Final answer:
The sampling distribution of the sample mean for 49 NFL games is normally distributed as N(21.1, 1.33), with calculations and probabilities based on z-scores and standard normal distribution tables. Decimal precision should be maintained to four decimal places.
Step-by-step explanation:
To determine the sampling distribution of the mean for 49 NFL games, we use the central limit theorem. The average score for games played in the NFL is 21.1 and the standard deviation is 9.3 points. The sampling distribution of the mean for a sample size of 49 is normally distributed due to the central limit theorem, which states that this distribution will approach normality as the sample size increases. Therefore, this distribution of the sample mean is N(21.1, 1.33), where 1.33 is the standard deviation of the sampling distribution, calculated by dividing the population standard deviation (9.3 points) by the square root of the sample size (49). When computing probabilities such as P( > 19.6071), we need to standardize this value to a z-score and use the standard normal distribution to find this probability. For the 70th percentile for the mean score, we use the z-table to find the corresponding z-score and then transform it back to the original scale using the mean and standard deviation of our sampling distribution. For parts c) and e) where specific probabilities are requested, even though the population distribution is normal, the central limit theorem assures us that the distribution of the sample means will be approximately normal, which allows us to use these probabilities. For decimal precision, all answers should be rounded to four decimal places where possible.