Compute the following probabilities:

1. Compute the normal approximation to the probability of having at least 196 and at most 205 heads in 400 tosses of a fair coin. Use the normal distribution table for your calculations.

2. A population of polar bears has a distribution of weight with a mean of 800 lbs and a standard deviation of 100 lbs. Compute, approximately and in terms of the standard normal cumulative distribution function (CDF), the probability that an arctic ship with a maximum weight capacity of 330,000 lbs can hold 400 bears chosen at random.

The answer involves using normal approximation to find probabilities. This involves standardization and using Z tables to find probabilities for coin toss results and a scenario of a ship carrying polar bears.

Explanation:

The question concerns two cases of the normal approximation to a probability distribution. In the first, we need to approximate the probability of getting between 196 and 205 heads in 400 tosses of a fair coin. In the second case, we need to approximate the probability that an arctic ship with a maximum weight capacity of 330,000 pounds can carry 400 polar bears with a mean weight of 800 pounds and a standard deviation of 100 pounds.

First Case:

For the first case, we take the expected number of heads (mean) as 200 (0.5 * 400), with the standard deviation sqrt(400 * 0.5 * 0.5) = 10. The number of heads is then standardised and looked up in the Z table for the values of 196 - 200 and 205 - 200.

Second Case:

The second case is a similar procedure with a weight problem. We calculate the mean total weight of 400 bears is 320,000 lbs, with a standard deviation of 2,000 lbs. Then we seek the Z-score for 330,000 - 320,000 lbs and look up this value in the standard normal cdf.

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