Questions: You need to compute the probability of 5 or fewer successes for a binomial experiment with 10 trials. The probability of success on a single trial is 0.57. Since this probability of success is not in the table, you decide to use the normal approximation to the binomial. Is this an appropriate strategy? Explain.

You need to compute the probability of 5 or fewer successes for a binomial experiment with 10 trials. The probability of success on a single trial is 0.57. Since this probability of success is not in the table, you decide to use the normal approximation to the binomial. Is this an appropriate strategy? Explain.

Transcript text: You need to compute the probability of 5 or fewer successes for a binomial experiment with 10 trials. The probability of success on a single trial is 0.57. Since this probability of success is not in the table, you decide to use the normal approximation to the binomial. Is this an appropriate strategy? Explain.

Solution

Solution Steps

Step 1: Calculate \( n \cdot p \) and \( n \cdot q \)

We start by calculating the expected number of successes and failures in the binomial experiment: \[ n \cdot p = 10 \cdot 0.57 = 5.7 \] \[ n \cdot q = 10 \cdot (1 - 0.57) = 10 \cdot 0.43 = 4.3 \]

Step 2: Determine the Appropriateness of Normal Approximation

To determine if the normal approximation is appropriate, we check the condition \( n \cdot p \cdot (1 - p) \): \[ n \cdot p \cdot (1 - p) = 10 \cdot 0.57 \cdot 0.43 = 2.451 \] Since \( 2.451 < 10 \), the normal approximation is not appropriate. Therefore, the conclusion is: \[ \text{No, normal approximation is not appropriate because } n \cdot p \cdot (1 - p) \leq 10. \]

Step 3: Calculate the Probability of 5 or Fewer Successes

Using the binomial cumulative distribution function, we find the probability of obtaining 5 or fewer successes: \[ P(X \leq 5) \approx 0.4436 \]