In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

What's your favorite ice cream flavor? For people who buy ice cream, the all-time favorite is still vanilla. About 25% of ice cream sales are vanilla. Chocolate accounts for only 11% of ice cream sales. Suppose that 173 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.)

To determine if we can use the normal approximation to the binomial distribution in this problem, let's review the criteria and then apply the approximation.

### Step 1: Verify Normal Approximation Criteria

The normal approximation to the binomial is appropriate when:
1. The sample size \( n \) is large.
2. Both \( np \) and \( n(1 - p) \) are greater than or equal to 5.

Let's define the parameters:
- Number of trials (customers) \( n = 173 \)
- Probability of success (buying vanilla) \( p = 0.25 \)

Let's calculate \( np \) and \( n(1 - p) \) to check the conditions:
- \( np = 173 \times 0.25 = 43.25 \)
- \( n(1 - p) = 173 \times (1 - 0.25) = 129.75 \)

Both \( np \) and \( n(1 - p) \) are greater than 5, so it is appropriate to use the normal approximation.

### Step 2: Apply the Normal Approximation

For a binomial distribution \( B(n, p) \), we approximate it with a normal distribution \( N(\mu, \sigma^2) \) where:
- \( \mu = np \)
- \( \sigma = \sqrt{np(1 - p)} \)

#### Calculate Mean and Standard Deviation
- \( \mu = 43.25 \)
- \( \sigma = \sqrt{43.25 \times 0.75} = \sqrt{32.4375} \approx 5.6963 \)

So, we use the normal distribution \( N(43.25, 5.6963^2) \).

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If you provide the specific probability ranges you'd like to find (e.g., "more than 40 customers" or "between 30 and 50 customers"), I can calculate the probabilities accordingly.

Would you like further details or have any specific questions?

Here are 5 related questions for further exploration:
1. What is the probability that fewer than 30 customers will buy vanilla ice cream?
2. What is the probability that more than 50 customers will buy vanilla ice cream?
3. How does changing the probability of vanilla purchases affect the distribution's mean and standard deviation?
4. Can we apply the normal approximation to chocolate sales as well?
5. How would we approach this problem if only 50 customers visited the store instead of 173?

**Tip:** When using a normal approximation for a binomial, consider applying a continuity correction by adjusting the interval by 0.5, which can improve accuracy.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What's your favorite ice cream flavor? For people who buy ice cream, the all-time favorite is still vanilla. About 25% of ice cream sales are vanilla. Chocolate accounts for only 11% of ice cream sales. Suppose that 173 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.)

Explore how to use the normal approximation to the binomial distribution for estimating probabilities in a real-world scenario. This problem involves calculating the probability of vanilla ice cream sales, with detailed steps on verifying normal approximation criteria and applying distribution formulas. Ideal for Grades 10-12 students learning probability and distributions.

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