estimating the value of a parameter using confidence intervals, Study notes of Statistics

Download estimating the value of a parameter using confidence intervals and more Study notes Statistics in PDF only on Docsity! CHAPTER 9: ESTIMATING THE VALUE OF A PARAMETER USING CONFIDENCE INTERVALS By: Wandi Ding 1 9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN 1. Point estimate: is the value of a statistic that estimates the value of a parameter. For example, the sample mean is a point estimate of the population mean, . Suppose: we want to estimate the average weight for all students in MTSU for this semester, we could take a random sample of 100 students and find the average weight of these students, say, 130 pounds, this kind of estimate is called a point estimate. Often, there is another question to be asked, it is, how good is a point estimate? There is no way of knowing how close a particular point estimate is to the population mean if the population is large. For this reason, statisticians prefer another type of estimate, called interval estimate. 2. Interval estimate: an interval for unknown parameter is an interval or a range of values used to estimate the parameter with the specific confidence level of estimate. It is also called confidence interval. 2 x  9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN Confidence interval estimates for the population mean can be written in this form, too. point estimate margin of error =E is margin of error, also called maximum error of estimate. There are three factors which affect the margin of error. 1) level of confidence . Note: α=1-level of confidence. 2) sample size, n. 3) standard deviation of the population. Note: The value of is called critical value of the distribution and the next slide shows common critical values used lot in confidence intervals. 5  )( 2 n z   2 z 9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN The common critical value for 90%, 95%, 99% confidence level: Interpretation of a confidence interval: A (1-α)*100% confidence interval indicates that (1-α)*100% of all simple random samples of size n from the population whose parameter is unknown will contain the parameter. 6 Level of confidence (1-α)*100% Area in each tail, α/2 Critical vale 90% 0.05 1.645 95% 0.025 1.96 99% 0.005 2.575 2 z 9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN Comments for confidence interval: Lower bound: Upper bound: 7 )()( 22 n zx n zx      )( 2 n zx   )( 2 n zx   9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN 5. Calculating the necessary sample size: 10 22 2 2 2 ) * ( * * E z n E z n znE n zE a a a a         9.1 WHEN THE POPULATION STANDARD DEVIATION KNOWN Ex: the college president asks the statistics teacher to estimate the average age of the students at their college. How large a sample is necessary? The statistics teacher would like to be 99% confident that the estimate should be accurate within 1 year. (the standard deviation of the age is known to be 3 years.) Ans: 11 α=1-99%=1%=0.01, α/2=0.01/2=0.005, Za/2 =2.575, E=1 60 toup round so, 67.59 ) 1 3*575.2 ( ) * ( 2 22    E z n a  The sample size at least is 60 students. 9.2 WHEN THE POPULATION STANDARD DEVIATION UNKNOWN As we know, if the standard deviation of population is known and 1)the sample is drawn from a normal distribution or 2) sample size n≥30 when parent population is unknown. We will use z-value based on standard normal distribution to construct confidence interval. If the population standard deviation is unknown and sample size n<30, then we will use t-value based on student’s t distribution NOT use z-value based on standard normal distribution. 12 9.2 WHEN THE POPULATION STANDARD DEVIATION UNKNOWN 6) As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because, as the sample size increases, the values of s get closer to the value of , by the law of large numbers. Page 426, figure 8 shows these properties. A. How to find t-values: Let’s go over the example 2 on page 426. After this example, you should know how to find the t-values. 15  9.2 WHEN THE POPULATION STANDARD DEVIATION UNKNOWN Constructing a (1-α)*100% confidence interval for unknown: 16  , )( :boundupper 2 n s tx  )( :boundlower 2 n s tx  )()( 22 n s tx n s tx    9.2 WHEN THE POPULATION STANDARD DEVIATION UNKNOWN Ex: the below data represents a sample of the number of home fires started by candles for the past several years, find the 99% confidence interval for the mean number of home fires started by candles each year. 5460, 5900, 6090, 6310, 7160, 8440, 9930 Answer: step 1: find the sample mean and standard deviation based on the sample Step 2: find the tα/2 sample size n=7, so degree of freedom is n-1=7-1=6 tα/2 =3.707 17 3.1610 4.7041  sx