Confidence intervals and hypothesis testing Part I


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  • The following topics will be covered in this lecture:
    • Random samples
    • Point estimators
    • The central limit theorem
    • Standard error


  • Suppose we take a sample of \( n = 10 \) observations \( \{x_{1,i}\}_{i=1}^{10} \) from a population and compute the sample average,

    \[ \overline{x}_1 = \frac{1}{n} \sum_{i=1}^n x_{1,i} = \frac{1}{10}\sum_{i=1}^{10} x_{1,i} \]

    getting the result \( \overline{x}_1 = 10.2 \).

  • Now we repeat this process, taking a second sample of \( n = 10 \) observations from the same population,

    \[ \{x_{2,i}\}_{i=1}^{10} \]

    and the resulting sample average is \( \overline{x}_2=10.4 \).

  • This discrepancy is what we call sampling error, in which the random variation in a sample of a fixed size \( n \) upon replication produces differences in the computation of a statistic.

  • The sample average depends on the observations in the sample, which differ from sample to sample because they are random variables.

  • Consequently, the sample average (or any other function of the sample data) is a random variable.

  • Because a statistic is a random variable, it has a probability distribution.

Random samples

  • Specifically, suppose that we want to obtain an estimate of a population parameter, where the population is modeled with a random variable \( X \).

  • We know that before the data are collected, the observations are considered to be random variables,

    • i.e., we treat an independent sequence of measurements of \( X \),

    \[ X_1, X_2, \cdots , X_n \]

    • as random variables all drawn from a parent distribution \( X \sim F_X(x) \) (where the CDF will define the distribution).
    Random sample
    The random variables \( X_1 , X_2, \cdots , X_n \) are a random sample of size \( n \) if the \( X_i \)’s are independent random variables and every \( X_i \) has the same probability distribution.
  • We then say that the measurements we obtain are possible outcomes of the sample variables \( \{X_i\}_{i=1}^n \); particularly, if we make a computation of the sample mean,

    \[ \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \]

    the above is treated as a random variable (a linear combination of random variables) which has a random outcome, dependent on the realizations of the \( X_i \).

Random samples

  • More generally, any function of the observations, i.e., any statistic, is also modeled as a random variable.

  • If \( h \) is a general function used to compute some statistic, we thus define

    \[ \tilde{X} = h(X_1, \cdots, X_n) \]

    to be a random variable that will depend on the particular realizations of \( X_1,\cdots, X_n \).

  • We call the probability distribution of a statistic a sampling distribution.

    Sampling Distribution
    The probability distribution of a statistic is called a sampling distribution.

Random samples

  • Given particular realizations of the sample random variables, we obtain a fixed numerical value.

  • Each numerical value in a data set is treated as the observed realization of a random variable.

  • Given particular realizations \( x_1,\cdots,x_n \) of the random variables \( X_1, \cdots, X_n \), the value

    \[ \overline{x} = \frac{1}{n}\sum_{i=1}^n x_i \]

    is not a random variable, as this is a fixed numerical value.

  • Given some particular, observed realizations \( x_1, \cdots,x_n \),

    \[ \tilde{x} = h(x_1, \cdots, x_n) \]

    is a fixed numerical value, based on the fixed, observed data values \( x_1, \cdots, x_n \).

Random samples

  • When discussing inference problems, it is convenient to have a general symbol to represent the parameter of interest – we use the Greek symbol \( \theta \) (theta) to represent the parameter.

  • The symbol \( \theta \) can represent the mean \( \mu \), the variance \( \sigma^2 \), or any parameter of interest to us.

  • The objective of point estimation is to estimate a single number based on sample data that is the most plausible value for \( \theta \).

  • The numerical value of a sample statistic is used as the point estimate.

  • Once we describe the process of point estimation, the next step is to describe how we quantify the uncertainty of the estimate.

  • If \( X \) is a random variable with probability distribution \( F_X(x) \), characterized by the unknown parameter \( \theta \),

    • and if \( X_1 , X_2, \cdots , X_n \) is a random sample of size \( n \) from \( X \),
  • the statistic \( \hat{\Theta} = h(X_1 , X_2 , ... , X_n ) \) given as a function of the sample is called a point estimator of \( \theta \).

  • Note that \( \hat{\Theta} \) is a random variable because it is a function of random variables.

  • After the sample has been selected, \( \hat{\Theta} \) takes on a particular numerical value \( \hat{\Theta} \) called the point estimate of \( \theta \).

  • The uncertainty of the point estimate \( \hat{\Theta} \) can be understood as how much will the sampling error cause a discrepancy between \( \hat{\Theta} \) and the true \( \theta \).

Point estimators

  • We will now introduce some formal definitions:
  • Point estimators
    A point estimate of some population parameter \( \theta \) is a single numerical value \( \hat{\theta} \) of a statistic \( \hat{\Theta} \). This is a particular realization of the random variable \( \hat{\Theta} \), viewed as a random variable; \( \hat{\Theta} \) is called the point estimator.
  • Estimation problems modeled as above occur frequently in engineering.

