Lab 5 - Department of Mathematics and Statistics



Lab 5

Time Between Successive Clicks

of a Geiger Counter

Introduction

In this lab, you will be recording random background radiation for five minutes. Sources of background radiation include cosmic rays, radon gas seeping out of the ground, very small contributions from nuclear facilities, and naturally occurring radioactive isotopes (for example, the principle source of radioactivity in the human body is potassium-40). Although the time of emission of any particular particle is random, there are some quantities related to radioactive emissions that that can be modeled and the associated probabilities predicted. One such variable is X = the length of the time between clicks on a Geiger counter monitoring background radiation. We can think of each “click” as an “event” for the Geiger counter, and then X = “waiting time between events.” How likely is a long or short wait?

X is a random variable, so it has a probability density function (p.d.f.), f. The p.d.f. is used to represent probabilities associated with X: it represents them as areas under its graph. More specifically, probability that X is between a and b is calculated by integrating f over the interval [a, b].

Of course, a formula for f, to model the probabilities in this situation, has to be based on data. For this particular random variable X, an exponential p.d.f. works pretty well (as will be indicated by the shape of a frequency chart, later). An exponential density function will have an equation like

f(x) = 0 for x < 0

f(x) = ce-cx for x ≥ 0

Such exponential distributions are often good to model random variables such as waiting times or time intervals between equipment failures.

One goal of this experiment is to use the data collected to estimate the probability density function of the random variable X = time gap between detections (= waiting time between events). In particular, you will use the observed mean time gap (in your data) as an estimate for the mean, μ, in the exponential model. Since c = 1/μ in the model (see textbook!), you will then have an estimated formula for f(x).

Note: In collecting the data, we will use time intervals of length ¼ second. The detection of any amount of radiation (1 or more clicks) in one of these time intervals will be considered to be one event (1 click). Since there will only be a few counts larger than 1 , this will not affect the model very much.

Materials

• Geiger counter

• LabPro interface

Since “nature” is providing the background radiation, we don’t need a separate radiation source (such as the barium solution used in Lab 3).

Procedure

Data Collection using LoggerPro

The Geiger counter may be resting anywhere on the tabletop. Make sure that the Geiger counter is connected to the Dig/Sonic Channel 1 of the LabPro. Turn the Geiger counter on. Open the experiment file Lab5_TimeGaps.

• To start the experiment, click Collect. The Geiger counter will collect data every quarter of a second for five minutes. If radioactivity is detected, the “count” will indicate the number of particles detected (clicks) during that quarter second of time. Otherwise, the count will be recorded as zero. There will be 1200 pieces of data, consisting of a time (in seconds) and the number of counts detected in the time interval. In past tests, background radiation in the data analysis lab, Room 203, at a rate of about 45 counts/minute), so the counts in most quarter second intervals will be 0.

A piece of the LoggerPro table might look like

Time(s) Count(s)

... ...

0.750 0

1.000 1

1.250 0

This would mean that 0 counts (clicks) occurred in the

time interval (0.500, 0.750], 1 count occurred in the

interval [0,750, 1.000], and no counts occurred in the

interval (1.000,1.250]. The time listed refers to the right

endpoint of a quarter-second interval of time.

• When the experiment stops, save the data. E-mail it to yourself or save your data on a diskette.

Unfortunately, LoggerPro cannot perform all the data analysis that we need. Therefore we will move the collected data into a spreadsheet (Microsoft Excel) and continue our analysis there. Most students are familiar with the basic ideas of a spreadsheet, and Microsoft Excel is one of the most widely used.

Data Analysis using Microsoft Excel

Transfer data from LoggerPro to Excel

• Start Microsoft Excel. (Use the Start icon (lower left hand corner of the screen), and choose Programs, Microsoft Excel) You should have a blank worksheet (if not, open a new one: from the File menu, select New, Blank Worksheet).

• Click on cell A1 if it is not highlighted already. Enter “Time (s)” in cell A1; then move to cell B1 and enter “Count”.

• Return to LoggerPro, by clicking on the LoggerPro label at the bottom of the screen (or by using the key combo Alt-Tab). Copy the contents of the table window (Use the All button in the Table Window to highlight all your data; copy the data using the Copy selection from the Edit menu (or use the keystroke combo Ctrl-C).

