39.539.5​% 

of consumers believe that cash will be obsolete in the next 20 years. Assume that 

77 

consumers are randomly selected. Find the probability that fewer than 

33 

of the selected consumers believe that cash will be obsolete in the next 20 years.

Question content area bottom

Part 1

The probability is 

enter your response here.

​(Round to three decimal places as​ needed.)

To solve this problem, we will use the binomial probability formula. The binomial probability formula is suitable here because the situation involves a fixed number of independent trials (consumers being surveyed), two possible outcomes (each consumer either believes or does not believe cash will be obsolete), and a constant probability of success.

### Given:
- **Probability of success** (\(p\)) = 39.5% = 0.395
- **Number of trials** (\(n\)) = 77
- **Number of successes** (\(X\)) = fewer than 33

We need to find the probability that fewer than 33 of the 77 consumers believe that cash will be obsolete.

### Steps to Calculate the Probability:

1. **Calculate the mean** (\(\mu\)) and **standard deviation** (\(\sigma\)) of the binomial distribution:

\[
\mu = n \times p = 77 \times 0.395 = 30.415
\]

\[
\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{77 \times 0.395 \times (1 - 0.395)} \approx 4.273
\]

2. **Convert the binomial problem to a normal distribution problem** using a continuity correction. This approximation is valid because \(n\) is large, and \(np(1-p) > 5\).

   - We want \(P(X < 33)\), so using the continuity correction, we calculate \(P(X < 32.5)\).

3. **Standardize the normal variable** using the z-score formula:

\[
z = \frac{X - \mu}{\sigma} = \frac{32.5 - 30.415}{4.273} \approx 0.489
\]

4. **Find the cumulative probability** for \(z = 0.489\) from the standard normal distribution table.

Let's calculate this now.The probability that fewer than 33 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately **0.687** (rounded to three decimal places).

Would you like more details or have any questions?

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Here are five related questions to deepen your understanding:
1. How would the probability change if the sample size was larger, say 100 consumers?
2. What is the probability that more than 40 consumers believe cash will be obsolete in the next 20 years?
3. How would the probability change if the percentage of consumers believing cash will be obsolete was 50%?
4. How do you interpret the z-score in the context of this problem?
5. Why is the continuity correction used when approximating a binomial distribution with a normal distribution?

**Tip:** When using the normal approximation to the binomial distribution, always apply the continuity correction by adding or subtracting 0.5 to account for the discrete nature of the binomial distribution.

Learn how to calculate the probability that fewer than 33 out of 77 randomly selected consumers believe cash will be obsolete using binomial and normal distributions. Detailed step-by-step solution with formulas and z-score application.

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