Emily tosses a fair die three times. The probability she gets an even number at least 2 times is

State-wide surveys indicates that 14.5% of toddlers in Ney York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York. Is this a binomial scenario?

已知 G(n, p) 模型，观察到 m 个边的概率呈： Given the G(n, p) model, the probability of observing m edges the

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单