Questions: Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain. In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 25, (b) at least 30, and (c) at most 24. A. Yes, because both np ≥ 5 and nq ≥ 5. B. No, because nq<5. C. No, because np<5. (a) Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is exactly 25. (Round to four decimal places as needed.)

Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain. In a survey of U.S. college students, 62% said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 25, (b) at least 30, and (c) at most 24. A. Yes, because both np ≥ 5 and nq ≥ 5. B. No, because nq<5. C. No, because np<5. (a) Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is exactly 25. (Round to four decimal places as needed.)
Transcript text: Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain. In a survey of U.S. college students, $62 \%$ said that if their college offers new programs tailored to the new economy, it will make them more likely to re-enroll. You randomly select 41 college students. Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is (a) exactly 25 , (b) at least 30 , and (c) at most 24 . A. Yes, because both $n p \geq 5$ and $n q \geq 5$. B. No, because $\mathrm{nq}<5$. C. No, because $\mathrm{np}<5$. (a) Find the probability that the number who are more likely to enroll if their college offers new programs tailored to the new economy is exactly 25. $\square$ (Round to four decimal places as needed.)
failed

Solution

failed
failed

Solution Steps

Step 1: Determine Normal Approximation Validity

To determine if we can use a normal distribution to approximate the binomial distribution, we calculate \( np \) and \( nq \):

\[ np = 41 \cdot 0.62 = 25.42 \] \[ nq = 41 \cdot (1 - 0.62) = 15.58 \]

Since both \( np \geq 5 \) and \( nq \geq 5 \), we conclude that we can use the normal distribution for approximation.

Step 2: Calculate Probability of Exactly 25

Using the binomial probability formula, we find the probability of exactly 25 successes:

\[ P(X = 25) = \binom{41}{25} \cdot (0.62)^{25} \cdot (0.38)^{16} \approx 0.1257 \]

Step 3: Calculate Probability of At Least 30

To find the probability of at least 30 successes, we sum the probabilities from 30 to 41:

\[ P(X \geq 30) = P(X = 30) + P(X = 31) + \ldots + P(X = 41) \approx 0.0926 \]

Step 4: Calculate Probability of At Most 24

To find the probability of at most 24 successes, we sum the probabilities from 0 to 24:

\[ P(X \leq 24) = P(X = 0) + P(X = 1) + \ldots + P(X = 24) \approx 0.3792 \]

Final Answer

  • Probability of exactly 25: \( \boxed{0.1257} \)
  • Probability of at least 30: \( \boxed{0.0926} \)
  • Probability of at most 24: \( \boxed{0.3792} \)
Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
How many moles of ammonia would be required to react exactly with 0.366 moles of copper(II) oxide in the following chemical reaction? 2 NH3(g) + 3 CuO(s) → 3 Cu(s) + N2(g) + 3 H2O(g)
failedAnswered
Outcomes are not probability Event A: Alternating hats and tails (with either coming first) Event B: A head on each of the first two tosses Event C: Two or more heads TTH THH HTT HTH HHT HHH TTT THT
failedAnswered
The is the temperature at which the vapor pressure above a liquid equals 760 mmHg. The pressure is the pressure above a liquid exerted by gas particles in equilibrium with the liquid phase. The is a measure of the resistance of a liquid to spread out.
failedAnswered
Homework 5.3 - Rational Function Question 2, 5.3.17 Find the domain of the following rational function. H(x)=-8 x^2/(x-7)(x+2) The domain is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
failedAnswered
Find the value of the following expression. -5.3