ABC Corporation of California publishes a variety of statistics,
including the number of individuals who got a new job during the past
12 months and the mean length of time the individuals have been on the
job. The Statistical Analysis Department of ABC Corporation reported
that the mean length of time of newly employed individuals in
California was 17.00 weeks.

A local Chamber of Commerce for the City of Riverside has commissioned
a study on the status of employment in the Riverside area. A sample of
16 employed residents of Riverside included data on the age and the
number of weeks on a job. A portion of the data collected in October
2001 is shown as follows:

 

Age  Weeks Employed     
55   21
30   18
23   11
52   36
41   19
25   12
42   7
45   25
25   6
40   21
25   13
25   11
59   34
49   27
33   18
35   20

a.       Based on the above data, summarize the data. Use EXCEL to
generate your statistical results.

b.      Develop a 99% confidence interval estimate of the mean age of
newly hired employees.

c.       How would you determine whether the mean duration of
employment in Riverside is greater than the California mean duration
of 17.00 weeks. When ? = .01, what is your conclusion?

Need this information 2 Aug by 10 p.m.

Catsabuggin ?

You can actually use Excel to do everything required here.

A.  The quickest way is to have Excel run your descriptive statistics
by entering the data, then doing the following:
?	click on Data Analysis, then the Descriptive Statistics option
?	Click on Summary Statistic; enter the Confidence Interval Level (in
decimals, not percentages).  Click okay and you?ll get an array of
statistics, as shown at the bottom of this spreadsheet linked below:
http://www.mooneyevents.com/confidence.xls


The 99% confidence level for both age and duration of employment uses
an ?alpha? or confidence level of 1% (0.01 in Excel?s statistical
format).

B.  As you can see the confidence interval for AGE is 37.75 years +/-
8.76, so you?re 99% confident that employees being hired are between
28.99 and 45.51 years.

Why are the results so bad?  There are 5 or almost one-third of our
new hires under 25 alone, not just 1%.

First, we have a small sample size, which is more prone to errors. 
But Excel has adjusted for this in using a T-distribution, meant for
small sample sizes of under 30.

The more fundamental problem is that these statistical tests are
designed for ?normal? or bell-shaped distributions.  The histogram
that I?ve included on the page for ?Age? on this page shows an unusual
number of new hires clumped at the bottom and another node in the
41-45 age range.

We don?t know why the distribution is skewed ? it may be that the
October, 2001 data reflected a large number of recent college
graduates.  Or that the population of Riverside is mostly young
people.  But what the test tells us is that it?s a non-normal
distribution.

Hofstra University
?Confidence Intervals? (Waner & Costenoble, September 2000)
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/finitetopic1/confint.html

C.	Because the age distribution is poor, it might have an impact on
the Weeks Employed.  Or it might not.

Here, the statistical mean is 18.69 weeks = / - 6.45 weeks or a range
of 12.24 to 25.14 weeks at the 99% confidence level.  The state
average of 17 weeks falls within that range ? so it?s impossible to
rule out a hypothesis that Riverside is significantly different than
the state data.

Here too it?s interesting to look at the histogram of the number of
Weeks Employed worked.  You can see that, while it?s not a classic
bell curve, it?s at least ?closer? to being bell-shaped than the Age
graph.

The summary statistics at the bottom of the page provide a wealth of
information.  For example we can see a measure of skew at the bottom
of the page ? and positive skew indicates a distribution with an
asymmetric tail extending toward more positive values.  (Negative skew
is a distribution with an asymmetric tail extending toward more
negative values.)

Similarly kurtosis measures flatness.  A positive number says that it
looks more like a normal distribution and a negative number says that
it is flatter.


Google search strategy:
?confidence interval? + ?small sample? + test



Best regards,

Omnivorous-GA

---

Request for Answer Clarification by catsabuggin-ga on 31 Jul 2005 06:48 PDT

In part C of the question, it ask "how would you determine whether the
mean distribution of employment in Riverside is greater than
California?"  Could you please word that question differently so I can
understand how you came to that conclusion?  I am clear about the
other parts, I had basically the same answers, just did not finsh the
problems.  Thanks.

---

Clarification of Answer by omnivorous-ga on 31 Jul 2005 07:15 PDT

Catsabuggin --

C.  Our statistical mean is 18.69 weeks.  If we want to know what
range falls within a 99% confidence interval, it's plus or minus 6.45
weeks, as we can see from the 6.448740214 statistic for alpha = .01.

That makes it a range of 12.24 to 25.14 weeks at the 99% confidence level.  

In statistical terms, we'd be asking ourselves, should be be accepting
the hypothesis H0 -- that there's no difference in the means of the
state and Riverside's mean employment period?  Or should we accept H1
-- the assumption that they're different?

Statistics Glossary
"Hypothesis Testing"
http://www.stats.gla.ac.uk/steps/glossary/hypothesis_testing.html

They're just too close too assume that Riverside's measure is
different from the state's.  So Riverside can't reasonably expect to
find a cause for length of employment being 1.69 weeks longer in the
city -- it just may be the small sample size.  And, from a practical
standpoint, it may be wise to predict future employees stay 17 weeks,
instead of using the 18.69 from its small sample.  Those are just some
conclusions a manager would reach.

On the other hand, a manager might well want to look at why
Riverside's hires in October 2001 had so many young people.  For
example, in an employment discrimination case due to age, this would
appear "outside the norm".

I know these last 2 paragraphs are nowhere in the scope of your
question: I'm simply trying to show the usefulness of the statistics.

Best regards,

Omnivorous-GA

Thank you for your answer, I beleive I understand it now.