To find the probability that a random sample of 7 has a mean between 564 and 587, we can use the z-score formula to standardize the values 564 and 587, and then use a z-table or a computer program to find the area under the normal curve between those two standardized values.
The z-score formula is:
z = (x - mean) / standard deviation
Where x is the value we are standardizing, mean is the mean of the population, and standard deviation is the standard deviation of the population.
We can use this formula to standardize 564 and 587:
z(564) = (564 - 555) / 72 = 0.97
z(587) = (587 - 555) / 72 = 1.67
We can use a z-table or a computer program to find the area under the normal curve between these two values. The probability that a random sample of 7 has a mean between 564 and 587 is the area under the curve between these two standardized values. This probability is approximately 0.3862, or 38.62%.