# Chapter 11- Confidence Intervals for Univariate Data - PowerPoint PPT Presentation

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Chapter 11- Confidence Intervals for Univariate Data. Math 22 Introductory Statistics. Introduction into Estimation. Point Estimate – the value of a sample statistic used to estimate the population parameter.

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Chapter 11- Confidence Intervals for Univariate Data

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## Chapter 11- Confidence Intervals for Univariate Data

Math 22

Introductory Statistics

### Introduction into Estimation

• Point Estimate – the value of a sample statistic used to estimate the population parameter.

• Interval Estimate – an interval bounded by two values calculated from the sample data, used to estimate a population parameter.

### Introduction into Estimation

• Level of Confidence – The probability that the sample to be selected yields an interval that includes the parameter being estimated.

• Confidence Interval – An interval estimate with a specified level of confidence.

• Assumption – a condition that needs to exist in order to properly apply a statistical procedure to be valid.

### Confidence Interval

• A confidence interval for a population parameter is an interval of possible values for the unknown parameter.

• The interval is computed in such a way that we have a high degree of confidence that the interval contains the true value of a parameter.

### Confidence Level

• The confidence, stated as a percent, is the confidence level.

• In practice, estimates of unknown parameters are given in the form:

estimate margin of error

### Developing a Confidence Interval

Three determinations must be made to develop a Confidence Interval:

• A good point estimator of the parameter.

• The sampling dist. (or approximate sampling dist.) of the point estimator.

• The desired confidence level, usually stated as a percentage.

### Standard Error of a Statistic

• The standard deviation of its sampling dist. when all unknown population parameters have been estimated.

### Interpreting Confidence Intervals

Q:What does a 99% C.I. really mean?

A:A 99% C.I. means that of 100 different intervals obtained from 100 different samples, it is likely 99 of those intervals will contain the true parameter and one will not.

### Validity and Precision of Confidence Levels

• Validity - Measured by the confidence level, which is the probability that the interval will contain the true value of the parameter.

• Precision - measured by the length of the interval

### Reducing the Margin of Error

Two ways to reduce the margin of error:

• Decrease z

(Problem - Reduces Validity)

• Increase n

(No Problem)

### Estimation of the Mean When the Standard Deviation is Known

• When the population standard deviation is known, a (1-a)100% confidence interval for based on m is given by the limits:

### Estimation of the Mean When the Standard Deviation is Unknown

• We must make sure that the sampled population is normally distributed.

• Normal Plots

### Student-t Distribution

• Many times we do not know what s is . In these cases, we use s as the standard deviation. The standard error of the sample mean is now

### Characteristics of the Student-t Distribution

• Bell shaped and symmetric, just like the normal distribution is bell shaped and symmetric. The t-distribution “looks” like the normal distribution but is not normal.

• The t-distributionis a family of distributions, each member being uniquely identified by its degrees of freedom (df) which is simply n-1 where n is the sample size.

### Characteristics of the Student-t Distribution

• As the sample size increases the t-distribution becomes indistinguishable from the standard normal curve.

### Using the t-Interval

For small sample sizes:

If the sample size is less than 30, construct a normal plot of your data. If your data appears to be from a normal distribution, then use the t-distribution. If the data does not appear to be normal, then use a non-parametric technique that will be introduced later.

### Using the t-Interval

For large sample sizes:

If the sample size is 30 or more, use the t-distribution citing the Central Limit theorem as justification for having satisfied the required assumption of normality.

### Confidence Interval for the Median

Large Sample Confidence Interval for the Median:

• Sample size must be 20 or more.

• We can construct a confidence interval for q based on p.

• We can then produce a confidence interval for p with a sample proportion of .50 (this is used to represent the definition of the median, 50% below this mark, 50% above this mark.)

### Large Sample Confidence Interval for the Median

Basic steps for conducting a large sample confidence interval for the median:

• Construct a normal plot to see if the data is normal.

• If the normal assumption is violated, construct a (1-a)100% for p based on a sample proportion of .50.

• Multiply the upper and lower bound of the C.I. by n, the sample size. Round up the lower bound and round down the upper bound.

### Large Sample Confidence Interval for the Median

• Sort the data and identify the data values in those positions identified by the previous step.

### Small Sample Confidence Interval for the Median

• Sample size must be less than 20.

• The method we will explore is based strictly on the binomial distribution.

### Small Sample Confidence Interval for the Median

Basic steps for conducting a small sample confidence interval for the median:

• Create a table that contains the discrete cumulative probability distribution for 0 to n for a binomial distribution where p = .50.

• Identify the position for the lower bound with a cumulative probability as near a/2 as possible.

### Small Sample Confidence Interval for the Median

• Identify the position for the upper bound with a cumulative probability as near 1-a/2 as possible.

• Sort the data and identify the data values corresponding to the position located in the last two steps.

• Report the actual confidence level by summing the tail probabilities associated with the positions chosen for the C.I. Bounds.