To solve this, we can model the number of people who like balloons as a binomial random variable. The binomial distribution is appropriate because each person is independent, and they can either like balloons (with probability $p = 0.60$) or not like them (with probability $1 - p = 0.40$).

Given:

• $n = 102$ (sample size),
• $p = 0.60$ (probability a person likes balloons),
• We need to find $P(58 \leq X \leq 63)$, where $X$ is the number of people in the sample who like balloons.

Step 1: Approximate the Binomial Distribution with a Normal Distribution

For large $n$, we can use the normal approximation to the binomial distribution, which is typically accurate when $n$ is large and $p$ is not too close to 0 or 1.

The parameters for the normal distribution approximation are:

• Mean $\mu = np = 102 \times 0.60 = 61.2$
• Standard deviation $\sigma = \sqrt{np(1-p)} = \sqrt{102 \times 0.60 \times 0.40} = \sqrt{24.48} \approx 4.948$

Step 2: Apply the Continuity Correction

Since we're using a continuous distribution (normal) to approximate a discrete one (binomial), we apply a continuity correction by adjusting the values for the desired range:

• For $X \geq 58$, we use $57.5$ (i.e., $58 - 0.5$).
• For $X \leq 63$, we use $63.5$ (i.e., $63 + 0.5$).

Thus, we need to compute $P(57.5 \leq X \leq 63.5)$.

Step 3: Convert to Standard Normal Distribution (Z-Scores)

We convert these values into standard normal $Z$-scores using the formula:

$Z = \frac{X - \mu}{\sigma}$

For $X = 57.5$: $Z_1 = \frac{57.5 - 61.2}{4.948} \approx \frac{-3.7}{4.948} \approx -0.747$

For $X = 63.5$: $Z_2 = \frac{63.5 - 61.2}{4.948} \approx \frac{2.3}{4.948} \approx 0.464$

Step 4: Use the Standard Normal Distribution Table or a Calculator

Now, we use the standard normal table or a calculator to find the cumulative probabilities corresponding to these $Z$-scores.

• $P(Z \leq -0.747) \approx 0.2276$
• $P(Z \leq 0.464) \approx 0.6772$

Step 5: Calculate the Desired Probability

The probability that $X$ is between 58 and 63 is the difference between these cumulative probabilities:

$P(58 \leq X \leq 63) = P(Z \leq 0.464) - P(Z \leq -0.747) = 0.6772 - 0.2276 = 0.4496$

Final Answer:

The probability that between 58 and 63 people like balloons is approximately 0.4496.

Would you like further details or explanations on any part of this? Here are a few related questions:

1. How do we calculate the mean and standard deviation for a binomial distribution?
2. Why is the continuity correction necessary in normal approximation?
3. What are the conditions for using the normal approximation to the binomial distribution?
4. How do we use the Z-score to find probabilities in a normal distribution?
5. Can this problem be solved directly using the binomial distribution instead of normal approximation?

Tip: When using the normal approximation, check that $np$ and $n(1-p)$ are both greater than 5 to ensure the approximation is valid.