The true proportion of respondents

[Akricatos]

Fellow, Maths lovers!

I am now trying to remember my early statistics course in order to solve the following problems:

1) During the survey, 40% of the population maintain interest in mathematics. 1000 people were interviewed. With what probability can it be argued that the proportion of respondents who support mathematics differs from the true proportion by no more than 0.05?
Should I use this statistical model: P((μn/n)−P≤0.4)≥Z ?

2) Is it more reasonable to use the Bayes' theorem or the full probability formula to solve the following problem: A product defect is 5%. Each product with the same probability can be checked by one of 2 controllers. The first of them detects an error with a probability of 0.7, the second - with a probability of 0.8. What is the probability that a recognized product is defective?

Thank you in advance for your assistance!

 

[afos]

1) Ill-posed question. It should say: "With which level of confidence can it be argued that the sample proportion of respondents who support mathematics differs from the true proportion by no more than 0.05?" This is a question about confidence intervals, not probabilities. That is an important distinction because the sample estimate of 0.4 and the true proportion are both fixed numbers, not random variables. So any talk about probabilities is meaningless.

The X% confidence interval for the true proportion is determined by formula

[p - Z * sqrt(p * (1 - p) / n), p + Z * sqrt(p * (1 - p) / n)],

where p = sample proportion, n = sample size, Z = quantile of standard normal distribution corresponding to probability 100% - X% / 2.

2) Again: "What is the probability that a recognized product is defective", conditional on what piece of information? Did a specific controller label the product defective? Did an unknown, random controller label it as defective?

Either way, the total probability formula is only one of the building blocks. You should also use formulas for conditional probability. Can be Bayes rule.

In general, ability to solve the above questions requires coherent training in the foundations of probability theory and statistics. One has to distinguish between several fundamental concepts. You may benefit from a coherent probability and statistics tutorial (the bottom part of the page, skip the beginning). The reading list covers both frequentist and Bayesian approaches.