MATH 115A Business Mathematics I

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MATH 115ABusiness Mathematics I

• Sections 5 and 7
• April 10, 2006

2
Announcements

• HW 10 due Friday
• Tuesday office hours canceled this week
• Regular office hours on Wednesday and Thursday

3
Focus on the Project

• Recall that we converted our first-hour arrival
  times to inter-arrival times and
• Plotted a histogram (March 20)
• Called the resulting r.v. A (April 3)
• Last Monday, we observed that the histogram of A
  closely resembles the graph of the p.d.f. of an
  exponential r.v.

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Focus on the Project, contd

• We now take our data and approximate the p.d.f.
  of A.
• Create a new column with the midpoints of the
  bins.
• We used bin widths of 0.3, so we take each
  relative frequency and divide it by 0.3 to
  convert it to Adjusted Relative Frequency (or
  just Height).
• Create a line graph.

5
Focus on the Project, contd

• We want to compare the line graph with an
  appropriate exponential r.v.
• Since our sample of size 573 has a sample mean of
  0.52, we know that E(A) ? 0.52.
• Thus, the parameter ? for the exponential r.v.
  will be approximately 0.52.
• Note in actual calculations, use cell reference
  rather than typing in 0.52.
• Create a third column giving the value of the
  exponential r.v. and plot a line graph.

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Focus on the Project, contd

• Note that the graphs are virtually identical.
• From now on, we will assume that A is an
  exponential r.v. with parameter ? ? 0.52.
• Thus, A has p.d.f. and c.d.f. given by
• We will use the c.d.f. to simulate values of A in
  the next section.

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Focus on the Project, contd

• We now do all the same steps for B, the
  inter-arrival times for the second hour.
• From now on, we will assume that B is an
  exponential r.v. with parameter ? ? 1.92.
• Thus, B has p.d.f. and c.d.f. given by
• We will use the c.d.f. to simulate values of B in
  the next section.

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Focus on the Project, contd

• For the service times S, we do the same steps but
  dont bother plotting the line graph of S with an
  exponential r.v. of any kind.
• From the histogram we made, S is obviously not an
  exponential r.v., and, in fact, doesnt resemble
  either type of continuous r.v. weve studied.
• Since S has an unknown distribution, we use
  bootstrapping to simulate S.

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Focus on the Project, contd

• Key to bootstrapping randomly selecting a bunch
  of observations of the r.v.
• An example is done at the bottom of sheet Random
  Sampling in Queue Focus.xls.
• We need the Excel functions IF, ISNUMBER,
  RANDBETWEEN, and VLOOKUP, as well as conditional
  formatting.

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Focus on the Project, contd

• Cell D80 is used to specify how many observations
  to generate.
• Cells C83C282 are an Index column.
• Used for reference to create the actual sample
• Cell D83 IF(C83ltD80,C83,"")
• Copies the numbers in Column C only up to the
  number of observations we need.

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Focus on the Project, contd

• Cell E83 IF(ISNUMBER(D83), VLOOKUP(RANDBETWEEN(1
  ,7634), Data!G45Data!H7678,2), "")
• ISNUMBER tests to see if we actually need to
  generate the entry here.
• If we do, then we generate a random number
  between 1 and 7,634, look in the list of service
  times for the row with the randomly chosen number
  in column 1, and take the observation in column 2
  of the same row.
• If not, we leave the cell blank.

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Focus on the Project, contd

• The conditional formatting in Columns DE is done
  by clicking on FormatConditional Formatting.
• Boxes Cell Value is, less than or equal to,
  and 200.
• Click on Format and select Border and
  Outline.
• Because blank cells have "" in them, rather than
  simply being empty, this works.

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Focus on the Project, contd

• Cells F83G83 show the sample mean and total time
  of all the observations in the sample.
• These are just two of many possible statistics
  that can be calculated from bootstrapped data.

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What to Do Now

• Answer the HW questions that refer to your teams
  data.
• Assuming A and B are exponential with appropriate
  parameters, find formulas for fA, FA, fB, and FB.

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Questions?