Unit 7Section 6.2

6.2: Confidence Intervals for the Mean (σ is unknown) When the population standard deviation

is unknown and our sample is less than 30…

We can use the sample standard deviation in place of the population standard deviation

We can use a t distribution to calculate our confidence interval in place of our standard normal distribution

Characteristics of a t Distribution (similarities to a standard normal distribution)

It is a bell shaped curve and symmetrical about the mean

The mean, median, and mode are equal to 0 and are located at the center of the distribution.

The curve never touches the x-axis

The area underneath the curve is equal to 1.

Section 6.2

Section 6.2

Characteristics of a t Distribution (different than a standard normal

distribution)

The variance and standard deviations are greater than 1

The t distribution is a family of curves based on its degree of freedom.

As the sample size increases, the t distribution approaches the standard normal distribution.

Section 6.2

Degrees of freedom - the number of free choices left after a sample statistic (such as the mean) is calculated

Symbol: d.f.

For Example: If the mean of 5 values is 10, then 4 of the values are free to vary. Once 4 values are selected, the 5th value must be a specific number to make a mean of 10.

Section 6.2

Formula for a Specific Confidence Interval of the Mean, whenαis

unknown and n<30

The degrees of freedom are n - 1

Section 6.2

Example 1:Find the t value for a 95%

confidence interval when the sample size is 22.

Section 6.2

Example 2:Ten randomly selected automobiles

were stopped, and the tread depths of the right front tire was measured. The mean was 0.32 inches, and the standard deviation was 0.08 inches. Find the 95% confidence interval of the mean depth. Assume that the variable is approximately normally distributed.

Section 6.2

Example 3:The data represents 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 63107160 8440 9930

Section 6.2

Homework: Pgs. 315 - 317: #’s 1 – 33 ODD

Section 6.2