To find the total number of potential customers Z*Z* that must be reached to be at least 50% certain of obtaining a total of 30 sign-ups, we'd need to calculate the **median** of the total distribution for Z*Z*. This involves finding the cumulative distribution function (CDF) of the total number of customers reached and solving for the point where P(Z≤Zmedian)=0.5*P*(*Z*≤*Z*median​)=0.5.

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To solve this problem and find the total number of potential customers \( Z^* \) such that we are at least 50% certain of obtaining 30 sign-ups, we will use a probabilistic approach based on cumulative distribution functions (CDFs). Here's the step-by-step breakdown:

### Step 1: Understanding the Problem
- We want to determine the number of potential customers, \( Z^* \), we must reach in order to be 50% certain of achieving 30 sign-ups.
- This requires us to find the **median** of the distribution, such that the probability of reaching this point corresponds to 50%, i.e., \( P(Z^* \leq Z_{\text{median}}) = 0.5 \).

We assume each customer has a probability \( p \) of signing up, and this process follows a **binomial distribution**, since each customer is either a sign-up or not.

### Step 2: Use of Binomial Distribution
The total number of sign-ups from \( Z^* \) customers follows a binomial distribution, which is given by:

\[
X \sim \text{Binomial}(Z^*, p)
\]

where:
- \( X \) is the number of sign-ups,
- \( Z^* \) is the total number of customers reached,
- \( p \) is the probability of any individual customer signing up.

The probability mass function (PMF) of the binomial distribution is:

\[
P(X = k) = \binom{Z^*}{k} p^k (1-p)^{Z^* - k}
\]

Here, \( k \) represents the number of sign-ups, and we want \( k = 30 \).

### Step 3: Cumulative Distribution Function (CDF)
We are asked to find the value of \( Z^* \) such that the probability of having 30 or fewer sign-ups is 50%. This means we need the median of the binomial distribution, where the CDF satisfies:

\[
P(X \leq 30) = 0.5
\]

The CDF of the binomial distribution is:

\[
P(X \leq 30) = \sum_{k=0}^{30} \binom{Z^*}{k} p^k (1-p)^{Z^* - k}
\]

### Step 4: Solve for \( Z^* \)
We now need to find \( Z^* \) such that this condition holds.

1. Assume a reasonable probability \( p \) for sign-ups, based on data (e.g., 5% sign-up rate \( p = 0.05 \)).
2. Use the binomial CDF and adjust \( Z^* \) until the cumulative probability reaches 50% for \( k = 30 \).

This involves numerical methods or software tools (such as Python or R) to compute the CDF iteratively and solve for \( Z^* \).

### Example with \( p = 0.05 \)
Suppose we assume that \( p = 0.05 \), the steps would look like this:

- Use a statistical tool to compute the binomial CDF for different values of \( Z^* \).
- Find the smallest \( Z^* \) such that \( P(X \leq 30) = 0.5 \).

Using numerical tools or a binomial CDF calculator, we can iteratively find the exact value of \( Z^* \). 

Would you like me to run the calculation for a specific value of \( p \), or explain further how to solve this numerically?

---

### Follow-up Questions:
1. How would the solution change if the sign-up probability \( p \) were different?
2. What is the interpretation of the median in the context of this problem?
3. How do we approximate solutions for binomial distributions using normal approximations?
4. Can we apply the same approach if the sign-ups follow a Poisson distribution instead?
5. What would happen if the desired number of sign-ups was 50 instead of 30?

**Tip:** When dealing with binomial distributions, using a normal approximation can often simplify the process for large \( Z^* \).

To find the total number of potential customers Z*Z* that must be reached to be at least 50% certain of obtaining a total of 30 sign-ups, we'd need to calculate the **median** of the total distribution for Z*Z*. This involves finding the cumulative distribution function (CDF) of the total number of customers reached and solving for the point where P(Z≤Zmedian)=0.5*P*(*Z*≤*Z*median​)=0.5.

Learn how to calculate the total number of potential customers needed to be 50% certain of achieving 30 sign-ups. This problem uses the binomial distribution and cumulative distribution function (CDF) to find the median. Suitable for college-level students studying probability and statistics.

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