It was reported by the National Forensic Center in the Guide to Experts Fees, 1998-99 that the average fee of forensic accountants for court testimony was 1500. Assume that the population standard deviation is 600. What is the probability that, in a random sample of 100 forensic accountants, the average fee falls between 1450 and 1600 per day?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to your Z scores.
To find the probability that the average fee falls between 1450 and 1600 per day in a random sample of 100 forensic accountants, we will use the Central Limit Theorem.
The Central Limit Theorem states that for a random sample size of at least 30, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the original population.
In this case, we have a random sample of 100 forensic accountants, which is larger than 30, so we can use the Central Limit Theorem.
Step 1: Find the mean and standard deviation of the sampling distribution of the sample mean.
The population standard deviation is given as 600. The sample size is 100. The mean of the sampling distribution is equal to the population mean, which is 1500.
The standard deviation of the sampling distribution, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size:
Standard Error (SE) = population standard deviation / sqrt(sample size)
SE = 600 / sqrt(100)
SE = 60
Step 2: Convert the values of 1450 and 1600 to z-scores using the formula:
z = (x - mean) / standard deviation
For 1450:
z1 = (1450 - 1500) / 60 = -0.8333
For 1600:
z2 = (1600 - 1500) / 60 = 1.6667
Step 3: Look up the z-scores in the standard normal distribution table or use a calculator to find the corresponding probabilities.
The probability that the average fee falls between 1450 and 1600 per day can be calculated as the difference between the two z-scores:
P(1450 ≤ x ≤ 1600) = P(-0.8333 ≤ z ≤ 1.6667)
Using a standard normal distribution table or a calculator, we can find the probabilities corresponding to the z-scores:
P(-0.8333 ≤ z ≤ 1.6667) = 0.7967 - 0.2033 = 0.5934
Therefore, the probability that in a random sample of 100 forensic accountants, the average fee falls between 1450 and 1600 per day is approximately 0.5934 or 59.34%.