Final answer:
To find the likelihood of the sample mean height of waves exceeding 339.5 cm, we calculate the standard error, determine the corresponding z-score, and then use the standard normal distribution to find the probability.
Step-by-step explanation:
The student is asking about the probability of the sample mean height of waves being more than 339.5 cm when given the mean height of waves is 320 cm and the standard deviation is 80 cm, with a sample size of 64. To answer this question, we use the standard normal distribution and z-scores. First, we calculate the standard error (SE) of the mean using the formula SE = \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size. Then, we find the z-score of the sample mean using the formula \( z = \frac{\bar{x} - \mu}{SE} \), where \( \bar{x} \) is the sample mean, and \( \mu \) is the population mean.
In this case, SE = \( \frac{80}{\sqrt{64}} = 10 \) and the z-score \( z = \frac{339.5 - 320}{10} = 1.95 \). Finally, to find the probability that the sample mean is greater than 339.5 cm, we look up the z-score in the standard normal distribution table or use a calculator with normal distribution functions to find the area to the right of this z-score. The probability that the sample mean height of waves is more than 339.5 cm is P(\( \bar{x} \)>339.5), which can be found using the standard normal distribution.