Step 1: Calculating the Mean and Standard Deviation

For a binomial distribution:

• Mean ($\mu$) = $n \cdot P$
• Standard Deviation ($\sigma$) = $\sqrt{n \cdot P \cdot (1 - P)}$

Given:

• $P = 0.15$
• $n = 50$

Calculations:

1. $\mu = 50 \cdot 0.15 = 7.5$
2. $\sigma = \sqrt{50 \cdot 0.15 \cdot (1 - 0.15)} = \sqrt{50 \cdot 0.15 \cdot 0.85} = \sqrt{6.375} \approx 2.53$

Answer for Question 1:

• Mean ($\mu$) = 7.5
• Standard Deviation ($\sigma$) = 2.53

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Step 2: Approximating the Binomial Distribution with a Normal Distribution

A binomial distribution can be approximated by a normal distribution when:

1. $n \cdot P \geq 5$
2. $n \cdot (1 - P) \geq 5$

Checking Conditions:

• $n \cdot P = 50 \cdot 0.15 = 7.5$ (satisfies $\geq 5$)
• $n \cdot (1 - P) = 50 \cdot 0.85 = 42.5$ (satisfies $\geq 5$)

Since both conditions are satisfied, the binomial distribution can be approximated by the normal distribution.

Answer for Question 2: Yes, the binomial probabilities can be approximated by the normal probability distribution because both $n \cdot P$ and $n \cdot (1 - P)$ are greater than or equal to 5.

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Step 3: Probability of 20 or More Successes

Using the normal approximation, the mean ($\mu$) and standard deviation ($\sigma$) calculated earlier are:

• $\mu = 7.5$
• $\sigma = 2.53$

To approximate:

1. Apply continuity correction: Use $X = 19.5$ instead of $X = 20$.
2. Convert to a z-score: $Z = \frac{X - \mu}{\sigma}$
3. Find $P(Z \geq Z_{\text{calculated}})$ using the standard normal distribution.

Calculations:

1. $Z = \frac{19.5 - 7.5}{2.53} = \frac{12}{2.53} \approx 4.74$
2. The probability corresponding to $Z = 4.74$ is nearly 0 from the standard normal table.

Thus, $P(X \geq 20) \approx 0$.

Answer for Question 3: The probability of 20 or more successes is approximately $0$.

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Summary of Answers:

1. Mean = $7.5$, Standard Deviation = $2.53$
2. Yes, the binomial probabilities can be approximated by the normal distribution since $n \cdot P$ and $n \cdot (1 - P)$ are both $\geq 5$.
3. $P(X \geq 20) \approx 0$.

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Do you want a detailed breakdown of these calculations or further clarification?

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Relative Questions:

1. How do continuity corrections improve normal approximations of binomial distributions?
2. What are the limitations of using normal approximation for small sample sizes?
3. Can other distributions besides the normal distribution approximate binomial probabilities?
4. How does changing $P$ in the binomial distribution affect the shape of its curve?
5. What tools can you use to calculate exact binomial probabilities?

Tip: Always use continuity correction when approximating binomial probabilities with a normal distribution for greater accuracy.