Introduction to Confidence Intervals using

Population Parameters

Chapter 10.1 & 10.3

Rate your confidence0 (no confidence) – 100 (very confident)

• Name my age within 10 years?• within 5 years?• within 1 year?

• What happens to your confidence as the interval (age range) gets smaller?

Would you agree?

As my age interval decreases your confidence decreases. On the other hand, your confidence increases as the interval widens, because you are given a greater margin of error.

Point Estimate• When we use a single statistic

based on sample data to estimate a population parameter

• Simplest approach• But not always very precise due to

variation in the sampling distribution

Confidence intervals

• Are used to estimate the unknown population parameter

• Formula:

estimate + margin of error

Margin of error• Shows how accurate we believe our

estimate is• The smaller the margin of error, m, the

more precise our estimate of the true parameter.

• Formula:

statistic theofdeviation standard

value

criticalm

Confidence level• Is the success rate of the method

used to construct the interval.

• Using this method, ____% of the intervals constructed will contain the true population parameter.

What does it mean to be 95% confident?

• 95% chance that the true p is contained in the confidence interval

• The probability that the interval contains the true p is 95%

• The method used to construct the interval will produce intervals that contain the true p 95% of the time.

• Found from the confidence level• The upper z-score with probability p lying to its right

under the standard normal curve

Confidence level(C) each tail area z*90% .10/2 =.05 1.64595% .05/2 =.025 1.9699% .01/2 =.005 2.576

• z* can be looked up in table or, by using 2nd VARS #3 invNorm(1.C/2 = Example: 2nd VARS #3 invNorm(1.95/2 = 2.575829

Critical value (z*)

ˆ ˆ(1 )ˆ * p pp zn

Confidence interval for a population proportion:

Steps for doing a confidence interval:

1) State the parameter of interest.

2) Name inference procedure & state assumptions. See assumptions for CI for population parameter on next slide.

3) Calculate the confidence interval using formula.

4) Write a statement about the interval in the context of the

problem.

CI assumptions for a pop. parameterStep 2: Name inference procedure and state assumptions:

1) SRS from population

2) Normality: The number of success and failures are both at least 10. • Note: On AP Test you must show the calculation

below, simply stating the number of successes and failures are both at least 10 isn’t enough.

• np > 10 & n(1-p) > 10.

3) Independence: Population size > 10n

Statement: (memorize!!)

We are ________% confident that the true proportion context lies within the interval ______ and ______.

Assumptions:• The voters were sampled randomly.• 330(.436)=144 & 330(.564)=186, both ≥ 10• Population of eligible voters must be at least 3300 =

10(330).

We are 95% confident that the true proportion of voters that will vote “yes” is between .382 and .490.

Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a 95 % confidence interval for actual sentiment of all voters. 1st Calculate p-hat = 144/330 = .436

.436 1.96.436(.564)

330

.382,.490

Assumptions:• The subjects were sampled randomly• 53 (.27)=14 and 53(.73)=39, both ≥10• The population of subjects using this new medicine must be

at least 530 = 10(53)

We are 95% confident that the true proportion of people that will improve after using the new medication is between .15 and .39.

An experiment finds that 27% of 53 randomly sampled subjects report improvement after using a new medicine. Create a 95% confidence interval for the actual cure rate.

..27 1.96.27(.73)

53

.15,.39

We are 90% confident that the true proportion of people that will improve after using the new medication is between .17 and .37.

90% confidence interval?

.27 1.645.27(.73)

53

.17,.37

How can you make the margin of error smaller?• z* smaller

(lower confidence level)

• s smaller(less variation in the population)

• n larger(to cut the margin of error in half, n, the

sample size must be 4 times as big)

In real life, you can’t adjust

Find a sample size:

• If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:

Always round up to the nearest person/object!

ˆ ˆ(1 )* p pm zn

Find the sample size required for ±5%, with 98% confidence. Consider the formula for margin of error. We believe the improvement rate to be .27 from our preliminary study.

.05 2.326.27(.73)n

.052 2.3262 .27(.73)n

n 2.3262 (.27)(.73)

.052 426.546

We need to run an experiment with at least 427 people receiving the new medication in order to have a margin of error of ±5%, with 98% confidence.

ˆ ˆ(1 )* p pm zn

When they don’t give you a % of confidence or p-hat:• Use 95% confidence and .5 for p-hat

What sample size does it take to estimate the outcome for an election with a margin of error of 3%?

.03 1.96.5(.5)n

.032 1.962 .5(.5)n

n 1.962 (.5)(.5)

.032 1068

We need to have a sample size of at least 1068 people to estimate the outcome for an election in order to have a margin of error of ±3%, with 95% confidence.

ˆ ˆ(1 )* p pm zn