The sample mean estimate Theta hat of the mean of a random variable with variance 1, based on 100 samples, happened to be 22. The 80% confidence interval provided by the CLT is of the form [a,b] with a=? and b=?
The central limit theorem (CLT) states that for sufficiently large sample sizes, the distribution of sample means follows an approximately normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the sample mean estimate Theta hat is equal to 22. Since the random variable has a variance of 1, the population standard deviation is also 1.
The CLT tells us that the standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size.
So, the standard deviation of the sample mean is 1 / sqrt(100) = 1/10.
To find the 80% confidence interval, we need to find the z-score corresponding to the desired confidence level. For a one-tailed 80% confidence level, the z-score is approximately 1.28.
The margin of error is equal to the z-score multiplied by the standard deviation of the sample mean.
Margin of error = 1.28 * (1/10) = 0.128
To calculate the lower bound of the confidence interval, subtract the margin of error from the sample mean estimate:
Lower bound = 22 - 0.128 = 21.872 (approximately)
To calculate the upper bound of the confidence interval, add the margin of error to the sample mean estimate:
Upper bound = 22 + 0.128 = 22.128 (approximately)
Therefore, the 80% confidence interval is approximately [21.872, 22.128].
To construct the confidence interval using the Central Limit Theorem (CLT), we need to consider the sample mean and the number of samples.
Given that the sample mean estimate (theta hat) is 22 and the variance is 1, we can calculate the standard error of the mean using the formula: standard error (SE) = sqrt(variance / number of samples).
SE = sqrt(1 / 100) = 0.1
The CLT establishes that the sampling distribution of the sample mean, as the number of samples (n) increases, approaches a normal distribution with mean equal to the population mean and variance equal to the population variance divided by the number of samples.
To calculate the confidence interval, we need to know the critical value associated with the desired confidence level (in this case, 80%). The critical value is obtained from the standard normal distribution.
For an 80% confidence interval, we need to find the critical value Z such that the area under the normal curve between -Z and Z equals 0.80. This critical value can be found using a standard normal distribution table or a statistical calculator.
Using the standard normal distribution table, we find that the critical value Z for an 80% confidence interval is approximately 1.28.
Now we can calculate the margins of error (ME) using the formula: ME = Z * SE.
ME = 1.28 * 0.1 = 0.128
Finally, the confidence interval is given by [theta hat - ME, theta hat + ME], which in this case becomes: [22 - 0.128, 22 + 0.128].
Therefore, the 80% confidence interval is approximately [21.872, 22.128].
So, a is approximately 21.872 and b is approximately 22.128.