Final answer:
The question pertains to finding the probability that the sample mean cost of shrimp bags will differ by less than $1.4 from the true mean, using the Central Limit Theorem and Z-scores. Calculate the Z-scores for both boundaries of the sample mean, then use the standard normal distribution to find the probability.
Step-by-step explanation:
The subject of the question is Statistics, a branch of mathematics that deals with collecting, analyzing, interpreting, and presenting data. The grade level is college since the question involves the concept of sampling distributions and probabilities, which is typically covered at the college level. The problem asks to find the probability that the sample mean would differ from the true mean by less than 1.4 dollars when sampling from a normally distributed population.
To solve the problem, we utilize the Central Limit Theorem which states that the distribution of sample means will be normally distributed if the sample size is large enough. In this problem, the sample size of 43 is sufficiently large. We use Z-scores to find this probability. The formula for the Z-score in this context is Z = (X - μ) / (σ / √n), where X is the sample mean, μ is the population mean, σ is the population standard deviation and n is the sample size. The population standard deviation is the square root of the variance, which is 6 in this case (since the variance is 36).
Calculating the Z-score for both boundaries of the sample mean (μ ± 1.4):
- Z1 = (47.0 + 1.4 - 47.0) / (6 / √43)
- Z2 = (47.0 - 1.4 - 47.0) / (6 / √43)
The Z-scores are equivalent in magnitude but opposite in sign since they are equidistant from the mean. After calculating these Z-scores, use the standard normal distribution table or a calculator to find the probability that the sample mean falls within these Z-scores. The desired probability is the area under the standard normal curve between these two Z-scores.