STAT 311: Confidence Intervals for Means

Y. Samuel Wang

Summer 2016

Y. Samuel Wang STAT 311: Confidence Intervals for Means

Logistics

Homework has been posted

Sampling Distribution of the Mean

From the Central Limit Theorem, we know that as n, the size ofmy sample, increases

√nx̄n − µσ

→ N (0, 1)

...But in almost every case, we don’t know σ

From the Central Limit Theorem, we know that as n, the size ofmy sample, increases

√nx̄n − µσ

→ N (0, 1)

...But in almost every case, we don’t know σ

So typically we consider using sx , the standard deviation of thedata, to estimate σ, the population standard deviation

√nx̄n − µsx

The sample standard deviation, sx , is also a random variable andchanges from sample to sample. Yesterday in lab we talked about adistribution which arises from the ratio of certain random variables.

√nx̄n − µsx

→ tn−1

Turns out, when we plug in sx for σ, we end up with a Tdistribution with n − 1 degrees of freedom

So typically we consider using sx , the standard deviation of thedata, to estimate σ, the population standard deviation

√nx̄n − µsx

The sample standard deviation, sx , is also a random variable andchanges from sample to sample. Yesterday in lab we talked about adistribution which arises from the ratio of certain random variables.

√nx̄n − µsx

→ tn−1

Turns out, when we plug in sx for σ, we end up with a Tdistribution with n − 1 degrees of freedom

T Distribution

The T Distribution was developed by William Gosset while workingat the Guiness Brewery in Dublin, Ireland.

Figure : William Gosset

In particular, Gosset was interested in estimating the “degree ofsaccharine” in malt extract

T Distribution

The T distribution looks very similar to the Normal, but givesmore weight to “extreme” observations

As the degrees of freedom increase, the distribution looks moreand more like a normal distribution

Normal Probabilities

In particular-

.95 = P

(−t?n−1 ≤

√nx̄n − µsx

≤ t?n−1

(−t?n−1

sx√n≤ x̄n − µ ≤ t?n−1

sx√n

(x̄ − t?n−1

sx√n≤ µ ≤ x̄ + t?n−1

sx√n

So there is a 95% chance that when I form sample a x̄ and formthe interval (x̄ − t?n−1

sx√n, x̄ + t?n−1

sx√n

), it will contain the trueparameter µ

Normal Probabilities

In particular-

.95 = P

(−t?n−1 ≤

√nx̄n − µsx

≤ t?n−1

(−t?n−1

sx√n≤ x̄n − µ ≤ t?n−1

sx√n

(x̄ − t?n−1

sx√n≤ µ ≤ x̄ + t?n−1

sx√n

So there is a 95% chance that when I form sample a x̄ and formthe interval (x̄ − t?n−1

sx√n, x̄ + t?n−1

sx√n

), it will contain the trueparameter µ

Example

Confidence Intervals in general

Hopefully you’ve picked up on a general pattern for Confidenceintervals

point estimate±multiplier× SE

Type Point Estimate Multiplier Standard Error

Single Proportion p̂ standard normal√

p̂(1−p̂)n

Diff in Proportions p̂1 − p̂2 standard normal√

p̂1(1−p̂1)n1

+ p̂2(1−p̂2)n2

Mean x̄ T distribution sxn

Means of paired differences

Now suppose that I want to estimate the average weight loss afterusing a specific work out regimen for 12 weeks. I record the weightof each individual before and after the program. For eachindividual, I denote the difference in weights as di .

I can still form confidence intervals for the mean of the differencesin the same exact way, except now the differences di are myobservations.

d̄ ± tn−1sd√n

Means of paired differences

Now suppose that I want to estimate the average weight loss afterusing a specific work out regimen for 12 weeks. I record the weightof each individual before and after the program. For eachindividual, I denote the difference in weights as di .

I can still form confidence intervals for the mean of the differencesin the same exact way, except now the differences di are myobservations.

d̄ ± tn−1sd√n

Difference in Means (independent samples)

Suppose I am interested in comparing the average IQ of studentsat UW agains the average IQ of students at WSU. My parameterof interest would be

µUW − µWSU

Can I create a confidence interval for the difference given a sampleof UW students and WSU students? In this case, the IQ of arandomly selected UW student is independent of the IQ of arandomly selected WSU student and it doesn’t quite make sense topair two randomly selected students together and take thedifference.

For two populations, if I am interested in estimating

µ1 − µ2

I can use the point estimate of

x̄1 − x̄2

which has a mean of µ1 − µ2. When they are independent, the

variance of the point estimate isσ2

, so we use a standarderror of √

So I can form a confidence interval

(x̄1 − x̄2)± tdf

The degrees of freedom we should be using can get complicated.For this class, we will simply use df = min(n1 − 1, n2 − 1).

(x̄1 − x̄2)± tdf

Example

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