39.839.8​% 

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

66 

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.

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Part 1

The probability is 

0.4500.450.

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

To find the probability that fewer than 33 of the 66 selected consumers believe that cash will be obsolete in the next 20 years, we can use the binomial probability distribution. 

Given:
- \( p = 0.398 \) (the probability that a consumer believes cash will be obsolete)
- \( n = 66 \) (the number of consumers)
- \( x = 32 \) (we want the probability of fewer than 33, so \( x \leq 32 \))

First, let's check if we can approximate this binomial distribution with a normal distribution. The criteria for using the normal approximation are:
1. \( np \geq 5 \)
2. \( n(1-p) \geq 5 \)

Calculating:
\[
np = 66 \times 0.398 = 26.268
\]
\[
n(1-p) = 66 \times (1 - 0.398) = 39.732
\]

Both values are greater than 5, so we can use the normal approximation.

The mean (\(\mu\)) and standard deviation (\(\sigma\)) of the binomial distribution are:
\[
\mu = np = 26.268
\]
\[
\sigma = \sqrt{np(1-p)} = \sqrt{26.268 \times 0.602} = \sqrt{15.801336} \approx 3.975
\]

Using the continuity correction factor, we adjust \( x \) by 0.5 for normal approximation:
\[
P(X < 33) \approx P\left(Z < \frac{32.5 - 26.268}{3.975}\right)
\]
\[
P\left(Z < \frac{32.5 - 26.268}{3.975}\right) = P\left(Z < \frac{6.232}{3.975}\right) = P(Z < 1.567)
\]

Using the standard normal distribution table or a calculator, we find:
\[
P(Z < 1.567) \approx 0.941
\]

So, the probability that fewer than 33 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately 0.941, not 0.450. The given probability of 0.450 might be a typo.

Do you want more details or have any questions?

Here are some related questions to expand on this topic:

1. How do you use the normal approximation to the binomial distribution?
2. What is the continuity correction factor and when is it used?
3. How do you find the mean and standard deviation of a binomial distribution?
4. What are the conditions for using the normal approximation for a binomial distribution?
5. How do you interpret a Z-score in a standard normal distribution?
6. How can you use a Z-table to find probabilities for a standard normal distribution?
7. What are the key differences between a binomial distribution and a normal distribution?
8. How do you calculate probabilities for a binomial distribution without using normal approximation?

**Tip:** When approximating a binomial distribution with a normal distribution, always apply the continuity correction factor to improve accuracy.

Calculate the probability that fewer than 33 out of 66 randomly selected consumers believe cash will be obsolete in the next 20 years using binomial distribution and normal approximation. Learn about key concepts like binomial probability, normal distribution, and the Central Limit Theorem.

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