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Unit 7
Section 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
Section 6.2
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
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
Section 6.2
 Degrees
of freedom - the number of free
choices left after a sample statistic (such
as the mean) is calculated
 Symbol:
 For
d.f.
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
X-E<m< X+E
s
E = tc ( )
n
 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
7160
5900
8440
6090
9930
6310
Section 6.2
Homework:
 Pgs.
315 - 317: #’s 1 – 33 ODD