Final answer:
The question seeks the proportion of daughters taller than the 90th percentile of daughters' heights, given that their mothers are at the 90th percentile, in a bivariate normal distribution with a correlation of 0.5.
Step-by-step explanation:
The question is asking about the proportion of daughters who are taller than the 90th percentile of all daughters, given their mothers are at the 90th percentile of mother's heights, with the additional information that the heights of mother-daughter pairs follow a bivariate normal distribution with a correlation of 0.5.
First, we need to understand that the 90th percentile of any normal distribution is the value below which 90% of the observations may be found. This equates to a z-score of approximately 1.28. Due to the correlation of 0.5, we expect that taller mothers are more likely to have taller daughters, but this does not guarantee that a mother's height determines that the daughter will also be in the top 90th percentile.
To find the specific proportion, we'd use the given correlation to calculate the conditional probability that a daughter is above the 90th percentile given her mother's height is at the 90th percentile. This is a more complex statistical problem that often requires the use of statistical software or tables that provide probabilities for the bivariate normal distribution.