2Overview 8.1 Z Interval for the Mean 8.2 t Interval for the Mean 8.3 Z Interval for a Population Proportion8.4 OMIT8.5 Sample Size Considerations
3Introduction to confidence intervals Confidence interval estimate - consists of an interval of numbers generated by a point estimate together with an associated confidence level specifying the probability that the interval contains the parameter
4Z Interval for the Population Mean μ May be constructed if EITHER of the followingtwo conditions are met:Case 1: The population is normally distributed, and the value of σ is known.Case 2: The sample size is large (n ≥ 30), and the value of σ is known.
5Z Interval for the Population Mean μ continued When a random sample of size n is taken from a population, a 100(1- a)% confidence interval for μ is given bylower bound = x – Za/2upper bound = x + Za/2where 1- a is the confidence level.The Z interval can also be written asx ± Za/2and is denoted (lower bound, upper bound)
6Z Interval for the Population Mean μ Used only under certain conditionsCase 1:Population is normally distributedThe value of σ is knownCase 2:n ≥ 30
7Margin of Error Denoted as E Measure of the precision of the confidence interval estimateFor the Z interval
8Interpreting the Margin of Error For a (1 - a)100% confidence interval for μ“We can estimate μ to within E units with(1 - a)100% confidence.”
9Example 8.8 - Interpreting the margin of error Find and interpret the margin of error E for the confidence interval for the mean sodium content of the 23 breakfast cereals containing sodium in Example 8.5 page 395.
10ExampleThe Washington State Department of Ecology reported that the mean lead contamination in trout in the Spokane River is 1 part per million (ppm), with a population standard deviation of 0.5 ppm. Suppose a sample of n = 100 trout has a mean lead contamination of = 1 ppm. Assume that σ = 0.5 ppm.Determine whether Case 1 or Case 2 applies.Construct a 95% confidence interval for μ, the population mean lead contamination in all trout in the Spokane River.Interpret the confidence interval.
11Example continuedSolution Case 1 or 2? Is the distribution of the population known? Is the sample size large enough?
12Example continued Use the formula for the confidence interval. We are given n = 100, x= 1, and σ = For a confidence level of 95%, Table 8.1 provides the value of Zα/2 = Z0.025 = Plugging in the formula.We are 95% confident that μ, the population mean lead contamination for all trout on the Spokane River, lies between ppm and ppm.
14t Distribution In real-world problems, σ is often unknown Use s to estimate the value of σFor a normal populationfollows a t distribution
15t Distribution continued n - 1 degrees of freedomWhere x is the sample meanμ is the unknown population means is the sample standard deviationn is the sample size
16Characteristics of the t Distribution Centered at zeroThe mean of t is zeroSymmetric about its mean zeroAs df decreases, the t curve gets flatter, and the area under the t curve decreases in the center and increases in the tails.As df increases toward infinity, the t curve approaches the Z curve, and the area under the t curve increases in the center and decreases in the tails.
18Procedure for Finding ta/2 Step 1Go across the row marked “Confidence level” in the t table (Table D in the Appendix, page T-11) until you find the column with the desired confidence level at the top.The ta/2 value is in this column somewhere.Step 2Go down the column until you see the correct number of degrees of freedom on the left.The number in that row and column is the desired value of ta/2.
19Example Finding ta/2Find the value of ta/2 that will produce a 95% confidence interval for μ if the sample size is n = 20.
20Example 8.11 continued Solution Step 1 Go across the row labeled “Confidence level” in the t table (Figure 8.15) until we see the 95% confidence level.ta/2 is somewhere in this column.
21Example 8.11 continued Solution Step 2 df = n - 1 = 20 - 1 = 19. Go down the column until you see 19 on the left.The number in that row is ta/2,
22Example 8.11 continuedFIGURE 8.15 t table (excerpt).
23t Interval for μ lower bound , Random sample of size n Unknown mean μ Confidence interval for μlower bound ,upper boundx is the sample meanta/2 is associated with the confidence leveln - 1 degrees of freedoms is the sample standard deviation.
24t Interval for μ continued The t interval may also be written asand is denoted(lower bound, upper bound)The t interval applies whenever either of the following conditions is met:Case 1:The population is normal.Case 2:The sample size is large (n ≥ 30).
25Margin of Error for the t Interval The margin of error E for a (1- a)100% t interval for μ can be interpreted as follows:“We can estimate to within E units with (1- a)100% confidence.”