• To transfer the data, return to Excel (click on the Microsoft Excel using the tab at the bottom of the screen, or Alt-Tab) and move to cell A2. Paste in the data (Use Paste from the Edit menu, or type Ctrl-V). You should see two columns of data. In A, the times in quarter-second intervals from 0 to 300 seconds; in B, the number of events (particles detected) in each interval.

• Relabel Worksheet 1. Select Sheet->Rename from the Format menu. Type “Original Data” and press ‘return’.

Analyze data with Excel

Caution: LoggerPro has been intentionally coded so that data cannot be fabricated or deleted. Changing or omitting real data would violate scientific procedure and nullify any “results” that were obtained by such methods—in addition to being a serious academic integrity violation.

Microsoft Excel is by nature much more flexible—if you are not careful, you can delete all or part of your data, possibly leading to incorrect results. It is important that you save the original LoggerPro file so that you can retrieve it if necessary.

We want to make a sorted list of the recorded “time gaps” between events.

1) We begin by separating out the time intervals in which there was an event (click) from those where there wasn’t.

Sort columns A and B so that Column B is in descending order. That will move all the intervals where Column B = 0 (“no event recorded”) to the bottom of the lists. (Put the cursor in cell A2 and from the Data Menu, use Sort to sort by Column B (Count) in descending order. Then click OK.)

2) We now move the list of times (Column A) in which an event (one or more clicks) was recorded to a separate worksheet. When we get it there, we’ll sort it into the correct time order again (ascending order). The current worksheet will retain all the original data; you’ll have that if something goes wrong later.

• Highlight and copy all of the data in Column A corresponding to nonzero counts in Column B. (For example, if the last nonzero entry in Column B is B75, select the data in Column A, Rows 2-75.)

• Click on the Sheet2 tab at the bottom of the document. In the new Worksheet 2, select cell A2 and use Ctrl-V (or Edit, Paste) to paste in the copied data.

Use menu Format , Sheet, Rename to Sheet 2 as “Data Analysis”.

• Select the data in Column A and then use Data, Sort to sort the times into ascending order. Column A then has the times, in increasing order, in which an event occurred; times when no event occurred have been discarded.

3) Now we want to find the time gaps (= waiting time between recorded events). For example, the time gap between the first and second observed event is A3-A2 (measured in quarter-seconds).

• Type the header “Time gaps” in cell B1.

• Compute the time intervals between detected events. To begin, click on

cell B2 and enter the formula “ =A3-A2 ”. (Don’t type the “ ” marks).

We want to continue in this way, putting “=A4-A3” in B3, “=A5-A4”

in B4, etc. Excel lets us do this efficiently, without needing to type the formula in each cell:

Select cell B2 and copy the formula “A3-A2” from there.

Then click on the lower right-hand corner of B2 and drag the cursor down column B to the cell beside the next-to-last row in column A. (For example, if the last number in column A is in row 103, drag the cursor in column B down to B102.) You should see a dark grey box surrounding the cells you selected. Type Ctrl-V (or Edit, Paste) to paste the formula for the consecutive cell differences of Column A to the selected cells in B.

(Excel assumes that you want the formula “updated” in each new row, so it puts “=A4-A3” in B3, “A5-A4” in B4, etc. There is an option (Edit, Paste Special) to force Excel to copy something verbatim, without updating, in each cell—but that’s not what we want at the moment.)

4) Finally, we want to sort the time gaps to easily see how often a time gap of each size was observed to occur. Informally speaking, this information will help us estimate how likely it is to find a long or short time gaps. In other words, since X = “time between events,” this information will help us estimate probabilities associated with X.

The time gaps you see in column B come from formulas you entered in column B. If we merely re-sorted a column of formulas, all sorts of unexpected things or recalculations might happen. To be safe, we first we copy the values (not the formulas) in column B over to column C:

• Put a header “Sorted Gaps” for Column C in cell C1.

• Highlight your data in column B (from B2 down to the end)

• Copy this information (use Edit, Copy, or use Ctrl-C). The cells you just copied should be highlighted now by a dotted line.

Move the cursor to cell C2 and use the menus Edit, Paste Special, Values. In

Column C you then have the values from column B rather than the formulas

from column B.

• Now, select column C, and use Data, Sort, to sort the time gaps in Column C into ascending order. You may get a “sort warning”—Excel is worried that you’re sorting one column independently of the others. (This would be an appreciated warning if someone were about to sort a list of telephone numbers in Column C without simultaneously sorting the associated names in Column B!). If this happens, just choose “continue with current selection” and move ahead.

