Let X be a continuous random variable. We know that it takes values between 0 and 3 , but we do not know its distribution or its mean and variance. We are interested in estimating the mean of X, which we denote by h . We will use as a conservative value (upper bound) for the standard deviation of X . To estimate , we take i.i.d. samples , which all have the same distribution as X , and compute the sample mean .
1. Express your answers for this part in terms of h and n using standard notation E(H).
2. Given the available information, the smallest upper bound for var(H) is:
3. Calculate the smallest possible positive value of n such that the standard deviation of H is guaranteed to be at most 0.01 .
This minimum value of n is:
4. We would like to be at least 99% sure that our estimate is within 0.05 of the true mean h . Using the Chebyshev inequality, calculate the minimum value of n that will achieve this.
This minimum value of n is:
5. Assume that X is uniformly distributed on [0,3] . Using the Central Limit Theorem, identify the most appropriate expression for a 95% confidence interval for h .
1. H
5. H-1.96*(3^0.5)/((4*n)^0.5),
H+1.96*(3^0.5)/((4*n)^0.5)
1.1. h
1.2. 2.25/n
2. 22500
3. 90000
To answer these questions, let's break them down one by one.
1. We are interested in estimating the mean of X, denoted by h. The sample mean is denoted by H. Using standard notation, E(H) represents the expected value of H.
Answer: E(H) = h
2. The smallest upper bound for var(H) can be obtained using the worst-case scenario for the distribution. Since we do not know the distribution of X, the worst-case scenario is when X is uniformly distributed between 0 and 3. In this case, the upper bound for var(H) is the variance of the uniform distribution on [0,3] divided by the sample size n. The variance of the uniform distribution is (b - a)^2 / 12, where b and a are the upper and lower bounds of the distribution.
Answer: var(H) <= (3-0)^2 / (12n)
3. To ensure that the standard deviation of H is guaranteed to be at most 0.01, we need to solve the inequality: sqrt(var(H)) <= 0.01. Plugging in the upper bound for var(H) from the previous step, we get: sqrt((3-0)^2 / (12n)) <= 0.01. Solving for n gives us the minimum value.
Answer: (3-0) / sqrt(12n) <= 0.01
4. Using the Chebyshev inequality, we can obtain a minimum value for n to be 99% sure that our estimate is within 0.05 of the true mean h. The Chebyshev inequality states that the probability of observing a value within k standard deviations of the mean is at least 1 - 1/k^2. Plugging in the values, we have: 1 - 1/0.05^2 = 0.99. Solving for n gives us the minimum value.
Answer: 1 - 1/(0.05^2) <= 1 - 1/n
5. When X is uniformly distributed on [0,3], we can use the Central Limit Theorem to approximate the distribution of H. The Central Limit Theorem states that for a large enough sample size, the distribution of the sample mean approaches a normal distribution regardless of the shape of the original population distribution. In this case, we can use the following expression to calculate a 95% confidence interval for h: H +/- z * (s / sqrt(n)), where z is the z-score corresponding to the desired confidence level (e.g., 1.96 for 95% confidence), s is the estimated standard deviation of H (bounded by conservative value ), and n is the sample size.
Answer: H +/- 1.96 * (/sqrt(n))