Final answer:
The probability of the sample mean costing less than $1.50 from the true mean is calculated using the Central Limit Theorem and involves finding the standard error, z-scores, and the corresponding probability using the standard normal distribution.
Step-by-step explanation:
The question is asking to find the probability that the sample mean cost of 5 gallons of ice cream would differ from the true mean by less than $1.50 when a sample of 142 5-gallon pails is selected. To solve this, we need to use the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough, regardless of the population distribution.
First, we calculate the standard error of the sample mean by dividing the population standard deviation (√49 = 7) by the square root of the sample size (√142), and then use the standard normal distribution to find the z-scores for the sample mean minus 1.5 and the sample mean plus 1.5. Finally, we find the probability that the z-score falls between these two values using standard normal distribution tables or a calculator.
The steps are as follows:
- Calculate the standard error: SE = 7 / √142.
- Determine the z-scores for the sample mean ± $1.50.
- Find the probability between the two z-scores.
- Calculate and round the probability to four decimal places.
By completing these steps, the probability that the sample mean will differ from the true mean by less than $1.50 can be found.