Using the Central Limit Theorem, the probability that the sample mean cost of 5-gallon pails of ice cream is less than $32.3 is approximately 0.8508 or 85.08%.
To find the probability that the sample mean would be less than $32.3, we use the Central Limit Theorem which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough. In this case, with a sample size of 31, we can assume normality. First, we calculate the standard error of the mean using the formula σ/√n, where σ is the standard deviation and n is the sample size. The variance of the cost is given as 49; hence the standard deviation is the square root of 49, which is 7.
Standard error (SE) = σ/√n = 7/√31 = 1.257.
Next, we find the z-score for $32.3 using the formula Z = (X - μ)/SE, where X is the value we're comparing the sample mean to, and μ is the population mean.
Z = (32.3 - 31) / 1.257 = 1.036.
Using a z-table, we find the probability corresponding to a z-score of 1.036. The probability of a z-score less than 1.036 is approximately 0.8508, so the probability that the sample mean is less than $32.3 is also 0.8508. Hence, we conclude there is an 85.08% chance that the sample mean cost of ice cream will be less than $32.3.