Confidence Interval Estimation of Population Mean, μ, when σ is Unknown Chapter 9 Section 2

The BIG Idea! Most of the time, the value for the populations standard deviation is NOT known. For example, what’s the likelihood anyone know the average number of kids in each family in this school or the standard deviation for that population? How do we develop a confidence interval when the standard deviation for the population is unknown????

Confidence Interval Estimation of Population Mean, μ, when σ is Unknown If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s This introduces extra uncertainty, since s varies from sample to sample So we use the student’s t distribution instead of the normal Z distribution

t-distribution The t – distribution is actually a family of curves based on the concept of degrees of freedom, which relate to sample size. As the sample increases, the t-distribution approaches the standard normal distribution.

Student’s t distribution Note: t Z as n increases Standard Normal t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal t (df = 5) t

Degrees of Freedom The degrees of freedom (denoted d.f.) are the number of values that are free to vary after a sample statistic has been computed, and tell the researcher which specific curve to use when a distribution consists of a family of curves. For example: If the mean of 5 values is 10, then 4 of the 5 values are free to vary. But once 4 values are selected, the fifth value must be a specific number to get the sum of 50, since 50/5 = 10. Hence, the degrees of freedom are 5 – 1 = 4, and this tells the researcher which curve to use.

d.f. Formula The formula for finding the degrees of freedom for the confidence interval of a mean is simply d.f. = n – 1

Using Table F Find the tα/2 value for a 95% confidence interval when the sample size is 22. Degrees of freedom are d.f. = 21.

Formula for a Specific Confidence Interval for the Mean When σ Is Unknown and n < 30 The degrees of freedom are n – 1.

Example #1- Commuting Time A sample of 14 commuters in Chicago showed the average of the commuting time was 33.2 minutes. If the standard deviation of the sample was 8.3 minutes, fine the 95% confidence interval of the true mean.

Example #2 - Sleeping Time Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.

Example #3 - Home Fires by Candles The data represent a sample of the number of home fires started by candles for the past several years. Find the 99% confidence interval for the mean number of home fires started by candles each year. 5460 5900 6090 6310 7160 8440 9930

Example #4 Poll Everywhere poll on sleep last night. Link to Poll What is the mean and standard deviation for our sample? What is the degrees of freedom for our sample? Find a 90% confidence interval of the mean time. Assume the situation is normally distributed.