CONFIDENCE INTERVALS

Types of Estimates Even when samples are taken using proper sampling techniques, there is still room for sampling error. Statistics are actually estimates of true population values (parameters)

Point Estimate Types of Estimates 2 basic types Point Estimate A specific numerical value that estimates a parameter Ex: is a point estimate for . “The average salary for a school nurse is $32000.”

Interval Estimate Types of Estimates An entire range of values used to estimate a parameter Ex: “The average salary for a school nurse is between $29,000 - $35,000 a year.” or “ The average salary for a school nurse is $32,000  3000.”

Interval Estimates  3000 in last example is refered to as the margin of error. An advantage to interval estimates compared to point estimates is that there is a better chance that the actual parameter falls within that range.

What is a Confidence Interval? An interval estimate of a parameter obtained from a sample with a certain probability the estimate will contain the parameter. For example: If the 95% confidence interval for the  salary of nurses is between $29000 - $35000 then that means there is a 95% chance that the true mean lies within that range.

Calculating confidence intervals for the mean (when  is known & n  30)

How is a confidence interval determined? Need 2 values: The Maximum Error of Estimate (E) The largest possible difference between a point estimate and the actual parameter itself. Formula: The sample mean:

Common Confidence Intervals () Most problems will ask for the 90%, 95%, or 99% confidence interval. Formula for a Confidence Interval:

Determining the z-scores: A 90% C.I. is comparable to the middle 90% on the normal distribution .45 .45 z = -1.64 z =1.64

Z-scores for common confidence intervals 90% C.I. - z = 1.64 95% C.I. - z = 1.96 99% C.I. - z = 2.58

Example 1: The president of a University wishes to estimate the average age of students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 50 students is selected and the mean is 23.2 years. Find the 95% C.I. and the 99% C.I. of the population mean.

Part 1:  = 95%

Part 2:  = 99%

Determining sample size for accuracy Is it large enough??? Determining sample size for accuracy

Determining sample size Sample size is the key element in determining accuracy when comparing sample means to population means. Example: If you want your sample mean to be within $1000 of your population mean then you must ensure to take a large enough sample.

How large is large enough? Formula to determine sample size:

Example 1: A college president wants a professor to estimate the average age of the students. The professor decides the estimate should be accurate within 1 year and 99% confident. From a previous study, the standard deviation is known to be 3 years. How large a sample is required?

Example 1: solution

Confidence intervals for  unkown & n < 30

Requires the use of the T-Distribution Comparison to the Normal Distribution: Similarities 1. Bell-Shaped 2. Symmetric About the mean 3.Mean, Median, Mode = 0 and located in the center of the distribution Differences 1. Variance is > 1 2. Actually a family of curves rather than just one curve 3. As n gets larger the t-distribution approaches the normal distribution

Degrees of Freedom Changes the shape of the t-distribution The # of values that are free to vary after a sample statistic has been computed. Tells the researcher (or the student) which specific curve from the family to use. The # of Degrees of Freedom is always equal to n - 1

Formula for Confidence Interval Same basic format of other formula but using different chart!

Example 1: A recent study of 25 students showed that they spent an average of $18.53 for gasoline per week. The standard deviation of the sample was $3. Find the 99% confidence interval of the true mean.

Example 1: Solution n = 25  D. of f. = 24 t = 2.797

When to use t or z????? Do you know the population standard deviation?  Use the normal distribution (z-scores) Is n  30? Use t-distribution (t-scores)

CONFIDENCE INTERVALS FOR PROPORTIONS

Example of a proportion In a study, 200 people were asked if they were satisfied with their job or profession. 162 said they were. Point estimate for a sample proportion:

Formula for Confidence Interval Where ‘p’ represents the true proportion of people from the population

Example 1: A survey of 80 recent fatal traffic accidents showed that 46 were alcohol- related. Find the 95% confidence interval of the true proportion of people who die in alcohol-related accidents.

Example 1: Solution n = 80 z = 1.96

DETERMINING SAMPLE SIZE FOR PROPORTIONS

THE FORMULA: If no approximation for p-hat is known, you should use .5

Example 1: An educator desires to estimate, within .03 the true proportion of high school students who study at least 1 hour each school night. He wants to be 98% confident. How large a sample is necessary? (From a previous study it is known that 60% of 250 students did)

Example 1: Solution

Example 2: We wish to estimate the proportion of students who own a cell phone. We want to be 95% confident and accurate within 5% of the true proportion. Find the minimum sample size necessary.

Example 2: Solution