ESTIMATION

• Estimation: process of using sample values to estimate population values

• Point Estimates: parameter is estimated as single point– Examples: x, s, p

• Careful statisticians dislike point estimates

Interval Estimates

• Example: there is a 90% probability that somewhere between 58 and 68% of Americans oppose same-sex marriage

• Draws explicit attention to the fact of variability in the sample results; avoids putting too much weight on one number

Think about the interval 42 – 1.64 * 1 to 42 + 1.64 * 1: This interval contains 90% of the sample means that could ever

be drawn from this population

37 42 47

• The interval 40.36 to 43.64 contains 90% of the sample means possible from this population

• No sample mean in this interval differs from μ by more than 1.64 grams

• Hence, there is a 90% probability that any arbitrary x differs from μ by no more than 1.64

• Thus, there is a 90% probability that μ is in the interval x 1.64

• Look at this intervalx 1.64 * 1

• 1.64 is a z value, chosen to correspond to 90%, the confidence level

• 1 is the standard error of the mean

• So the width of the interval is set by the confidence level (which determines number of standard errors in interval) and the standard error, the measure of variation in sample means

A C% Confidence Interval for the Population Mean When σ Is Known

xCzx

Examples:

• A population of Christmas trees has unknown mean with σ = 4. For a sample of 25 trees, the sample mean = 16.6 ft. Calculate a 95% confidence interval for the population mean.

• Same data: calculate a 90% confidence interval

• Same data: suppose that we increase sample size to 81

Width of Confidence Interval Depends On

• Confidence level: as C increases, width decreases

• Sample size: as n increases, width decreases

• Variability in population: as σ increases, standard error increases and width of interval increases

• The quantity zC * σX is called the maximum error in the estimate

• The quantity 2 * zC * σX is called the precision in the estimate– this quantity is the width of the confidence

interval

FINDING THE RIGHT SAMPLE SIZE

• Sometimes we wish to hold the error in the estimate within some limit

• Define e = zC * σX or substituting

ne

Solve this expression for n, yielding

2

e

zn C

Example: With σ = 4 and 95% confidence level, we require that the maximum error in the estimate be no more than 0.5 ft. What sample size is necessary?

Examples:

• Expectations of inflation are known to be normally distributed with standard deviation = 1.2%. A survey of sixty households found a sample average expectation of 4% inflation for the coming year. Calculate a 98% confidence interval for the population’s expectation of inflation in the coming year.

• If we require a maximum error in the estimate of 0.1%, how large a sample must we take?

• Cigarette filters have a “process” standard deviation of 0.3 mm with normal distribution. The current mean is unknown, but a sample of 25 filters have a mean of 20 mm. – Calculate a 90% confidence interval for the

population mean– Find the sample size necessary to hold the

error in the estimate to 0.04 mm

Student’s t distribution

• Suppose σ is NOT known; then we are not entitled to use a z value in calculating confidence intervals

• If, however– The population is known to be normally

distributed OR– The sample size is large enough to invoke the

Central Limit Theorem, then we use

• A value drawn from the t distribution

Hey, Prof, what’s a t distribution?

• Characteristics– Symmetric about its mean of zero– Values tend to cluster in the center, producing

a bell shaped curve

• Differences from z:– Fatter tails and less mass in the center– There is a family of t distributions, based on

“degrees of freedom”

• Degrees of freedom: the sample size minus number of parameters to be estimated before estimating a variance

1

)( 22

n

xxs

Before estimating the variance, we must first calculate x-bar, an estimate of the population mean: we lose one degree of freedom, leaving us with n – 1 degrees of freedom

Confidence Intervals with the t distribution:

n

ss

stx

x

x

Where t is chosen for the desired confidence level and has n – 1 degrees of freedom

Examples:

• Seven male students are allowed to imbibe their favorite beverage until they are visibly inebriated. The amounts consumed in ounces are: 3.7, 2.9, 3.2, 4.1, 4.6, 2.3, 2.5. Calculate a 95% confidence interval for the amount of the drink it would take to get the average member of the population drunk.

• Calculate x-bar and s

• Then calculate the sample standard error

• Find t for 6 degrees of freedom and = 0.025

• Finally, calculate the confidence interval

• In a sample of 41 students who work, the sample mean is 16.561 hours and s = 5.7128 hours. The distribution appears to be somewhat skewed upwards. Find a 90% confidence interval for the average hours worked by all ASU students who work.

USE OF THE t DISTRIBUTION

• Footnote: Who was “Student”? A pseudonym for William Gosset

• The t is often thought of as a small-sample technique

• But, STRICTLY SPEAKING, the t should be used whenever the population standard deviation σ is NOT KNOWN

• Some practitioners use z whenever the sample is large– Central Limit Theorem– There isn’t much difference between t and z

Population standard deviation known?

Yes

Population normal?

Yes No

Sample Size

n >= 30 n < 30

No

Population normal?

z value

z or t (see note)

Yes No

Sample Size

n >= 30 n < 30

t value z or t (see note) ERROR

ERROR

Notes:

• For large samples with σ unknown, different practitioners may proceed differently. Some argue for using a z, appealing to CLT. Others use a t since it gives a less precise estimate. For this course: use a t whenever the population standard deviation is not known.

• Small samples from non-normal populations are beyond the scope of this course

Confidence intervals for the population proportion

• Sample proportion p = x/n• E(p) = and

np)1(

In general is not know, so must be estimated with p and we use

n

ppsp

)1(

• Then the confidence interval is

• p zC sp

• Note that proportion problems always use a z value– Normal approximates binomial

• EXAMPLE: Of 112 students in a sample, 70 have paying jobs. Calculate a 95% confidence interval for the proportion in the population with paying jobs.

• p = 70/112 = 0.625

• 0.625 1.96 * 0.045 etc.• 0.625 0.089660819 or 0.625 0.09• We are 95% confident that

0.54 0.71

045745315.0112

375.0625.0

ps

• EXAMPLE:

• In a sample of 320 professional economists, 251 agreed that “offshoring” jobs is good for the American economy. Calculate a 90% confidence interval for the proportion in the population of professional economists who hold this view.

Finding the Right Sample Size

• The error in the estimate is given by zC σp or, substituting

nze C

)1(

Solving for n yields:

2

2 )1(

e

zn C

• In general is not known

• Two solutions:– Assume = 0.5

• Result is the largest sample that would ever be needed

– Conduct a pilot study and use the resulting p as an estimate of

• May give a somewhat smaller sample size if p is much different from 0.5

• Saves sampling cost

Example:

• Above we had a 95% confidence interval with n = 112 of 0.625 0.09 or a 9% error. Suppose we require a maximum error of 3%.

• Approach 1: let = 0.5

106811.106703.

5.05.096.12

2

n

• Approach 2: assume = 0.625

100141.100003.

375.0625.096.12

2

n

The difference is more dramatic if p is much different from 0.5. In a random sample of 300 students in NC, 30 have experienced “study” abroad. A 95% confidence interval for the population proportion is 10% 3.4%. Suppose we require a maximum error of 2%. Approach 1 gives _______ and approach 2 gives _________.