Download Section 9.1 Inference for the Mean NOTES

Section 9.1 A confidence Interval for a Mean DISCUSSION
Statistics
Our last lesson involved finding a confidence interval for a proportion. We
found that we can substitute the proportion for a sample, _______, for the
population proportion, _______, and obtained the relationship:
In this lesson we are investigating the confidence interval for a ____________.
Remember that the mean for the population is represented as ________. The
mean for a sample is represented as ______.
The standard error of the
sampling distribution of a sample mean is ______
______
, where ______ is
______
the sample size and ________ is the standard deviation of the population. If the
sample size is large enough or the population is _________________
distributed the middle 95% would be represented as:
As in our analysis with a confidence interval for a sample proportion, we often
have only 1 sample to work with, and we do not know what the standard
deviation is for the population. What we can do is use the standard deviation of
the sample, _______. There is one problem with this, however. The sampling
distribution of s is ________________ right. Although the average value of
______ is approximately equal to ________, because it is skewed right the
______________ interval will be too narrow, so the capture rate will be less
than ________ if we use z = __________.
To compensate for this, we increase the width of the ________________
interval by using a _______ value rather than _______. ______ is called a
___________ t value. It was developed by ______________________ in 1915
when he worked for the ___________________ Brewery. The company did
not allow him to publish under his own name. In his articles, then, he used the
pen name _________________. As a result, the _______ values are often
called the ________________________________.
The value to use for t* can be found from a table. In our book, the table is on
page 826. Unlike z-scores, which give probabilities less than a certain value,
the t-distribution gives values of the tail above the value. At the top of the table,
we have the ___________ probability, ______. The bottom gives the
confidence level between the upper and lower tail. For example, the 95%
confidence level, C, corresponds with the tail probability, p, of ________ (2.5%
upper and 2.5% lower). There are a number of values down the column for
each tail probability (confidence level). The value used is determined by the
sample size, _____. On the left side of the table, each row has a _____ value,
which stands for _____________________________. The degrees of freedom
is always 1 less than the __________________, _____. For example, if our
sample size is 21 and we want to determine what t* value to use for a 95%
confidence interval, we would go across the row where df is equal to
___________ = ________. The value we would use is _______________.
Notice that as the sample size increases, the value of t* becomes closer and
closer to the z value, for a confidence interval of 95% being _________. The t*
value at a sample size of ________ is equal to the z value.
The book example deals with comparing body temperatures for 10 males and 10
females. The goal is to determine reasonably likely temperature values for
males and females. The data is given in the table on the next page:
Body Temperature (0F)
Male
Female
96.8
97.8
97.4
98.0
97.5
98.2
97.8
98.2
97.8
98.2
97.9
98.6
98.0
98.8
98.1
98.8
98.6
99.2
98.8
99.4
We are told that the distributions for each are approximately normally
distributed. The mean for the sample of 10 males is 97.88 and the sample
standard deviation, s, is 0.555. We want to determine a 95% confidence
interval for the males.
You are 95% confident that the mean body temperature of men in this
population is in the interval ___________________.
For the females, the sample mean is 98.52 and the sample standard deviation, s,
is 0.527. We want to determine a 95% confidence interval for the females.
You are 95% confident that the mean body temperature of women in this
population is in the interval ___________________.
We might be tempted to say that the average body temperature of females is
___________ than that of males. We need to be careful, though, because the
range of likely values has some _____________, so this might be an incorrect
assumption.
In drawing a conclusion, it is good practice to say something like “I am 95%
confident that the mean of whatever you are studying is in the interval from
lower value to higher value.
The Meaning of “95% Confident”
Saying you are 95% confident means that if you could take random samples
repeatedly from the population and compute a confidence interval from each
sample, in the long run 95% of these different ____________ would
__________________ or ________________ _______.
Margin of Error
The margin of error for a confidence interval for a mean is
Like that of the proportion, it is half the width of the _______________
interval. From the relation, you can see that as the sample size ______
increases, the margin of error _________________.
Expressing the margin of error for our male and female temperatures, we would
say that for men:
______________________
and for women:
_______________________