Confidence Intervals

Chapter 9

Rate your confidenceRate your confidence0 - 1000 - 100

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

• Shooting a basketball at a wading pool, will make basket?

• Shooting the ball at a large trash can, will make basket?

• Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller?

The larger your confidence, the wider the interval.

Point Estimate

• Use a singlesingle statistic based on sample data to estimate a population parameter

• Simplest approach

• But not always very precise due to variationvariation in the sampling distribution

Confidence intervalsConfidence intervals

• Are used to estimate the unknown population mean

• Formula:

estimate + margin of error

Margin of errorMargin of error

• Shows how accurate we believe our estimate is

• The smaller the margin of error, the more precisemore precise our estimate of the true parameter

• Formula:

statistic theof

deviation standard

value

criticalm

Confidence levelConfidence level

• Is the success rate of the method used to construct the interval

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

What does it mean to be 95% What does it mean to be 95% confident?confident?

• 95% chance that is contained in the confidence interval

• The probability that the interval contains is 95%

• The method used to construct the interval will produce intervals that contain 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 tail area z*

.05 1.645

.025 1.96

.005 2.576

Critical value (z*)Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

Confidence interval for a Confidence interval for a population mean:population mean:

n

zx

*

estimate

Critical value

Standard deviation of the statistic

Margin of error

ActivityActivity

Steps for doing a z-interval Steps for doing a z-interval for means:for means:1) Assumptions –

• SRS from population• Sample is < 10% of the population• Independence among data values is plausible• Sampling distribution is normal (or approximately

normal)• Given (normal)• Large sample size (n>30)• Graph data (unimodal and relatively symmetric)

• is known2) Calculate the interval3) Write a conclusion about the interval in the

context of the problem.

Conclusion:Conclusion:

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

• The NAEP (National Assessment of Educational Progress) includes a short test of quantitative skills, covering basic arithmetic and the ability to apply it. The standard deviation of the test is 60. Suppose a random sample of 50 young adult men are taken from a large population. If the sample mean of their scores is 265, what is a 95% confidence interval for the true mean score for young adult men on this test?

• What about a 90% confidence interval?

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

3899.3,0101.33

2.645.12.3

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

95% confidence interval?

4263.3,9737.23

2.96.12.3

99% confidence interval?

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

4974.3,9026.23

2.576.22.3

What happens to the interval as the What happens to the interval as the confidence level increases?confidence level increases?

the interval gets wider as the confidence level increases

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

(lower confidence level)

• smaller(less variation in the population)

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

must be 4 times as big)

Really cannot change!

A random sample of 50 PWSH students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of PWSH students?

We are 95% confident that the true mean SAT score for PWSH students is between 1220.9 and 1279.1

Suppose that we have this random sample of SAT scores:

950 1130 1260 1090 1310 1420 1190

What is a 95% confidence interval for the true mean SAT score? (Assume = 105)

We are 95% confident that the true mean SAT score for PWSH students is between 1115.1 and 1270.6.

Find a sample size:Find a sample size:

n

zm

*

• 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!

The heights of PWSH male students is normally distributed with = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?

n = 43

In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.

Can you find a z-interval for this Can you find a z-interval for this problem? Why or why not?problem? Why or why not?

Student’s t- distributionStudent’s t- distribution

• Developed by William Gosset

• Continuous distribution

• Unimodal, symmetrical, bell-shaped density curve

• Above the horizontal axis

• Area under the curve equals 1

• Based on degrees of freedom

t- curves vs normal curve

-4 -3 -2 -1 0 1 2 3 4

df = 2

df = 5

df = 10

df = 25

Normal

Comparison of normal and t distibutions

How does How does tt compare to compare to normal?normal?• Shorter & more spread out

• More area under the tails

• As n increases, t-distributions become more like a standard normal distribution

How to find How to find tt**

• Use Table for t distributions• Look up confidence level at bottom &

df on the sides• df = n – 1

Find these t*90% confidence when n = 595% confidence when n = 15

t* =2.132

t* =2.145

Can also use invT on the calculator!

Need upper t* value with 5% is above – so 95% is below

invT(p,df)

Formula:Formula:

n

stx * :Interval Confidence

estimate

Critical value

Standard deviation of statistic

Margin of errorMargin of error

Assumptions for Assumptions for tt-inference-inference

• unknown• Have an SRS from population• Sample is < 10% of the population• Independence among data values is

plausible• Sampling distribution is normal (or

approximately normal.– Given (population normal)– Graph data (unimodal and relatively symmetric

with no outliers) or large sample size

For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.Assumptions:

• Have an SRS of healthy, white males

• 27 white males (placebo group) is <10% of white males

• We assume blood pressures are independent

• Systolic blood pressure is normally distributed (given).

• is unknown, so we will construct a t-interval

We are 95% confident that the true mean systolic blood pressure of healthy white males is between 111.22 and 118.58.

)58.118,22.111(27

3.9056.29.114

26 ,*

dfn

stx

RobustRobust

• An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated.

• t-procedures can be used with some skewness, as long as there are no outliers.

• Larger n can have more skewness.

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.

(70.883, 74.497)

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain.

The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:

160 200 220 230 120 180 140

130 170 190 80 120 100 170

Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.

(126.16, 189.56)

A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?

Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.

Some Cautions:Some Cautions:

• The data MUST be a SRS from the population

• The formula is not correct for more complex sampling designs, i.e., stratified, etc.

• No way to correct for bias in data

Cautions continued:Cautions continued:

• Outliers can have a large effect on confidence interval

• Must know to do a z-interval – which is unrealistic in practice