Assume that when human resource managers are randomly​ selected, 4242​% say job applicants should follow up within two weeks. If 77 human resource managers are randomly​ selected, find the probability that at least 55 of them say job applicants should follow up within two weeks.
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Part 1
The probability is enter your response here.

We are dealing with a binomial probability problem, where we need to find the probability that at least 55 out of 77 human resource managers say job applicants should follow up within two weeks.

Here are the steps to solve this problem:

### 1. **Identify Parameters**
   - \( n = 77 \): The number of human resource managers randomly selected.
   - \( p = 0.4242 \): The probability that a human resource manager says that job applicants should follow up within two weeks.
   - We need to find the probability that **at least 55** say job applicants should follow up, i.e., \( P(X \geq 55) \).

### 2. **Use Normal Approximation to Binomial Distribution**
   Since \( n \) is large, we can approximate the binomial distribution using a normal distribution with:
   - Mean \( \mu = np \)
   - Standard deviation \( \sigma = \sqrt{np(1 - p)} \)

#### Calculate the mean and standard deviation:
   \[
   \mu = 77 \times 0.4242 = 32.6634
   \]
   \[
   \sigma = \sqrt{77 \times 0.4242 \times (1 - 0.4242)} \approx 4.350
   \]

### 3. **Convert to Z-Score**
   We convert the binomial probability to a normal probability. Since we are looking for \( P(X \geq 55) \), we apply a continuity correction by finding \( P(X \geq 54.5) \) and then convert this to a Z-score:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   For \( X = 54.5 \):
   \[
   Z = \frac{54.5 - 32.6634}{4.350} \approx 5.02
   \]

### 4. **Find Probability Using Z-Table**
   A Z-score of 5.02 is extremely large, meaning the probability that at least 55 managers respond affirmatively is extremely small.

Thus, the probability is nearly zero.

Would you like more details on any of these steps?

Here are some related questions:
1. How do we decide when to use normal approximation for binomial distributions?
2. What is the continuity correction, and why is it necessary?
3. How do you interpret a Z-score in probability?
4. How can we calculate binomial probabilities without using normal approximation?
5. What are some conditions that must be satisfied to use the normal approximation?

**Tip:** When the Z-score is greater than 3, the probability is usually very close to 0.

Assume that when human resource managers are randomly​ selected, 42​% say job applicants should follow up within two weeks. If 77 human resource managers are randomly​ selected, find the probability that at least 55 of them say job applicants should follow up within two weeks.

Calculate the probability that at least 55 out of 77 HR managers suggest job applicants follow up within two weeks, using binomial distribution and normal approximation. Learn the step-by-step process including the use of Z-scores and continuity correction, suitable for high school and college students.

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