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 * 1: This interval contains 90% of the sample means that could ever be drawn from this population

The interval to 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 interval  x  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

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 z C * σ X is called the maximum error in the estimate The quantity 2 * z C * σ X 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 = z C * σ X or substituting Solve this expression for n, yielding

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 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: 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  = Finally, calculate the confidence interval

In a sample of 41 students who work, the sample mean is hours and s = 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

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

Then the confidence interval is p  z C  s p 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 =  1.96 * etc  or  0.09 We are 95% confident that 0.54    0.71

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 z C  σ p or, substituting Solving for n yields:

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  0.09 or a 9% error. Suppose we require a maximum error of 3%. Approach 1: let  = 0.5

Approach 2: assume  = 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 _________.