Chapter 7 Inferences Based on a Single Sample: Estimation with - PowerPoint PPT Presentation

Chapter 7 Inferences Based on a Single Sample: Estimation with Confidence Intervals

Large-Sample Confidence Interval for a Population Mean How to estimate the population mean and assess the estimate’s reliability?  is an estimate of , and we use CLT to x assess how accurate that estimate is According to CLT, 95% of all from sample x size n lie within of the mean   1 . 96 x We can use this to assess accuracy of x  as an estimate of

Large-Sample Confidence Interval for a Population Mean  1 . 96      1 . 96 . 95 x x x n We are 95% confident, for any from sample  x  size n, that will lie in the interval  x 1 . 96 n

Large-Sample Confidence Interval for a Population Mean We usually don’t know , but with a large  sample s is a good estimator of .  We can calculate confidence intervals for different confidence coefficients Confidence coefficient – probability that a randomly selected confidence interval encloses the population parameter Confidence level – Confidence coefficient expressed as a percentage

Large-Sample Confidence Interval for a Population Mean  The confidence coefficient is equal to 1- , and is split between the two tails of the distribution

Large-Sample Confidence Interval for a Population Mean The Confidence Interval is expressed more generally as      x z x z   2 2 x n For samples of size > 30, the confidence interval is expressed as   s    x z  2   n Requires that the sample used be random

Large-Sample Confidence Interval for a Population Mean Commonly used values of z  /2 Confidence level 100(1-  )   /2 z z  /2  2 90% .10 .05 1.645 95% .05 .025 1.96 99% .01 .005 2.575

Small-Sample Confidence Interval for a Population Mean 2 problems presented by sample sizes of less than 30 – CLT no longer applies – Population standard deviation is almost always unknown, and s may provide a poor estimation when n is small

Small-Sample Confidence Interval for a Population Mean If we can assume that the sampled population is approximately normal, then the sampling distribution of can be assumed x to be approximately normal     x x Instead of using we use   z t  n s n This t is referred to as the t-statistic

Small-Sample Confidence Interval for a Population Mean The t-statistic has: a sampling distribution very similar to z Variability dependent on n, or sample size. Variability is expressed as (n-1) degrees of freedom (df). As (df) gets smaller, variability increases

Small-Sample Confidence Interval for a Population Mean Table for t-distribution contains t-value for various combinations of degrees of freedom and t  Partial table below shows components of table Need Table 7.3 from text inserted here.

Small-Sample Confidence Interval for a Population Mean Comparing t and z distributions for the same  , with df=4 for the t-distribution, you can see that the t-score is larger, and therefore the confidence interval will be wider. The closer df gets to 30, the more closely the t-distribution approximates the normal distribution

Small-Sample Confidence Interval for a Population Mean When creating a Confidence interval around  for a small sample we use   s    x t  2   n basing t  /2 on n-1 degrees of freedom We assume a random sample drawn from a population that is approximately normally distributed

Large-Scale Confidence Interval for a Population Proportion Confidence intervals around a proportion are confidence intervals around the probability of success in a binomial experiment Sample statistic of interest is ˆ p ˆ Mean of sampling distribution of is p. p is an p ˆ p unbiased estimator of Standard deviation of the sampling distribution is  p  where q=1-p pq n ˆ ˆ p For large samples, the sampling distribution of is approximately normal

Large-Scale Confidence Interval for a Population Proportion   ˆ Sample size n is large if falls between 0 p 3 ˆ p and 1 Confidence interval is calculated as ˆ ˆ pq p q       ˆ ˆ ˆ p z p z p z    2 2 2 p n n x   p  ˆ ˆ ˆ where and 1 q p n

Large-Scale Confidence Interval for a Population Proportion When p is near 0 or 1, the confidence intervals calculated using the formulas presented are misleading An adjustment can be used that works for any p, even with very small sample sizes   ~ ~  1 p p ~  p z  2  n 4

Determining the Sample Size When we want to estimate  to within x units with a (1-  ) level of confidence, we can calculate the sample size needed We use the Sampling Error (SE), which is half the width of the confidence interval To estimate  with Sampling error SE and 100(1-  )% confidence,   2 2  z  2  n   2 SE where  is estimated by s or R/4

Determining the Sample Size Assume a sample with  =.01, and a range R of .4 What size sample do we need to achieve a desired SE of .025?   2   2  2 2 z ( 2 . 575 ) . 1  2    n 106 . 09     2 2 . 025 SE

Determining the Sample Size Sample size can also be estimated for population proportion p     2 z pq  2  n   2 SE Since pq is unknown you must estimate. Estimates with a value of p being equal or close to .5 are the most conservative

Finite Population Correction for Simple Random Sampling Used when the sample size n is large relative to the size of the population N, when n/N >.05 Standard error calculation for  with correction  s N n  ˆ  x N n Standard error calculation for p with correction     ˆ ˆ p 1 p N n  ˆ  p n N

Sample Survey Designs • Simple Random Sample • Stratified Random Sampling – separation into two or more groups of sampling units – Produce estimators with smaller standard errors – Increase representativeness – Can reduce cost

Sample Survey Designs • Systematic Sampling – Sampling of every nth unit – Samples are easier to select – Can lead to systematic bias, particularly if there is periodicity in the data you are drawing the data from

Sample Survey Designs • Randomized Response Sampling – Used when questions in the survey are of a sensitive nature and likely to result in false answers

Sample Survey Designs Nonresponse – when units in the sample do not produce observations Nonresponse can produce bias in the results of the survey if there is a relationship between the type of response and whether or not a response is achieved. If a random sample is called for, any nonresponse means that your sample is no longer random