Chapter 7 Confidence Intervals and Sample Sizes 7. Estimating a Proportion p 7.3 Estimating a Mean µ (σ known) 7.4 Estimating a Mean µ (σ unknown) 7.5 Estimating a Standard Deviation σ In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adult population that believe in global warming? TRICK QUESTION! We only know the sample proportion s, We do not know the population proportion σ. BUT The proportion of the sample (0.7) is our best point estimate (i.e. best guess). 1 Point Estimate A single value (or point) used to approximate a population parameter Confidence Interval : CI The range (or interval) of values to estimate the true value of a population parameter. Proportion Mean Std. Dev. Population Parameter p µ σ Best Point Estimate p x s It is abbreviated as CI In, the 95% confidence interval for the population proportion p is CI = (0.677, 0.73) 3 4 Confidence Level : 1 α The probability that the confidence interval actually contains the population parameter. The most common confidence levels used are 90%, 95%, 99% 90% : α = 0.1 95% : α = 0.05 99% : α = 0.01 Margin of Error : E The maximum likely difference between the observed value and true value of the population parameter (with probability is 1 α) The margin of error is used to determine a confidence interval (of a proportion or mean) In, the Confidence level is 95% In, the 95% margin of error for the population proportion p is E = 0.03 5 6
Continued In a recent poll, 70% or 1501 randomly selected adults said they believed in global warming. Q: What is the proportion of the adult population that believe in global warming? A: 0.7 is the best point estimate of the proportion of all adults who believe in global warming. The 95% confidence interval of the population proportion p is: CI = (0.677, 0.73) ( with a margin of error E = 0.03 ) What does it mean, exactly? Interpreting a Confidence Interval For the 95% confidence interval CI = (0.677, 0.73) we say: We are 95% confident that the interval from 0.677 to 0.73 actually does contain the true value of the population proportion p. This means that if we were to select many different samples of size 1501 and construct the corresponding confidence intervals, then 95% of them would actually contain the value of the population proportion p. 7 8!!! Caution!!!!!! Caution!!! Know the correct interpretation of a confidence interval It is incorrect to say the probability that the population parameter belongs to the confidence interval is 95% because the population parameter is not a random variable, its value cannot change Do not confuse the two percentages The proportion can be represented by percents (like 70% in ) The confidence level may be represented by percents (like 95% in ) Proportions can be any value from 0% to 100% The population is set in stone Confidence levels are usually 90%, 95%, or 99% 9 10 Confidence Interval Formula ( y E, y + E ) y = Best point estimate E = Margin of Error Finding the Point Estimate and E from a Confidence Interval Point estimate : y y = (upper confidence limit) + (lower confidence limit) Centered at the best point estimate Width is determined by E The value of E depends the critical value of the CI Margin of Error : E E = (upper confidence limit) (lower confidence limit) 11 1
Critical Value The number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. A critical value is dependent on a probability distribution the parameter follows and the confidence level (1 α). Normal Dist. Critical Values For a population proportion p and mean µ (σ known), the critical values are found using z-scores on a standard normal distribution The standard normal distribution is divided into three regions: middle part has area 1 α and two tails (left and right) have area α/ each: 13 14 Normal Dist. Critical Values Normal Dist. Critical Values The z-scores z a/ and z a/ separate the values: Likely values ( middle interval ) Unlikely values ( tails ) Use StatCrunch to calculate z-scores (see Ch. 6) The value z a/ separates an area of a/ in the right tail of the z-dist. The value z a/ separates an area of a/ in the left tail of the z-dist. The subscript a/ is simply a reminder that the z- score separates an area of a/ in the tail. z a/ z a/ 15 16 Section 7. Estimating a Population Proportion s The best point estimate for a population proportion p is the sample proportion p Best point estimate : p Objective Find the confidence interval for a population proportion p Determine the sample size needed to estimate a population proportion p The margin of error E is the maximum likely difference between the observed value and true value of the population proportion p (with probability is 1 α) 17 18
