Lecture 7 Dustin Lueker

2  Point Estimate ◦ A single number that is the best guess for the parameter  Sample mean is usually at good guess for the population mean  Interval Estimate ◦ Point estimator with error bound  A range of numbers around the point estimate  Gives an idea about the precision of the estimator  The proportion of people voting for A is between 67% and 73% STA 291 Winter 09/10 Lecture 7

3  Inferential statement about a parameter should always provide the accuracy of the estimate ◦ How close is the estimate likely to fall to the true parameter value?  Within 1 unit? 2 units? 10 units? ◦ This can be determined using the sampling distribution of the estimator/sample statistic ◦ In particular, we need the standard error to make a statement about accuracy of the estimator STA 291 Winter 09/10 Lecture 7

4  Range of numbers that is likely to cover (or capture) the true parameter  Probability that the confidence interval captures the true parameter is called the confidence coefficient or more commonly the confidence level ◦ Confidence level is a chosen number close to 1, usually 0.90, 0.95 or 0.99 ◦ Level of significance = α = 1 – confidence level STA 291 Winter 09/10 Lecture 7

5  To calculate the confidence interval, we use the Central Limit Theorem ◦ Substituting the sample standard deviation for the population standard deviation  Also, we need a that is determined by the confidence level  Formula for 100(1-α)% confidence interval for μ STA 291 Winter 09/10 Lecture 7

 90% confidence interval ◦ Confidence level of 0.90  α=.10  Z α/2 =1.645  95% confidence interval ◦ Confidence level of 0.95  α=.05  Z α/2 =1.96  99% confidence interval ◦ Confidence level of 0.99  α=.01  Z α/2 = STA 291 Winter 09/10 Lecture 7

 This interval will contain μ with a 100(1-α)% confidence ◦ If we are estimating µ, then why it is unreasonable for us to know σ?  Thus we replace σ by s (sample standard deviation)  This formula is used for large sample size (n≥30)  If we have a sample size less than 30 a different distribution is used, the t-distribution, we will get to this later 7STA 291 Winter 09/10 Lecture 7

 Compute a 95% confidence interval for μ if we know that s=12 and the sample of size 36 yielded a mean of 7 8STA 291 Winter 09/10 Lecture 7

 “Probability” means that in the long run 100(1-α)% of the intervals will contain the parameter ◦ If repeated samples were taken and confidence intervals calculated then 100(1-α)% of the intervals will contain the parameter  For one sample, we do not know whether the confidence interval contains the parameter  The 100(1-α)% probability only refers to the method that is being used 9STA 291 Winter 09/10 Lecture 7

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 Incorrect statement ◦ With 95% probability, the population mean will fall in the interval from 3.5 to 5.2  To avoid the misleading word “probability” we say that we are “confident” ◦ We are 95% confident that the true population mean will fall between 3.5 and STA 291 Winter 09/10 Lecture 7

 Changing our confidence level will change our confidence interval ◦ Increasing our confidence level will increase the length of the confidence interval  A confidence level of 100% would require a confidence interval of infinite length  Not informative  There is a tradeoff between length and accuracy ◦ Ideally we would like a short interval with high accuracy (high confidence level) 12STA 291 Winter 09/10 Lecture 7

 The width of a confidence interval ◦ as the confidence level increases ◦ as the error probability decreases ◦ as the standard error increases ◦ as the sample size n decreases  Why? 13STA 291 Winter 09/10 Lecture 7

 Start with the confidence interval formula assuming that the population standard deviation is known  Mathematically we need to solve the above equation for n 14STA 291 Winter 09/10 Lecture 7

15  About how large a sample would have been adequate if we merely needed to estimate the mean to within 0.5, with 95% confidence? Assume Note: We will always round the sample size up to ensure that we get within the desired error bound. STA 291 Winter 09/10 Lecture 7

 To account for the extra variability of using a sample size of less than 30 the student’s t- distribution is used instead of the normal distribution 16STA 291 Winter 09/10 Lecture 7

 t-distributions are bell-shaped and symmetric around zero  The smaller the degrees of freedom the more spread out the distribution is  t-distribution look much like normal distributions  In face, the limit of the t-distribution is a normal distribution as n gets larger STA 291 Winter 09/10 Lecture 717

 Need to know α and degrees of freedom (df) ◦ df = n-1  α=.05, n=23 ◦ t α/2 =  α=.01, n=17 ◦ t α/2 =  α=.1, n=20 ◦ t α/2 = 18STA 291 Winter 09/10 Lecture 7

 A sample of 12 individuals yields a mean of 5.4 and a variance of 16. Estimate the population mean with 98% confidence. 19STA 291 Winter 09/10 Lecture 7

 The sample proportion is an unbiased and efficient estimator of the population proportion ◦ The proportion is a special case of the mean 20STA 291 Winter 09/10 Lecture 7

 As with a confidence interval for the sample mean a desired sample size for a given margin of error (ME) and confidence level can be computed for a confidence interval about the sample proportion ◦ This formula requires guessing before taking the sample, or taking the safe but conservative approach of letting =.5  Why is this the worst case scenario? (conservative approach) 21STA 291 Winter 09/10 Lecture 7

 ABC/Washington Post poll (December 2006) ◦ Sample size of 1005 ◦ Question  Do you approve or disapprove of the way George W. Bush is handling his job as president?  362 people approved  Construct a 95% confidence interval for p  What is the margin of error? 22STA 291 Winter 09/10 Lecture 7

 If we wanted B=2%, using the sample proportion from the Washington Post poll, recall that the sample proportion was.36 ◦ n=2212.7, so we need a sample of 2213  What do we get if we use the conservative approach? 23STA 291 Winter 09/10 Lecture 7