Final answer:
To find 65% of 90, use the proportion 65/100 = x/90, which results in x = 58.5. Sample size for population proportions relies on a formula involving the z-score, estimated proportion, and margin of error. The 90th percentile is the value below which 90% of data in a distribution lies, calculable using statistical methods or technology.
Step-by-step explanation:
To find 65% of 90 using a proportion, we can set up a ratio that compares the part to the whole. Since percentages are out of a total of 100, this will translate to 65/100 = x/90, where x represents the part of 90 that is 65%.
To solve for x, you would cross-multiply and divide: (65)(90) = 100x. This gives us 5850 = 100x, and when you divide both sides by 100 to isolate x, you get x = 58.5. Therefore, 65% of 90 is 58.5.
If you are considering a population parameter and aiming for a certain level of confidence, the sample size necessary can be calculated using a formula that includes the z-score for the confidence level, the estimated proportion (p'), the margin of error (E), and the standard deviation of the population. This can be represented by the formula n = (z² p'q') / E².
Regarding the percentage of a distribution, if you are seeking the 90th percentile, you are looking for a value k, such that 90% of the data falls below k. This involves using the area under the curve of a probability distribution, potentially utilizing technology like the TI-83 or TI-84 calculator to find this exact number.