# Confidence Intervals Confidence Interval for a Mean Confidence Interval for a Proportion Sigma...

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Confidence IntervalsConfidence Interval for a Mean

Confidence Interval for a ProportionSigmaSample Size

Inferential Statistics:INFERENTIAL STATISTICS: Uses sample data to make estimates, decisions, predictions, or other generalizations about the population. The aim of inferential statistics is to make an inference about a population, based on a sample (as opposed to a census), AND to provide a measure of precision for the method used to make the inference.An inferential statement uses data from a sample and applies it to a population.

Some TerminologyEstimation is the process of estimating the value of a parameter from information obtained from a sample.Estimators sample measures (statistics) that are used to estimate population measures (parameters).

Terminology (contd.)Point Estimate is a specific numerical value estimate of a parameter.Interval Estimate of a parameter is an interval or range of values used to estimate the parameter. It may or may not contain the actual value of the parameter being estimated.

Terminology (contd.)Confidence Level of an interval estimate of a parameter is the probability that the interval will contain the parameter.Confidence Interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using a specific confidence level.

Margin of Error, EThe term is called the maximum error

of estimate or margin of error. It is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter. It is represented by a capital E;

95%.025.025Za/2 : Areas in the TailsObtaining a: Convert the Confidence Level to a decimal, e.g. 95% C.L. = .95. Then:-z (here -1.96)z (here 1.96)

Situation #1: Large Samples or Normally Distributed Small Samples

A population mean is unknown to us, and we wish to estimate it.Sample size is > 30, and the population standard deviation is known or unknown.OR sample size is < 30, the population standard deviation is known, and the population is normally distributed. The sample is a simple random sample.

Confidence Interval for (Situation #1)

ConsiderThe mean paid attendance for a sample of 30 Major League All Star games was $46,970.87, with a standard deviation of $14,358.21. Find a 95% confidence interval for the mean paid attendance at all Major League All Star games.

95% Confidence Interval for the Mean Paid Attendance at the Major League All Star Games

Minimum Sample Size NeededFor an interval estimate of the population mean is given by

Where E is the maximum error of estimate (margin of error)

Situation #2: Small Samples

A population mean is unknown to us, and we wish to estimate it.Sample size is < 30, and the population standard deviation is unknown.The variable is normally or approximately normally distributed.The sample is a simple random sample.

Student t DistributionIs bell-shaped.Is symmetric about the mean.The mean, median, and mode are equal to 0 and are located at the center of the distribution.Curve never touches the x-axis.Variance is greater than 1.As sample size increases, the t distribution approaches the standard normal distribution.Has n-1 degrees of freedom.

Student t Distributions for n = 3 and n = 12

Confidence Interval for (Situation #2)A confidence interval for is given by

ConsiderThe mean salary of a sample of n=12 commercial airline pilots is $97,334, with a standard deviation of $17,747. Find a 90% confidence interval for the mean salary of all commercial airline pilots.

90% Confidence Interval for the Mean Salary of Commercial Airline Pilots

- t or z????Is Known?yesnoUse z-values no matter whatthe sample size is.*Is n greater thanor equal to 30?Use z-values and s in place of in the formula.yesnoUse t-values ands in the formula.***Variable must be normally distributed when n
A confidence interval for a population proportion p, is given by

Where is the sample proportion .

n = sample sizenp and nq must both be greater thanor equal to 5.

Situation #3: Confidence Interval for a Proportion

ConsiderIn a recent survey of 150 households, 54 had central air conditioning. Find the 90% confidence interval for the true proportion of households that have central air conditioning.

Here

We can be 90% confident that the true proportion, p, ofall homes having central air conditioning is between 29.6%and 42.5%

Minimum Sample Size NeededFor an interval estimate of a population proportion is given by

Where E is the maximum error of estimate (margin of error)

End of slides

07/16/96*##*##In what may be more understandable terms: We want to be able to make a statement about a group as a whole, by examining just a portion of the group, and we want to be able to say just how good or accurate our statement is.

Lets look at some terminology that we will be using throughout the course:*Recall: Parameter numerical characteristic of a population.

We will first be looking at confidence intervals for mu. Estimation Example: One out of four Americans is currently dieting.72% of Americans have flown commercial airlinesTwo out of 100 college students say Statistics is their favorite subject.*The best point estimate for mu, the population mean, is x-bar, the sample mean.

*Show 90%, 98%, and 99%**EXPLAIN:1 alpha

Z, alpha/2*We can be 95% confident that mu, the mean paid attendance for all Major League All Star games is between $41,832.85 and $52,108.89*Suppose we wish to estimate with 95% confidence, the mean paid attendance at Major League All Star games, and we want to be within $5,000 of the actual amount. *HOWEVER, the Std. Nrml. Dist. (z), is no longer appropriate to use.*n-1 is sample size minus 1.*Student t distributions have the same general shape and symmetry as the standard normal distribution, but reflect a greater variability that is expected with small samples.*Demonstrate use of t-table*We can be 90% confident that mu, the mean salary of all commercial airline pilots is between $88,132.88 and $106,535.12*If no approximation of p-hat is known, you should use p-hat = 0.5.This will give a sample size sufficiently large enough to guarantee an accurate prediction.