Overview 8.1 Z Interval for the Mean 8.2 t Interval for the Mean 8.3 Z Interval for a Population Proportion 8.4 OMIT 8.5 Sample Size Considerations

Introduction 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

Z Interval for the Population Mean μ May be constructed if EITHER of the following two 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.

Z 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 by lower bound = x – Za/2 upper bound = x + Za/2 where 1- a is the confidence level. The Z interval can also be written as x ± Za/2 and is denoted (lower bound, upper bound)

Z Interval for the Population Mean μ Used only under certain conditions Case 1: Population is normally distributed The value of σ is known Case 2: n ≥ 30

Margin of Error Denoted as E Measure of the precision of the confidence interval estimate For the Z interval

Interpreting 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.”

Example 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.

Example The 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.

Example continued Solution Case 1 or 2? Is the distribution of the population known? Is the sample size large enough?

Example continued Use the formula for the confidence interval. We are given n = 100, x= 1, and σ = 0.5. For a confidence level of 95%, Table 8.1 provides the value of Zα/2 = Z0.025 = 1.96. Plugging in the formula. We are 95% confident that μ, the population mean lead contamination for all trout on the Spokane River, lies between 0.902 ppm and 1.098 ppm.

8.2 t Interval for the Mean

t Distribution In real-world problems, σ is often unknown Use s to estimate the value of σ For a normal population follows a t distribution

t Distribution continued n - 1 degrees of freedom Where x is the sample mean μ is the unknown population mean s is the sample standard deviation n is the sample size

Characteristics of the t Distribution Centered at zero The mean of t is zero Symmetric about its mean zero As 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.

FIGURE 8.14 ta/2 has area a/2 to the right of it

Procedure for Finding ta/2 Step 1 Go 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 2 Go 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.

Example 8.11 - Finding ta/2 Find the value of ta/2 that will produce a 95% confidence interval for μ if the sample size is n = 20.

Example 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.

Example 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, 2.093.

Example 8.11 continued FIGURE 8.15 t table (excerpt).

t Interval for μ lower bound , Random sample of size n Unknown mean μ Confidence interval for μ lower bound , upper bound x is the sample mean ta/2 is associated with the confidence level n - 1 degrees of freedom s is the sample standard deviation.

t Interval for μ continued The t interval may also be written as and 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).

Margin 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.”

Example 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.7 22.4 25.5 23.7 21.7 23.4 22.8 24.1 22 22.5 24.1 21 22.7 23.2 25 21.6 24.7 23.1 24 20.9 23.5 Construct a 95% confidence interval for μ, the population mean length of all fourth graders’ feet.

Example 8.14 continued Solution n = 20, x = 23.095 s = 1.280. For a confidence level of 95%, ta/2 = 2.093. The margin of error of fourth-grade foot length is We are 95% confident that the population mean length of fourth graders’ feet lies between 22.496 and 23.694 cm.

8.3 Z Interval for a Population Proportion

Point Estimate The sample proportion of successes is a point estimate of the population proportion p.

Central Limit Theorem for Proportions The sampling distribution of the sample proportion p follows an approximately normal distribution with mean μp = p standard deviation When both the following conditions are satisfied: (1) np ≥ 5 and (2) n(1 - p) ≥ 5.

Z Interval for p May be performed only if both the following conditions apply: np ≥ 5 and n(1 - p) ≥ 5 Random sample of size n is taken from a binomial population with unknown population proportion p The 100(1 - a)% confidence interval for p is given by

Z Interval for p continued Alternatively Where p is the sample proportion of successes, n is the sample size, and Za/2 depends on the confidence level

Margin 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.”

Example 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.

Example 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% confidence interval for the population proportion of all American adults who think there should not be such a law.

Example 8.19 continued Solution Sample size is n = 1012 Observed proportion is so

Example 8.19 continued Solution a. We next check the conditions for the confidence interval: and

Example 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

Example 8.19 continued Solution c. The 95% confidence interval is point estimate ± margin of error Thus, 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%.

8.5 Sample Size Considerations

Sample 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 100(1- a)% is given by

Sample 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.

Sample Size Estimating a Population Proportion with No Prior Information About p Within a margin of error E with confidence 100(1- a)% is given by Where 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.

Example 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?

Example 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:

Example 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!

Sample Size Estimating a Population Proportion When Prior Information About p Is Available Within a margin of error E with confidence 100(1- a)% is given by Where 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.