Download 7 - Tarleton State University

7.2: Parameters and statistics:
DEFINITION:
The population is the entire group of objects or individuals under study, about which
information is wanted.
A unit is an individual object or person in the population. The units are often called subjects
if the population consists of people.
A sample is a part of the population that is actually used to get information.
A variable is a characteristic of interest to be measured for each unit in the sample.
The size of the population is denoted by the capital letter N.
The size of the sample is denoted by the small letter n.
Population
Unit
Population size N = 16
Sample size n=4
Sample
DEFINITION:
A parameter is a numerical value that would be calculated using all of the values of the
units in the population.
A statistic is a numerical value that is calculated using all of the values of the units in a
sample.
Tip: One way to remember this distinction is this: The letter p is for population and
parameter, while the letter s is for statistic and sample.
Let's Do It! 1 (1min.) Parameter or Statistic?
According to the Campus Housing Fact Sheet at a Big-Ten University,
60% of the students living in campus housing are in-state residents. In
a sample of 200 students living in campus housing, 56.5% were found
to be in-state residents. Circle your answer.
(a)
In this particular situation, the value of 60% is a (parameter,
statistic).
(b) In this particular situation, the value of 56.5% is a (parameter,
statistic).
ESTIMATION:
 What is the mean number of delay hours of Northwest Airline
Flights to Chicago?
 What is the mean weight of actresses living in Hollywood?
 What is the mean of times per day a person in the U.S. uses a
pain reliever?
Each of these questions is asking, “What is the value of the parameter
?”
DEFINITIONS:
The sample average is a point (single number) estimate for the population
mean .
A confidence interval estimate for the population mean is an interval of
values, computed from the sample data, for which we can be quite confident
that it contains the population mean .
The confidence level is the probability that the estimation method will give an
interval that contains the parameter ( in this case). The confidence level is
denoted by , where common values of are 0.10, 0.05, and 0.01, for 90%, 95%,
and 99% confidence.
Let's Do It! 2 Point estimates
What is a possible point estimate of the population mean  ?_____.
What is a possible point estimate of the population standard deviation
 ?______.
7.2: Confidence Interval for the Population Mean (  known
OR n  30 ):
In class, you were requested to construct the distribution of the average
of all possible samples of size 2 from a small population of size 4. At
the end of the activity we had the following conclusions:
-The distribution of the x ' s has a bell shape( Normal).
- The mean of the x ' s is the same as the population mean.
-The standard deviation of the x ' s is related to the standard deviation of
the population.
A java applet was demonstrated in the class to show the approximate
normality of the averages of the as the sample size increased.
This Lead us to the conclusion that X ~N(  ,  / n )
Let’s Think About It!
Let X ~N(  ,  / n ), using the Empirical Rule, answer the following
questions:
a. 68% of the x ' s fall with within 1 standard deviation of the mean.
This is equivalent to saying that the mean  that the mean is
within 1standard deviation of the average of a sample 68% of the
times.
Based on this fact, can you construct the interval that has 68%
chance of containing the mean?
b. 95% of the x ' s fall with within almost 2 standard deviation of the
mean. This is equivalent to saying that the mean  is within
almost 2 standard deviation of the average of a sample 95% of
the times.
Based on this fact, can you construct the interval that has 95%
chance of containing the mean?
c. 99.7% of the x ' s fall with within almost 3 standard deviation of the
mean. This is equivalent to saying that the mean  is within
almost 3 standard deviation of the average of a sample 99.7% of
the times.
Based on this, can you construct the interval that has 99.7%
chance of containing the mean?
In statistics, any sample that has a size greater than or equal 30
subjects is considered a large sample.
Confidence Interval for the Population Mean using a large sample
If the sample used to construct the confidence interval is large, then, we can use the
following formula for a (1  ) % confidence interval for the mean  :
( x  z / 2
s
s
, x  z / 2
)
n
n
Where,
x : is the sample average.
 : is called type I error. This is predetermined.
s : The population standard deviation, if known.
n: The sample size.
z /2 : is the z score that has an area ( 1   / 2 ) to the left of it.( which means  / 2
area o the right)
Let's Do It! 3
What is the 95% confidence interval for  using a sample having the
following statistics n  75, x  25, s  10 .?
What is the 90% confidence interval for  using a sample having the
. ?
following statistics n  35, x  2.8, s  12
Let's Do It! 4 Life of light bulbs
An electrical firm manufactures light bulbs that have a length of life with
a standard deviation of 40 hours. A sample of 35 bulbs has an average
life of 780 hours.
a. Construct a 95% confidence interval for the mean length of life
of bulbs manufactured by this firm.
b. If the manufacturer claimed that the life of a light bulb is 900
hour. What would be your reaction? Explain.
Let's Do It! 5
The height of a random sample of 50 college students showed a mean
of 174.5 cm and a standard deviation of 6.9 cm. Construct a 99%
confidence interval for the mean height of college students.
HW7.2 page 358: 1,6, 9-20 all
7.3: Small Sample Confidence Interval for the Population Mean
A sample is said to be statistically small if the size of the sample is less
than 30.
When sampling a small sample from a normally distributed population
with an unknown variance, an alternative confidence interval must be
used using a distribution called the t-distribution.
Properties of the t-distribution
 The t-distribution has a symmetric bell-shaped density centered at 0,
similar to the N(0,1) distribution.
 The t-distribution is “flatter” and has “heavier tails” than the N(0,1)
distribution.
 As the sample size increases, the t-distribution approaches the
N(0,1) distribution.
Small Sample Confidence Interval for the Population Mean
The (1  )% confidence interval for the mean of a normally distributed
population when the sample is small and the population variance is
unknown is given by:
x  t / 2,n1
s
n
Where,
x : is the sample average.
 : type I error. This is predetermined.
s : The sample standard deviation.
n : The sample size.
t / 2,n 1 : is the t score that has an area 1   / 2 to the left of it. (This value
can easily be obtained from table F).
How to use Table F to find
t / 2,n 1
:
Example : Find the ta_2 value for a 95% confidence interval when the
sample size is 22.
Solution
The d.f. = 22  1, or 21. Find 21 in the left column and 95% in the row
labeled Confidence Intervals. The intersection where the two meet
gives the value for ta_2, which is 2.080.
Let's Do It! 6 Skin Cancer
: A dermatologist is investigating a certain skin cancer. Twenty five rats
have this cancer and are treated with a new drug. The dermatologist is
interested in the number of hours until the cancer is gone. He found that
the sample produced an average of 322 hours and a standard deviation
of 101 hours. Assuming normality,
a. Compute a 90% confidence interval for the mean number of
hours.
b. Interpret the confidence interval constructed above.
Let's Do It! 7 Jogging and Pulse Rate
A random sample of 21 US adult males who jog at least 15 miles per
week is taken and their pulse rate is measured. The sample had an
average pulse rate of 52.6 beats /minute with a standard deviation of
3.22 beats /minute.
a. Find a 95% confidence interval for the mean pulse rate of all US
males who jog at least 15 miles per week. Assuming pulse rate is
normally distributed.
b. Interpret the interval obtained above.
c. If the mean pulse rate of all US adult males is approximately 72
beats/minute. Does it appear that jogging at least 15 miles per
week reduces the mean pulse rate? Explain
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