Interval Estimation

Interval Estimation

1. inferential statistics

   • A range of values within which, we believe, the true parameter lies with high probability
   • This interval will be named a confidence interval
     • A type of interval to estimate the true value of the population parameter, used to indicate the reliability of an estimate,
     • It is an observed interval, different from sample to sample, which potentially includes the unobservable true parameter of population
   • Confidence Level
     • The confidence level is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of the corresponding parameter.
     • How frequently the observed interval contains the parameter if the experiment is repeated, is called the confidence level
       • if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level
     • Gives us the success rate of the procedure used to construct the confidence interval.
       • Confidence interval
         • The confidence interval is different from random sample to sample, which is therefore also random.
         • Be careful to interpret confidence intervals correctly
           • Example: 0.456 < p < 0.954 at 95% confidence level
             • Correct: We are 95% confident that the interval from 0.456 to 0.954 actually does contain the true value of p
             • Incorrect: There is a 95% chance that the true value of p will fall between 0.456 and 0.954
         • Constructing a Confidence Interval
           • 1. Verify basic requirements are satisfied (see previous definition)
             • 2. Find the critical value zα/2 from the z tables that corresponds to the desired confidence leve
               • 3. Evaluate the margin of error using formula for E
                 • 4. Find the values of:
     • The confidence level is a kind of probability.
     • A confidence level is always represented as 1- a
       • a is called the significance level (is the rejecting probability of the study)

2. Point estimation

   • an estimate based on only a single random sample, expected to vary from sample to sample for example, the mean of some quantity.

3. Estimating a Population Proportion

   • A proportion is a statistic/parameter of a sample/population containing binomial measurement data
   • Requirements
     • The sample is a simple random sample
     • Binomial distribution conditions are satisfied
     • There are at least 5 successes and 5 failures
       • This condition ensures that np ≥ 5 and nq ≥ 5
   • Notations
     • Considering a sample of size n consistent with binomial distribution, where the number of success is x.
       • Sample proportion of success
       • Sample proportion of failure
       • Population proportion of success: P
       • Assumption
         • when n is large enough, the sample proportion of success is in a normal distribution

4. Estimating a Population Mean

   • Requirements
     • Sample is a simple random sample
     • Either n > 30, or the population is normally distributed
     • Standard deviation for the population is known
   • Notations:
     • μ = population mean
     • σ = population standard deviation (known)
     • x̄ = sample mean
     • n = sample size
   • Confidence Interval for Estimating a Population Mean (with σ known)
     • Margin error
   • Assumptions
     • When n is large enough, the sample mean is assumed to be consistent with a normal distribution
   • σ not known
     • If we do not know the value of the population’s Standard deviation σ. We need to use Student’s t-distribution instead of the normal distribution that we used as an assumption for interval estimation
     • We do not know the standard deviation of the population, so we estimate it with the value of the sample standard deviation s,
       • This introduces another source of unreliability
         • In order to maintain a desired confidence level, we compensate for this additional unreliability by making the confidence interval wider.

5. Student t-Distribution

   • If a population has a normal distribution, the distribution of all samples of size n collected from the population forms:
   • Properties of the t-Distribution
     • 1. It is different for different sample sizes
     • 2. Same general bell shape as the standard normal distribution, but it reflects the greater variability that is expected with small samples
     • 3. It has a mean of t = 0
     • 4. The standard deviation of the Student tdistribution varies with the sample size, but it is greater than 1
     • 5. As the sample size n gets larger, the Student t-distribution gets closer to the standard normal distribution
   • Interval estimation of population mean when σ is unknown
     • Step 1: Check requirements (For estimating population mean)
       • Step 2: Confidence interval
         • We use ta/2 as the critical value, because we assume that the sample mean is consistent with a Student t-distribution around the population mean
   • t-Table
     • In order to facilitate the interval estimation, mathematicians and statistics pioneers developed a t-table which lists the corresponding critical values against the significance level and the degree of freedom of sample used for estimation
     • Confidence level: 1- a
     • Significance level: a
     • Degree of freedom: n - 1 (n is sample size)
     • t-tables are used to provide a t-score
     • Degrees of Freedom is given in the left-hand column
     • the body of the table tests the level of significance for either onetailed test or two-tailed test
     • Degrees of Freedom
       • The degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.
         • Often abbreviated as df
         • where df = n - 1

6. Estimating a Population Variance

   • Chi-Square Distribution
     • The following sample statistic has a sampling distribution called the chi-square distribution
       • where n is the sample size; s2 is the sample variance; and (sigma)2 is the population variance
     • It is not symmetric, unlike the normal and Student t-distributions
     • 2. Values of chi-square can be positive or zero, but not negative,
     • 3. It is different for each number of degrees of freedom
     • Because chi-square is not symmetric, we cannot apply a confidence interval formula of s2 ± E
       • We must do separate calculations for the upper and lower confidence interval limits
     • Notation
       • σ2 is the population variance
       • s2 is the sample variance
       • χ2 L is the left-tail critical value of χ2
       • χ2 R is the right-tail critical value of χ2
     • Chi-square Distribution Table
       • Critical values of the chi-square distribution are contained in chi-square tables,
       • Degrees of freedom, as with the Student t-Distribution, degrees of freedom is n-1
       • How to use Chi-square Distribution Table
       • Constructing a confidence interval for σ2
         • Verify that the requirements are satisfied
           • Sample is simply random sample
           • Population is normally distributed
           • Using n – 1 degrees of freedom, refer to the chi-square tables and find the critical values χR 2 and χL 2 that corresponds to the desired confidence level
             • Evaluate the upper/lower confidence interval limits using this format:

Bilder und Dateien der Mindmap