Final answer:
To find the probability that the total number of pasta packages sold by the store for a random sample of 58 days is less than 1862, we can use the Central Limit Theorem. First, we calculate the mean and standard deviation for the sample mean. Then, we standardize the value of 1862 using the formula z = (x - mean) / standard error. Finally, we use a z-table or calculator to find the probability associated with the standardized value.
Step-by-step explanation:
To find the probability that the total number of pasta packages sold by the store for a random sample of 58 days is less than 1862, we need to use the Central Limit Theorem. First, we need to calculate the mean and standard deviation for the sample mean. The mean of the sample mean is equal to the mean of the original data, which is 32. The standard deviation of the sample mean (also known as the standard error) is equal to the standard deviation of the original data divided by the square root of the sample size. In this case, the standard deviation is 4 and the sample size is 58, so the standard error is 4 / sqrt(58) = 0.527.
Next, we need to standardize the value of 1862 using the formula z = (x - mean) / standard error. Plugging in the values, we get z = (1862 - 32) / 0.527 ≈ 3510.
Finally, we can use a z-table or a calculator to find the probability associated with a z-score of 3510. This probability represents the probability that the total number of pasta packages sold is less than 1862.