Final answer:
To approximate the resulting binomial distributions, we can use the appropriate normal distribution. For this question, a fair coin is tossed 130 times and we want to find the probability of obtaining between 55 and 69 tails, inclusive. The probability can be calculated by converting the values to z-scores and finding the areas under the normal curve using the mean and standard deviation of the binomial distribution.
Step-by-step explanation:
To approximate the resulting binomial distributions, we can use the appropriate normal distribution. In this case, a fair coin is tossed 130 times, and we want to find the probability of obtaining between 55 and 69 tails, inclusive. To solve this, we need to first calculate the mean and standard deviation of the binomial distribution using the formula: mean = n * p and standard deviation = sqrt(n * p * (1-p)). For a fair coin, the probability of getting tails is 0.5, so the mean is 130 * 0.5 = 65 and the standard deviation is sqrt(130 * 0.5 * (1 - 0.5)) = sqrt(32.5) ≈ 5.7. Now, we can use the normal distribution with a mean of 65 and a standard deviation of 5.7 to find the probability of obtaining between 55 and 69 tails. We can convert these values to z-scores using the formula: z = (x - mean) / standard deviation. For 55 tails, z = (55 - 65) / 5.7 ≈ -1.75, and for 69 tails, z = (69 - 65) / 5.7 ≈ 0.70. Next, we can use a standard normal distribution table or a calculator to find the area under the normal curve between these two z-scores. The probability of obtaining between 55 and 69 tails, inclusive, is the sum of these two areas. Therefore, the correct answer is option c) 0.7221.