Estimations from Sample Data

Point Estimates

Population Parameter

Unbiased estimator Formula

Mean, µ

Variance,

Proportion, π p

Interval Estimates

•Interval estimate for the mean describes a range of values that is likely to include the population mean.

•Interval limits – the lower and upper values of the interval estimate

•Confidence interval – An interval for which there is a specified degree of certainty that the actual population parameter will fall with the interval▫Margin of error

Interval Estimates

•Confidence coefficient/Confidence level – express the degree of certainty that an interval will include the actual value of the population parameter. ▫Coefficients are expressed as a number

between 0 and 1 (.95) ▫Levels are expressed as percentages (95%)

•Accuracy – the difference between the sample statistic and the actual parameter▫Sometimes called sampling error.

Interval Estimates - Example

•In a random sample of 2000 households, the average income is $65,000 with a standard deviation of $12,000. Based on these data, we are 95% confident the population mean is between $64,474 and $65,526.Point estimate of µ = 65,000 Point estimate of σ = 12,000

Interval estimate of µ =64,474 to 65,526

Lower and upper interval limits for µ = 64,474 and 65,526

Confidence coefficient = 0.95 Confidence level = 95%

Accuracy: For 95% of such intervals, the sample mean would not differ from the actual population mean by more than $526

Special election 2010

Poll n Error +/- Coakley Brown

A 804 3.5% 43 %39.5 46.5

52% 48.5 55.5

B 574 4.2% 42%37.8 46.2

52% 47.8 56.2

C 1231 2.8% 46%43.2 48.8

51% 48.2 53.8

D 600 4.1% 45%40.9 49.1

52% 47.9 56.1

E 500 4.5% 48%43.5 52.5

48% 43.5 52.5

Source: Real Clear Politics, 19 Jan 2010: http://www.realclearpolitics.com/epolls/2010/senate/ma/massachusetts_senate_special_election-1144.html

Result Coakley 47.1% Brown 51.9%

ExampleProblem 9.11: In surveying a simple random sample of 1000 employed adults, we found that 450 individuals felt they were underpaid by at least $3000. Based on these results, we have 95% confidence that the proportion of employed adults who share this sentiment is between 0.419 and 0.481.

What is the point estimate for the population proportion?

What is the confidence interval estimate for the population proportion?

What is the confidence level and the confidence coefficient?

What is the accuracy of the sample result?

Confidence Interval: s known

where x = sample mean ASSUMPTION:

s = population standard infinite population

deviationn = sample sizez = standard normal score for area in tail = a/2a/2 a/21- a

nzxx

nzxx

zzzss

×+×

+

–:

0–:

© 2008 Thomson South-Western

Confidence interval practice

A simple random sample of 25 is collected from a normally distributed population.• σ = 17.0• = 342

Construct and interpret a 95% confidence interval for the population mean.

Construct and interpret a 99% confidence interval for the population mean.

• t distribution is the probability distribution for the random variable, t:

•The t distribution has a mean of 0, but its shape is determined by degrees of freedom.▫Specifically, for this distribution, df = n-1

Confidence Interval: s unknown

Using the t Distribution Table

•For a sample size of n = 15, what t values would correspond to an area centered at t = 0 and having an area beneath the curve of 95%?

•For a sample size of n = 99, what t values would correspond to an area centered at t = 0 and having an area beneath the curve of 90%?

Confidence IntervalsUsing the t distribution• Like before, only using t instead of z

• Where▫Sample mean▫Sample standard deviation▫Sample size▫ t value corresponding to the desired level of

confidence▫Estimated standard error of the mean

Confidence Intervals - Example

•A random sample of 90 employees has been selected from those working in a company. The average number of overtime hours last week was 8.46 with a sample standard deviation of 3.61 hours.

•What is the 98% confidence interval for the population mean?

Confidence Intervals - Practice

•The service manager for Appliance Universe conducted a random sample of 50 service calls from last year’s records. The same mean is 25 minutes and the sample standard deviation is 10 minutes.

•Construct a 95% confidence interval.

Confidence Interval - Proportion•Π unknown

Correction for finite population, when required

•Then, back to business as normal

Confidence Interval - Proportion•A major metropolitan newspaper selected

a simple random sample of 1,600 readers from their list of 100,000 subscribers. They asked whether the paper should increase its coverage of local news. Forty percent of the sample wanted more local news. What is the 99% confidence interval for the proportion of readers who would like more coverage of local news?

Sample Size - Mean• Let’s consider a case where the population

standard deviation is known.

• Where:▫n = required sample size▫z = z value for which ± z corresponds to the

desired level of confidence▫σ = known or estimated value of the standard

deviation▫e = maximum likely error that is acceptable

Sample Size - Example

•The governor wants to know the average amount teenagers earn during their summer vacation. He wants to be 95% confident the sample mean is within $50 of the population mean. Standard deviation is estimated at $400.

•What sample size is required to achieve the desired results?

Sample Size Mean - Practice

•A package-filling machine has been found to have a standard deviation of 0.65 ounces. A random sample will be conducted to determine the average weight of product being packed by the machine.

•To be 95% confident that the sample mean will not differ from the actual population mean by more than 0.1 ounces, what sample size is required?

Sample Size - Proportion

•Required sample size for a population proportion.

•When we have no idea what the population proportion is, set p=0.5

•When we think the proportion might fall within some range, chose the value closest to 0.5.

Sample Size Proportion - Practice•A tourist agency would like to determine

the proportion of U.S. adults who have vacationed in Mexico and wants to be 95% confident the sampling error will be no greater than 3%.

•What is the required sample size to satisfy these parameters?