  • We often need to estimate

    • The mean \( \mu \) of a single population
    • The variance \( \sigma^2 \) (or standard deviation \( \sigma \)) of a single population
    • The proportion \( p \) of items in a population that belong to a class of interest
    • The difference in means of two populations, \( \mu_1 - \mu_2 \)
    • The difference in two population proportions, \( p_1 − p_2 \)

Point estimators

  • Reasonable point estimates of these parameters are as follows:

    • For \( \mu \),
      • the estimate is \( \hat{\mu}=\overline{x} \), the sample mean.
    • For \( \sigma^2 \),
      • the estimate is \( \hat{\sigma}^2 = s^2 \), the sample variance.
    • For \( p \),
      • the estimate is \( \hat{p}=\frac{x}{n} \), the sample proportion, where \( x \) is the number of items in a random sample of size \( n \) that belong to the class of interest.
    • For \( \mu_1 -\mu_2 \),
      • the estimate \( \hat{\mu}_1 - \hat{\mu}_2 = \overline{x}_1 - \overline{x}_2 \), the difference between the sample means of two independent random samples.
    • For \( p_1 − p_2 \) ,
      • the estimate is \( \hat{p}_1 - \hat{p}_2 \) , the difference between two sample proportions computed from two independent random samples.
  • Although a point estimate may be the “best” estimate for a population parameter given a single sample, it is critically important to understand how far this estimate might be from the true value.

  • In order to determine the accuracy of this estimate, we use the concept of the sampling distribution to derive hypothesis tests and confidence intervals.

Central limit theorem

  • Let's consider a simple argument for the sampling distribution of the sample mean \( X \).

  • Suppose that a random sample of size \( n \) is taken from a normal population with mean \( \mu \) and variance \( \sigma^2 \).

  • By definition of a random sample each observation in this sample, say, \( X_1, X_2, \cdots, X_n \), is a normally and independently distributed random variable with mean \( \mu \) and variance \( \sigma^2 \).

  • A special property of the normal distribution is that it can be translated and rescaled while remaining normal;

    • similarly, a sum of independent, normally distributed random variables are also normally distributed.
  • We conclude that the sample mean

    \[ \overline{X}= \frac{X_1 + X_2 + \cdots + X_n}{n} \]

    has a normal distribution with mean

    \[ \mu_\overline{X} = \frac{\mu + \mu + \cdots + \mu}{n} = \mu \]

    • and variance

    \[ \sigma^2_\overline{X} = \frac{\sigma^2 + \sigma^2 + \cdots + \sigma^2}{n^2} = \frac{\sigma^2}{n} \]

Central limit theorem continued

  • More generally, if we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately normal with mean \( \mu \) and variance \( \frac{\sigma^2}{n} \) if the sample size \( n \) is large.

  • This is one of the most useful theorems in statistics, called the central limit theorem:

    The central limit theorem
    Let \( X_1 , X_2 , \cdots , X_n \) be a random sample of size \( n \) taken from a population with mean \( \mu \) and finite variance \( \sigma^2 \) and \( \overline{X} \) be the sample mean. Then the limiting form of the distribution of \[ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \] as \( n \rightarrow \infty \) is the standard normal distribution.
  • Put another way, for \( n \) sufficiently large, \( \overline{X} \) has approximately a \( N\left(\mu, \frac{\sigma^2}{n}\right) \) distribution – this says the following.

    • Suppose we take a sample of size \( n \) and compute the sample mean \( \overline{X} \).
    • Then suppose we replicate this sample and record the observed realizations for the sample mean \( \overline{x}_1, \overline{x}_2, \cdots \).
    • If the sample size \( n \) is lage, these data points \( \overline{x}_1, \cdots \) will be approximately bell shaped with the following properties:
      • the bell will be centered approximately at \( \mu \), the true population mean;
      • the spread of the data around the center will be given by approximately by the standard deviation \( \frac{\sigma}{\sqrt{n}} \).
    • Particularly, if \( n \) is very large, the observed sample means will be very close to the center (the true mean).

Central limit theorem continued

  • As a visualization of the concept, suppose again that we have a random sample indexed by \( j \) \[ X_{j,1}, \cdots, X_{j,n}. \]
  • We will make replications for \( j=1,\cdots,m \) and get a random variable for sample mean indexed by \( j \), \[ \overline{X}_j = \frac{1}{n}\sum_{i=1}^n X_{j,i}. \]
  • When we observe a realization of \( \overline{X}_j=\overline{x}_j \) or respectively the sample \[ X_{j,1}=x_{j,1}, \cdots, X_{j,n}=x_{j,n}, \] we record these fixed numerical values.