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Estimating ๐ when ๐ is unknown Quantitative Methods II Plan for Today โข Studentโs t distribution โข Confidence Intervals โข Examples 1 Estimating a population mean โข One of the purposes of randomly sampling a population is to get an estimate of the mean of the population. โข If the population standard deviation ๐ is not known, then we will use a sample standard deviation s instead. โข However, instead of the normal distribution we will use Studentโs t distribution (discovered by English statistician William Gosset, who worked at a Guinness brewery and published under a pseudonym). Interval Estimates โข Interval estimate: an interval bounded by two values that is calculated from the sample and that is used to estimate the value of a population parameter. โข Level of confidence 1 โ ๐ผ : the proportion of all interval estimates that include the parameter being estimated. โข Confidence interval: an interval estimate with a specified level of confidence. 2 Confidence Intervals Summary : Calculating Confidence Intervals โข โข โข โข Sample Mean: ๐ฅาง Sample Size: n Sample standard deviation: s Level of confidence we wish to have: 1 โ ๐ผ 1 โ ๐ผ โ 100% gives us an estimate of how confident you can be that your mean falls within this interval 0.95 *100% = 95%: you are 95% confident that the population mean falls within this interval 3 Studentโs t distribution Also represented by a bell-shaped curve, but it is wider and lower than the normal curve. The actual shape depends on the sample size, or, degrees of freedom df = ๐ โ 1 . The bigger is the number of degrees of freedom, the closer the t curve gets to the normal curve. We have a separate table for the critical values of Studentโs t distributions. Studentโs t distribution 4 Examples of table values ๐ก 7, 0.1 = 1.415 ๐ก 18, 0.025 = 2.101 ๐ก 25, 0.05 = 1.708 ๐ก 5, 0.01 = 3.365 Step by step Estimation of Mean ฮผ (ฯ unknown) Same assumptions as before: either the general population has the bell-shaped symmetric distribution, or the sample size is at least 25. 5 Confidence Coefficient ๐(๐๐, ๐ถฮค๐) c c Constructing a Confidence Interval โข Step 1: Set-Up โ Describe the population parameter of interest โข Step 2: The Confidence Interval Criteria โ Check the assumptions โ Identify the probability distribution and the formula to be used โ State the level of confidence ๐ โ ๐ถ โข Step 3: The Sample Evidence โ Collect the sample information 6 Constructing a Confidence Interval โข Step 4: The Confidence Interval โ Determine the confidence coefficient ๐ก(df, ๐ผ ฮค2) โ Find the error bound for a population mean ๐ธ๐ต๐ = ๐ก(df, ๐ผ ฮค2) โ ๐ ๐ โ Find the lower and upper confidence limits โข Step 5: State the confidence interval from ๐ฅาง โ ๐ธ๐ต๐ to ๐ฅาง + ๐ธ๐ต๐ (units) How to decrease the error? โข To decrease the value of EBM (and thus, to decrease the size of the confidence interval for ๐) there are two possibilities: (A) Decrease the confidence level. A smaller confidence level will result in a smaller ๐ก(df, ๐ผ/2) ะฐnd thus, youโll get a smaller EBM. (B) Increase the size of a sample. A larger value of n means a larger value of ๐ and thus, youโll get a smaller value of EBM. โข Tradeoffs: (A) less certain, (B) more costly 7 Example: garbage A sample of 11 families of four people has produced the following amount of garbage during a randomly selected week (in kg): 20.4 18.1 17.9 12.8 24.6 18.8 29.2 17.0 16.5 28.0 17.7 Construct a 95% confidence interval for the mean amount of garbage produced by a family of four, assuming that the population is normally distributed. Example: garbage Step 1: What is the population parameter of interest? Step 2: ๐ is unknown, the sample size is 11, and the population is assumed to be normally distributed. Thus, we will be using Studentโs t distribution with 1 โ ๐ผ = 0.95 (95%). Step 3: ๐ = 11, and we compute ๐ฅาง = 20.1 kg, ๐ = 5.0765 kg, df = 10 and ๐ผ ฮค2 = 0.025. 8 Example: garbage Step 4: ๐ก(10, 0.025) = 2.228 (table) ๐ธ๐ต๐ = ๐ก df, ๐ผฮค2 โ ๐ ๐ = 2.228 โ 5.0765 11 = 3.4 ๐ฅาง โ ๐ธ๐ต๐ = 16.7 , ๐ฅาง + ๐ธ๐ต๐ = 23.5 Step 5: The 95% confidence interval for the population mean ๐ is: from 16.7 kg to 23.5 kg (same precision as the data) Example: exercise 15 randomly selected students at a University were asked how many hours they exercise per week. Their answers are as follows: 12 1.5 3 6 4.5 10.5 15 7.5 3 10 1 8 4 3.5 5 Construct a 90% confidence interval for the mean amount of hours of exercise per week for students at this University, assuming that these numbers are normally distributed. 9 Example: exercise Step 1: What is the population parameter of interest? Step 2: ๐ is unknown, the sample size is 15, and the population is assumed to be normally distributed. Thus, we will be using Studentโs t distribution with 1 โ ๐ผ = 0.90 (90%). Step 3: ๐ = 15, and we compute ๐ฅาง = 6.3 hours, ๐ = 4.096 hours, df = 14 and ๐ผ ฮค2 = 0.05. Example: exercise Step 4: ๐ก(14, 0.05) = 1.761 (table) ๐ธ๐ต๐ = ๐ก df, ๐ผฮค2 โ ๐ ๐ = 1.761 โ 4.096 15 = 1.9 ๐ฅาง โ ๐ธ๐ต๐ = 4.4 , ๐ฅาง + ๐ธ๐ต๐ = 8.2 Step 5: The 90% confidence interval for the population mean ๐ is: from 4.4 hours to 8.2 hours 10 Example: practice Nine hotel maids in Nashville have the following weekly incomes (in dollars): 415 565 430 510 580 450 435 505 475 Construct a 99% confidence interval for the mean weekly income of all hotel maids in Nashville, assuming that the incomes are normally distributed. Answer: from $419 to $551 11