Final answer:
The symmetrical limits for the sample percentage to contain the population percentage with a 90% probability are approximately 64.923% and 65.077%.
Step-by-step explanation:
In this question, we are asked to find the symmetrical limits of the sample percentage that will contain the population percentage with a 90% probability. Given that the institute reported that 65% of its members indicate that lack of ethical culture within financial firms has contributed most to the lack of trust in the financial industry, we want to find the range within which the sample percentage will fall with 90% probability.
To find the symmetrical limits, we can use the formula for the margin of error (E) in a proportion estimate:
E = z*sqrt((p*(1-p))/n)
Where z is the z-score for the desired level of confidence (in this case, 90%), p is the population proportion (65%), and n is the sample size (100).
Using a standard normal distribution table or a calculator, we can find the z-score for a 90% confidence level, which is approximately 1.645.
Plugging the values into the formula:
E = 1.645*sqrt((0.65*(1-0.65))/100)
Simplifying the equation, we get:
E = 1.645*sqrt((0.22175)/100)
E = 1.645*sqrt(0.0022175)
E ≈ 1.645*0.04701
E ≈ 0.07749
So, the margin of error is approximately 0.07749.
To find the symmetrical limits, we add and subtract the margin of error from the sample percentage:
65% + 0.07749 = 65.077%
65% - 0.07749 = 64.923%
Therefore, with a 90% probability, the sample percentage will be contained within the symmetrical limits of approximately 64.923% and 65.077%.