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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

- 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.

- 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.

- 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.

- 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

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.

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

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 - 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

Two ways to reduce the margin of error:

- Decrease z
(Problem - Reduces Validity)

- Increase n
(No Problem)

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

- We must make sure that the sampled population is normally distributed.
- Normal Plots

- 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

- 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.

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

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.

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.

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.)

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.

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

- Sample size must be less than 20.
- The method we will explore is based strictly on the binomial distribution.

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.

- 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.