Margin of Error for Proportions E z a pq ˆˆ n E = margin of error p = sample proportion q = 1 p number sample values Confidence Interval for a Population Proportion p ( p ˆ E, p ˆ + E ) E where z a pq ˆˆ n 1 α = Confidence Level 19 0 Finding the Point Estimate and E from a Confidence Interval Round-Off Rule for Confidence Interval Estimates of p Point estimate of p: p = (upper confidence limit) + (lower confidence limit) Margin of Error: E = (upper confidence limit) (lower confidence limit) Round the confidence interval limits for p to three significant digits. 1 Direct Computation Stat Proportions One Sample with Summary 3 4
Enter Values Click Next 5 6 Select Confidence Interval Enter Confidence Level, then click Calculate 7 8 Sample Size Standard Error Lower Limit Upper Limit Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained? From the output, we find the Confidence interval is CI = (0.578, 0.76) 9 30
Determining Sample Size E = z a / (solve for n by algebra) ˆ ( Z a / ) p ˆ q E p ˆ q ˆ n Sample Size for Estimating Proportion p When an estimate of p is known: z ˆ ˆ ˆ ( a / ) p q E ˆ ˆ ˆ ( a / ) 0.5 z When no estimate of p is known: use p = q = 0.5 E 31 3 Round-Off Rule for Determining Sample Size A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet. If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. s: 310.67 round up to 311 95.3 round up to 96 113.01 round up to 114 How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points when (a) In 006, 73% of adults used the Internet. (b) No known possible value of the proportion. 33 34 (a) Given: (b) Given: Given a sample has proportion of 0.73, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 84 adults. 35 For any sample, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 1068 adults. 36
Summary Confidence Interval of a Proportion E = margin of error p = sample proportion number sample values 1 α = Confidence Level E z a pq ˆˆ n ( p E, p + E ) 37 Summary Sample Size for Estimating a Proportion When an estimate of p is known: z ˆ ˆ ( a / ) p q E When no estimate of p is known (use p = q = 0.5) z ( a / ) 0.5 E 38 Section 7.3 Estimating a Population mean µ (σ known) Objective Find the confidence interval for a population mean µ when σ is known Determine the sample size needed to estimate a population mean µ when σ is known Best Point Estimation The best point estimate for a population mean µ (σ known) is the sample mean x Best point estimate : x 39 40 Notation Requirements = population mean = population standard deviation x = sample mean number of sample values E = margin of error z a/ = z-score separating an area of α/ in the right tail of the standard normal distribution (1) The population standard deviation σ is known () One or both of the following: The population is normally distributed or n > 30 41 4
Margin of Error Confidence Interval ( x E, x + E ) where 43 44 Round-Off Rules for Confidence Intervals Used to Estimate µ The two values x E and x + E are called confidence interval limits. 1. When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data.. When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample mean. 45 46 size 4 has a mean of 38.4 size 4 has a mean of 38.4 Direct Computation Stat Z statistics One Sample with Summary 47 48
size 4 has a mean of 38.4 size 4 has a mean of 38.4 Enter Parameters Click Next 49 50 size 4 has a mean of 38.4 size 4 has a mean of 38.4 Select Confidence Interval Enter Confidence Level, then click Calculate 51 5 size 4 has a mean of 38.4 Sample Size for Estimating a Population Mean = population mean Standard Error Lower Limit Upper Limit σ = population standard deviation x = sample mean E = desired margin of error z α/ = z score separating an area of a/ in the right tail of the standard normal distribution From the output, we find the Confidence interval is CI = (35.86, 40.938) 53 (z a/ ) E 54
Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. s: 310.67 round up to 311 95.3 round up to 96 113.01 round up to 114 We want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean? What we know: a = 0.05 E = 3 = 15 a / = 0.05 z a/ = 1.96 (using StatCrunch) 1.96 15 = 96.04 = 97 3 With a simple random sample of only 97 statistics students, we will be 95% confident that the sample mean is within 3 IQ points of the true population mean. 55 56 Summary Confidence Interval of a Mean µ (σ known) σ = population standard deviation x = sample mean number sample values 1 α = Confidence Level ( x E, x + E ) 57 Summary Sample Size for Estimating a Mean µ (σ known) E = desired margin of error σ = population standard deviation x = sample mean 1 α = Confidence Level (z a/ ) E 58