Chapter 7

Estimation Procedures

Basic Logic

In estimation procedures, statistics calculated from random samples are used to estimate the value of population parameters.

Example: If we know 42% of a random sample

drawn from a city are Republicans, we can estimate the percentage of all city residents who are Republicans.

Basic Logic Information from

samples is used to estimate information about the population.

Statistics are used to estimate parameters.

POPULATION

SAMPLE

PARAMETER

STATISTIC

Basic Logic Sampling Distribution

is the link between sample and population.

The value of the parameters are unknown but characteristics of the S.D. are defined by theorems.

POPULATION

SAMPLING DISTRIBUTION

SAMPLE

Estimation Procedures A point estimate is a sample statistic

used to estimate a population value. Both sample means and sample

proportions are unbiased estimates of the population mean or proportion. (explain bias)

For both means and proportions we can use characteristics of their respective sampling distributions to establish confidence intervals around the statistic.

Shape of sampling distributions

For both means and proportions, as N becomes large, the sampling distribution will be normal (for our purposes, N=100 is large enough)

The standard deviation of the sampling distribution is called the standard error.

Standard error and mean of sampling distribution

1. If we take a large sample, we can use the mean of the sample as an unbiased estimate of the mean of the sampling distribution.

2. The standard error of the mean = (the standard deviation of the sample)/(the square root of N-1)

Efficiency of confidence intervals

Efficiency is determined by the dispersion in the sampling distribution.

The smaller the standard deviation of the sampling distribution, the greater the efficiency of our estimate

Efficiency is therefore maximized as N gets larger

Constructing Confidence Intervals For Means Set the alpha level(probability that the interval will be

wrong). Setting alpha equal to 0.05, a 95% confidence level,

means the researcher is willing to be wrong 5% of the time.

Find the Z score associated with alpha. Z-scores are expressed in standard deviation units. Statistics texts always have a table that correlates z scores and areas under a normal curve (show example on overhead projector) If alpha is equal to 0.05, we would place half (0.025)

of this probability in the lower tail and half in the upper tail of the distribution.

Confidence Intervals For Means: Problem 7.5c

For a random sample of 178 households, average TV viewing was 6 hours/day with s = 3. Alpha = .05. c.i. = 6.0 ±1.96(3/√177) c.i. = 6.0 ±1.96(3/13.30) c.i. = 6.0 ±1.96(.23) c.i. = 6.0 ± .44

Confidence Intervals For Means

We can estimate that households in this community average 6.0±.44 hours of TV watching each day.

Another way to state the interval: 5.56≤μ≤6.44 We estimate that the population mean is greater

than or equal to 5.56 and less than or equal to 6.44.

This interval has a .05 chance of being wrong.

Confidence Intervals For Means

Even if the statistic is as much as ±1.96 standard deviations from the mean of the sampling distribution the confidence interval will still include the value of μ.

Only rarely (5 times out of 100) will the interval not include μ.

Sampling error for proportions

As N becomes large (100 or more), the proportion in the sample is an unbiased estimate of the proportion in the sampling distribution (and the population)

The standard error of proportions = the square root of (.25/N)

Constructing Confidence Intervals For Proportions

Procedures: Set alpha. Find the associated Z score. For an alpha of .05, we put .025 in each

tail of the normal distribution, and using our table of normal curve areas, Z = 1.96.

Confidence Intervals For Proportions

If 42% of a random sample of 764 from a Midwestern city are Republicans, what % of the entire city are Republicans?

Don’t forget to change the % to a proportion. c.i. = .42 ±1.96 (√.25/764) c.i. = .42 ±1.96 (√.00033) c.i. = .42 ±1.96 (.018) c.i. = .42 ±.04

Confidence Intervals For Proportions

Changing back to %s, we can estimate that 42% ± 4% of city residents are Republicans.

Another way to state the interval: 38%≤Pu≤ 46% We estimate the population value is greater than

or equal to 38% and less than or equal to 46%. This interval has a .05 chance of being

wrong.

SUMMARY

In this situation, identify the following: Population Sample Statistic Parameter

SUMMARY

Population = All residents of the city.

Sample = the 764 people selected for the sample and interviewed.

Statistic = Ps = .42 (or 42%) Parameter = unknown. The % of all

residents of the city who are Republican.