Estimation Point Estimates estimates Confidence Intervals Assumptions about

Estimation

Point Estimates estimates

Confidence Intervals

Assumptions about Confidence Intervals The types of confidence intervals in this course make the assumption that the sample means are normally distributed.

Confidence Intervals

Confidence Interval for Mean If variance of population is known • Confidence Interval (1 - )100% bagi If variance of population is unknown • Confidence Interval (1 - )100% bagi

The t distribution • The t distribution is a statistical distribution that varies according to the number of degrees of freedom (Sample size – 1) • As df gets larger, the t approximates the normal distribution.

Selecting the critical value – t-dist • Selecting the critical value of the t-distribution requires these steps. – – Select α level Determine degrees of freedom (n-1) Find value for t in appropriate column Critical value of t is at intersection of df row and αlevel column.

Example 1. 2. 3. 4. Five measurements of the pressure in a vessel are recorded. They are: 1 , 2 , 1, 3, 2 Estimate the variance of pressure. Estimate the variance of the mean pressure. Write an expression for a confidence interval for a pressure measurement. Write an expression for a confidence interval for the mean pressure measurement.

Example Let X denote the pressure in a vessel mean of x = 2 standard deviation of x = 0. 71 standard deviation of mean = 0. 32

Large Sample Estimation for a Proportion • Confidence Interval (1 - )100% for p

Example • The 1994 General Social Survey “ Please tell me whether or not you think it should be possible for a pregnant woman to obtain a laegal abortion if the woman wants it for any reason” • Said yes = 895, said no = 1039 • Let p is the population proportion that would respond yes

Continued Example • Estimation of p = = 895/1934 =. 46 • 1=. 54 • The estimated standard error of : • A 95% confidence interval for p is : (. 44, . 48)

Sample size required for Estimating a Proportion p • Let B denote the desired bound of error. The sample size n ensuring that, with fixed probability, the error estimation of p by is no greater than B, is : • The z score is the one for a confidence interval with confidence coefficient equal to The fixed probability

Example • A group of social scientist wanted to estimate the proportion of school children in Boston who were living with only one parent • The wanted the sample proportion to fall within. 04 of the true value, with probability. 95 • The question is how much the sample (n)?

Continued Example • Identification: = 0. 05 z 0. 025 = 1. 96 B =. 04 • Asked : n? • Solving problem: Since p is unknown the variance p(1 -p) maximun when p=0. 5 can be alternative, so:

Sample size required for Estimating a Mean ( ) • Let B denote the desired bound of error. The sample size n ensuring that, with fixed probability, the error estimation of by is no greater then B, is: • The z score is the one for a confidence interval with confidence coefficient equal to The fixed probability