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Transcript
Aim: What are confidence intervals
for means that have unknown
standard deviations and sample
sizes less than 30?
Quiz Friday
T distribution
• Used when sample size is less than 30 and the
variable is normally or approximately normally
distributed
• Important Characteristics:
– It is a bell-shape
– It is symmetric 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
How T distribution differs from
standard normal distribution
• The variance is greater than 1
• The t distribution is actually a family of curves
based on the concept of degrees of freedom,
which is related to sample size
• As the sample size increases, the t distribution
approaches the standard normal distribution
Degrees of freedom
• Degrees of freedom: the number of values
that are free to vary after a sample statistics
has been computed
– Tells the researcher which specific curve to sue
when a distribution consists of a family of curves
d.f. = n -1
The t family of Curves
Formula for a specific confidence
interval for the mean
• Use this when standard deviation is unknown and
n < 30
 s 
 s 
X  t 
   X  t 


2
2
n
 n
• The values of tα/2 are found in table F. The top
row of table F, labeled confidence intervals is
used to get these values
Using the F Table
Example
• Ten randomly selected automobiles were
stopped and the tread depth of eh right front
tire was measured. The mean was 0.32 inch,
and the standard deviation was 0.08 inch. Find
the 95% confidence interval of the mean
depth. Assume the variable is approximately
normally distributed.
Solution
• Since σ is unknown and s must replace it, the t
distribution must be used for 95% confidence
interval. Hence the d.f. = 9, tα/2 =2.262 (from
table F).
 s 
 s 
X  t 
   X  t 


2
2
n
n
 .08 
 .08 
.32   2.262  
    .32   2.262  

 10 
 10 
.32  0.057    .32  .057
.26    .38
Class Work
1. Find the values for each
1.
2.
tα/2 and n = 18 for the 99% confidence interval for the mean
tα/2 and n =20 for the 95% confidence interval for the mean
2. 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 $3.00. Find the 95%
confidence interval of the true mean.
3. The number of unhealthy days based on the Air Quality
Index for a random sample of metropolitan areas is
shown. Construct a 98% confidence interval based on the
data.
61
12
6
40
27
38
93
5
13
40