Final answer:
To answer the given question, we need to find the mean sample weight and the standard deviation of the sample weights of the candies in the table. Then, we calculate the sum of the sample weights and the standard deviation of the sum of the weights. Next, we find the probability that the weights sum to at least 396.9 g. Finally, we compare the claimed proportion of red candies to the actual proportion in the sample bag to determine if the labeling is accurate.
Step-by-step explanation:
In order to answer the given question, we need to find the mean sample weight and the standard deviation of the sample weights of the candies in the table.
a. To find the mean sample weight, we sum up all the weights and divide it by the number of candies. The standard deviation can be found using the formula for sample standard deviation. We subtract the mean weight from each weight, square the result, sum up all the squares, divide by the number of candies minus 1, and take the square root of the result.
b. To find the sum of the sample weights in the table, we simply sum up all the weights. The standard deviation of the sum of the weights can be found using the formula for the standard deviation of the sum of independent random variables. We multiply the standard deviation of the individual weights by the square root of the number of candies.
c. To find the probability that the weights sum to at least 396.9 g, we need to calculate the z-score of this value using the formula for z-score. We then use a standard normal distribution table to find the probability associated with the z-score.
d. To determine if the candy company's labeling is accurate, we compare the claimed proportion of red candies (33%) to the actual proportion of red candies in the sample bag. If the actual proportion is close to the claimed proportion, we can conclude that the labeling is accurate.