Search
STATLECT > Fundamentals of statistics > Set estimation of the mean

Set estimation of the mean - Exercise set 1

This exercise set contains some solved exercises on set estimation of the mean. The theory needed to solve these exercises is introduced in the lecture entitled Set estimation of the mean.

Exercise 1.1

Suppose you observe a sample of $100$ independent draws from a normal distribution having unknown mean mu and known variance $sigma ^{2}=1$. Denote the $100$ draws by X_1, ..., $X_{100}$. Suppose their sample mean $overline{X}_{100}$ is equal to 1, i.e.:[eq1]

Find a confidence interval for mu, using a set estimator of mu having $90%$ coverage probability.

nav_button Solution

For a given sample size n, the interval estimator[eq2]has coverage probability[eq3]where Z is a standard normal random variable and [eq4] is a strictly positive constant. Thus, we need to find $z$ such that[eq5]But[eq6]where the last equality stems from the fact that the standard normal distribution is symmetric around zero. Therefore $z$ must be such that:[eq7]or:[eq8]Using normal distribution tables or a computer program to find the value of $z$ (see the lecture entitled Normal distribution - Values), we obtain:[eq9]Thus, the confidence interval for mu is:[eq10]

Exercise 1.2

Suppose you observe a sample of $100$ independent draws from a normal distribution having unknown mean mu and unknown variance sigma^2. Denote the $100$ draws by X_1, ..., $X_{100}$. Suppose their sample mean $overline{X}_{100}$ is equal to 1, i.e.:[eq11]and their adjusted sample variance $s_{100}^{2}$ is equal to $4$, i.e.:[eq12]

Find a confidence interval for mu, using a set estimator of mu having $99%$ coverage probability.

nav_button Solution

For a given sample size n, the interval estimator[eq13]has coverage probability[eq14]where $Z_{n-1}$ is a standard Student's t random variable with $n-1$ degrees of freedom and [eq15] is a strictly positive constant. Thus, we need to find $z$ such that[eq16]But[eq6]where the last equality stems from the fact that the standard Student's t distribution is symmetric around zero. Therefore $z$ must be such that:[eq18]or:[eq19]Using a computer program to find the value of $z$ (for example, with the MATLAB command tinv(0.995,99)), we obtain:[eq20]Thus, the confidence interval for mu is:[eq21]

by
About | Contacts | Privacy and terms of use | Sitemap