Final answer:
To find the probability of a sample mean differing from the true mean by less than $1.4, find the standard error of the sample mean, calculate the z-score, and use a z-table to find the corresponding probability.
Step-by-step explanation:
To find the probability that the sample mean would differ from the true mean by less than $1.4, we need to calculate the standard error of the sample mean. The standard error is equal to the standard deviation divided by the square root of the sample size. In this case, the standard deviation is sqrt(49) = 7, and the sample size is 54. So, the standard error is 7/√54 = 0.9487.
Next, we need to find the z-score for a difference of $1.4 from the mean. The z-score formula is (x - μ) / σ, where x is the difference, μ is the mean, and σ is the standard error. Plugging in the values, we get z = (1.4 - 0) / 0.9487 ≈ 1.4746.
Finally, we can use a z-table or a calculator to find the probability corresponding to the z-score. The probability of a difference of less than $1.4 is the area under the normal curve to the left of the z-score. From the z-table, we find that the probability is approximately 0.9306.