Baruch College, 11-8-19 Prof. L. Tatum
Lecture Notes 8 Solutions
Section 2
1. A sample of n=30 bags of Nestlé's Raisinets were analyzed in August of 2012 by a
Baruch class. The net weights in grams of the 30 bags are given, below. The data can be
readily copied into Excel.
Below, a photo of a bag of Raisinets is shown with the stated net weight visible.
Using that net weight (in grams) as the null hypothesis, conduct a t-test of that claim as
follows.
(a) State the null and alternate hypotheses, both in words and in symbols. In those
statements, reference the subject matter of this test. That is, refer not just to a generic
population average, but specify what population is under study.
Ho: μ=44.7 g
Ha: μ ≠44.7 g.
The null hypothesis claims that the population average of the net weights of all the
bags of Raisinets in this production run is 44.7 grams. The alternate hypothesis says
that this population average is not equal to 44.7 grams.
(b) Compute the raw distance between the sample average and the value of μ claimed by
the null.
Using Excel, the sample average was found to be 46.32 grams, and μo has been given
as 44.7 grams. Thus, the raw distance between the value of the population average
claimed by the null and the observed sample average is
x - μo= 46.32 - 44.7 = 1.62.
(c) With respect to the distance between x and μo, under what conditions do we reject
the null?
If the distance between x and μo is large, then reject the null hypothesis. Or, put
differently, if x is far from μo, then reject the null hypothesis.
(d) Compute the standard error of the average. Show your work.
s.e. of average = s / √n = 1.192 / √30 = 0.2176.
(e) Why are we going to divide the raw distance by the standard error of the average?
1
, I am going to expand on this topic to include the class discussion, and to review the
deeper material in the Lecture Notes.
We need to translate the raw distance into a statistical distance for which we can
define the meaning of "large" (or "far"). We do that by comparing the x outcome to
the bell curve distribution from which we think it came. The outcome will be judged
to be far from μ if it falls in the tails of the bell curve or beyond: in Jane's terms, if it
is unusual or exceptional.
What is the "outcome" we are dealing with, here? The outcome is a value of x .
From what underlying distribution does x come from? There a zillion different
samples of size n=30 that we could take, and each has its own sample average. The
complete set of these x 's constitutes a new population, a population of sample
averages! Now, remember that the sample average is a kind of portfolio bet. And, like
the portfolios we considered earlier, the population of sample averages is narrower
than that of the original population. In particular, the population standard deviation of
the population of sample averages is smaller by a factor of √n, so
σX = σ / √n.
The next problem is that don't know σ, here. But, we can estimate it using the sample
standard deviation, s. Here, s=1.192. How, do we translate that into an estimate of the
width parameter for the population of sample averages? Divide it square root of n:
sx = s / √n.
This estimate of σX is also called "the standard error of the average", although that
term is does not help anyone to understand this process!
Now, the upshot is this: when we divide the raw distance by the standard error of the
average to get a "t-statistic." The t-statistic comes from a t-distribution with n-1 d.f. if
the null hypothesis were true. That t-distribution looks a lot like a standard Normal Z
distribution as long as the sample size is large, say larger than 15, but it always has
longer tails than a Z. (Those tails are ridiculously long if our sample size is only 2 or
3.) So, if the t-statistic comes from the ordinary range of outcomes under the bell
curve for that t-distribution, then we do not reject the null hypothesis because this
would be an ordinary outcome if the null were true. But, if the t-statistic falls out in
the tails of that bell curve, then we take that as evidence that the null hypothesis was
false.
Whew! Now you know why most people who use the t-test do not understand it!
(f) Compute the t-statistic. Show your work.
x - μo 46.32 - 44.7
t-statistic = = = 7.45
s/ n 1.
(g) Find the degrees of freedom.
2
Lecture Notes 8 Solutions
Section 2
1. A sample of n=30 bags of Nestlé's Raisinets were analyzed in August of 2012 by a
Baruch class. The net weights in grams of the 30 bags are given, below. The data can be
readily copied into Excel.
Below, a photo of a bag of Raisinets is shown with the stated net weight visible.
Using that net weight (in grams) as the null hypothesis, conduct a t-test of that claim as
follows.
(a) State the null and alternate hypotheses, both in words and in symbols. In those
statements, reference the subject matter of this test. That is, refer not just to a generic
population average, but specify what population is under study.
Ho: μ=44.7 g
Ha: μ ≠44.7 g.
The null hypothesis claims that the population average of the net weights of all the
bags of Raisinets in this production run is 44.7 grams. The alternate hypothesis says
that this population average is not equal to 44.7 grams.
(b) Compute the raw distance between the sample average and the value of μ claimed by
the null.
Using Excel, the sample average was found to be 46.32 grams, and μo has been given
as 44.7 grams. Thus, the raw distance between the value of the population average
claimed by the null and the observed sample average is
x - μo= 46.32 - 44.7 = 1.62.
(c) With respect to the distance between x and μo, under what conditions do we reject
the null?
If the distance between x and μo is large, then reject the null hypothesis. Or, put
differently, if x is far from μo, then reject the null hypothesis.
(d) Compute the standard error of the average. Show your work.
s.e. of average = s / √n = 1.192 / √30 = 0.2176.
(e) Why are we going to divide the raw distance by the standard error of the average?
1
, I am going to expand on this topic to include the class discussion, and to review the
deeper material in the Lecture Notes.
We need to translate the raw distance into a statistical distance for which we can
define the meaning of "large" (or "far"). We do that by comparing the x outcome to
the bell curve distribution from which we think it came. The outcome will be judged
to be far from μ if it falls in the tails of the bell curve or beyond: in Jane's terms, if it
is unusual or exceptional.
What is the "outcome" we are dealing with, here? The outcome is a value of x .
From what underlying distribution does x come from? There a zillion different
samples of size n=30 that we could take, and each has its own sample average. The
complete set of these x 's constitutes a new population, a population of sample
averages! Now, remember that the sample average is a kind of portfolio bet. And, like
the portfolios we considered earlier, the population of sample averages is narrower
than that of the original population. In particular, the population standard deviation of
the population of sample averages is smaller by a factor of √n, so
σX = σ / √n.
The next problem is that don't know σ, here. But, we can estimate it using the sample
standard deviation, s. Here, s=1.192. How, do we translate that into an estimate of the
width parameter for the population of sample averages? Divide it square root of n:
sx = s / √n.
This estimate of σX is also called "the standard error of the average", although that
term is does not help anyone to understand this process!
Now, the upshot is this: when we divide the raw distance by the standard error of the
average to get a "t-statistic." The t-statistic comes from a t-distribution with n-1 d.f. if
the null hypothesis were true. That t-distribution looks a lot like a standard Normal Z
distribution as long as the sample size is large, say larger than 15, but it always has
longer tails than a Z. (Those tails are ridiculously long if our sample size is only 2 or
3.) So, if the t-statistic comes from the ordinary range of outcomes under the bell
curve for that t-distribution, then we do not reject the null hypothesis because this
would be an ordinary outcome if the null were true. But, if the t-statistic falls out in
the tails of that bell curve, then we take that as evidence that the null hypothesis was
false.
Whew! Now you know why most people who use the t-test do not understand it!
(f) Compute the t-statistic. Show your work.
x - μo 46.32 - 44.7
t-statistic = = = 7.45
s/ n 1.
(g) Find the degrees of freedom.
2
