Final answer:
The sample mean of the normally distributed TOEFL Writing scores is the appropriate statistic with its sampling distribution also being normal. To calculate the probability of an average score over 18, we use the Z-score and standard normal distribution, which gives an exact probability, not an approximation due to the known population standard deviation.
Step-by-step explanation:
The TOEFL Writing score is normally distributed with a mean (μ) of 16 and a standard deviation (σ) of 3. Given a random sample of five test-takers, we can consider the appropriate statistic to be the sample mean for evaluating their average scores.
Sampling Distribution
The sampling distribution of the sample mean will also be normally distributed according to the Central Limit Theorem, with a mean equal to the population mean (μ = 16) and a standard deviation equal to the population standard deviation divided by the square root of the sample size, σ/√n (standard error), which is 3/√5.
Probability Calculation
To find the probability that the average score is higher than 18, we calculate the Z-score for 18 using the formula Z = (X - μ) / (σ/√n). Then, we look up this Z-score in the standard normal distribution table to find the probability. Assuming the calculation gives a Z-score of z, the probability that the average score is higher than 18 is P(Z > z).
Numeric Answer Justification
Since the TOEFL score distribution is normal and we have the population standard deviation, the use of the Z-score to find the probability is not an approximation but an exact calculation under the normal distribution assumption.