26Example 8.14 - Margin of error for the fourth-grader foot lengths Suppose a children’s shoe manufacturer is interested in estimating the population mean length of fourth graders’ feet. A random sample of 20 fourth graders’ feet yielded the following foot lengths, in centimeters.7Construct a 95% confidence interval for μ, the population mean length of all fourth graders’ feet.
27Example 8.14 continued Solution n = 20, x = 23.095 s = 1.280. For a confidence level of 95%, ta/2 =The margin of error of fourth-grade foot length isWe are 95% confident that the population mean length of fourth graders’ feet lies between and cm.
29Point Estimate The sample proportion of successes is a point estimate of the population proportion p.
30Central Limit Theorem for Proportions The sampling distribution of the sample proportion p follows an approximately normal distribution with mean μp = pstandard deviationWhen both the following conditions are satisfied: (1) np ≥ 5 and (2) n(1 - p) ≥ 5.
31Z Interval for pMay be performed only if both the following conditions apply: np ≥ 5 and n(1 - p) ≥ 5Random sample of size n is taken from a binomial population with unknown population proportion pThe 100(1 - a)% confidence interval for p is given by
32Z Interval for p continued AlternativelyWhere p is the sample proportion of successes, n is the sample size, and Za/2 depends on the confidence level
33Margin of Error for the Z Interval for p The margin of error E for a (1- a)100% Z interval for p can be interpreted as follows:“We can estimate p to within E with (1- a)100% confidence.”
34Example 8.19 - Polls and the famous “plus or minus 3 percentage points” There is hardly a day that goes by without some new poll coming out. Especially during election campaigns, polls influence the choice of candidates and the direction of their policies. In October 2004, the Gallup organization polled 1012 American adults, asking them, “Do you think there should or should not be a law that would ban the possession of handguns, except by the police and other authorized persons?” Of the 1012 randomly chosen respondents, 638 said that there should NOT be such a law.
35Example 8.19 continued a. Check that the conditions for the Z interval for p have been met.b. Find and interpret the margin of error E.c. Construct and interpret a 95% confidenceinterval for the population proportion of allAmerican adults who think there should not besuch a law.
36Example 8.19 continued Solution Sample size is n = 1012 Observed proportion isso
37Example 8.19 continued Solution a. We next check the conditions for theconfidence interval:and
38Example 8.19 continued Solution b. The confidence level of 95% implies that our Za/2 equals 1.96 (from Table 8.7).The margin of error equals
39Example 8.19 continued Solution c. The 95% confidence interval is point estimate ± margin of errorThus, we are 95% confident that the population proportion of all American adults who think that there should not be such a law lies between 60% and 66%.
41Sample Size for Estimating the Population Mean The sample size for a Z interval that estimates the population mean μ to within a margin of error E with confidence (1- a)% is given by
42Sample Size for Estimating the Population Mean continued Where Za/2 is the value associated with the desired confidence level (Table 8.1), E is the desired margin of error, and σ is the population standard deviation.Round sample sizes calculations that are decimals to the next whole number.
43Sample Size Estimating a Population Proportion with No Prior Information About pWithin a margin of error E with confidence 100(1- a)% is given byWhere Za/2 is the value associated with the desired confidence level, E is the desired margin of error, and 0.5 is a constant representing the most conservative estimate.
44Example 8.25 - Required sample size for polls Suppose the Dimes-Newspeak organization would like to take a poll on the proportion of Americans who will vote Republican in the next Presidential election. How large a sample size does the Dimes-Newspeak organization need to estimate the proportion to within plus or minus three percentage points (E = 0.03) with 95% confidence?
45Example 8.25 continued Solution The 95% confidence implies that the value for Za/2 is 1.96.Since there is no information available about the value of the population proportion of all Americans who will vote Republican in the next election, we use 0.5 as our “worst case scenario” value of p:
46Example 8.25 continued Solution To estimate the population proportion of American voters who will vote Republican to within 3% with 95% confidence, they will need a sample of 1068 voters.Don’t forget to round up!
47Sample Size Estimating a Population Proportion When Prior Information About p Is AvailableWithin a margin of error E with confidence 100(1- a)% is given byWhere Za/2 is the value associated with the desired confidence level, E is the desired margin of error, and p is the sample proportion of successes available from some earlier sample.