Column C now starts with the smallest time gaps recorded between events

and ends with the largest gaps recorded. If column C, for example, began with

0.25

0.25

0.25

0.50

0.50

0.75

this would mean means that 3 times happened an interval of 0.25 seconds

between one recorded event (click) and the next; that and interval of 0.50

seconds occurred 2 times, etc.

Question based on the data: do Lab Report 5, # 1, 2

Make charts and tables summarizing the time-gap information. This gives us tabular and visual impressions of how frequently various time gaps were observed.

A frequency table organizes data into “bins” and records the number of pieces of data put into each bin. A frequency chart presents the same information as a bar chart.

To make a frequency table/chart of our time gaps, we need to decide on the number of “bins” to use. For example, we could put our data into bins ¼-second wide: [0,0.25], (0.25, 0.50], (0.50, 0.75], ... and count the number of time gaps put in each bin. If we choose bins that are too wide or too narrow, important patterns can be hidden. There is no “right” way to choose. Here, bins of width 1 second seem to work pretty well.

To put the time-gaps into a frequency table/chart with bins of width 1:

Look at your data in column C. Find the largest value (at the bottom of C) and round it up, if necessary, to the nearest integer—let’s say the result = 12. (Instead of 12, use the correct number for your own data.) If bin-width is 1, we want to use the bins [0,1], (1,2], (2,3], ..., (11,12] for our frequency chart/table.

Put a header “Rt Endpts” in cell D1. In D2, D3, etc., enter the right endpoint of your bins. In the illustration in the preceding paragraph, we’d put 1,2,...,12 in cells D2, D3, ..., D13.

From the menus, select Tools, Data Analysis, Histogram and click OK.

Click in the Input Range window, then use the cursor to highlight list to time gaps in column C, from C2 down to the last one.

Click in the Bin Range window, then use the cursor to highlight your list of endpoints in column D

Check the option Output Range, and then click in the Output Range window, then use the cursor to highlight cell E1.

Check the option Chart Output. Then press OK.

The result should be a frequency table with its upper left corner in cell E1, and a frequency chart off to the side.

By clicking inside the frequency chart,

i) you can drag it into a different position if you like.

ii) you can “grab” one of the black dots on the sides of the

frame and drag in order to make the frequency chart bigger.

Add a columns showing the “relative frequency of time-gaps” and the cumulative frequencies” for each bin. (If necessary, drag the frequency chart out of the way so that columns G and H are not covered by the chart)

Put a header for Column G, “Rel Freq” in cell G1. Let “r” be the number of time gaps in your data (See Lab Report 5, #1). Using your value for r, enter “ =F2/r “ in cell G2. (If F2/r = 0.22, say, that would mean that about 22% of the observed time gaps were in the first bin [0,1].)

Copy the formula in G2 and paste it into the other cells in column G beside the frequencies in column F. (Check that the formulas in each cell have automatically been updated to compute the correct relative frequency in each cell.)

Put a header for Column H, “Cum Freq” in cell H1.

In H2, enter “= G2” ; in H3, enter “=H2 + G3”

Copy the formula in H3, and paste it into the unfilled cells in the table for column H. (In each new row, the relative frequency from that row should have been added onto the preceding cumulative frequency. Therefore the last cell in your column H should contain a “1”).

Lab Report 5, #3. Use the cursor to highlight the headers and data in the 4 columns“Bin,” “Frequency,” “Rel Freq,” “Cum Freq” and print the highlighted material. (Use the menu File, Print and when the print dialog box is on screen, check Selection under “Print What” )

Also print the frequency chart. (Use the menu File, Print and in the print dialog box, check Selected Chart under “Print What”)

Do Lab Report 5, #4 a,b,c

The relative frequency that a value of X = “time between events” falls in a bin

(a,b] is an estimate (based on this limited data) for P(a < X ≤ b).

In our exponential model for X = “waiting time between events,”

P(a < X ≤ b) = area under density function f(t) over the interval from a to b = ∫abf(t) dt.

The general shape of an appropriate density function is suggested roughly by the shape of the frequency chart—a decaying exponential—with perhaps some irregularities created by our limited amount of data and the way we chose the bin widths.

Estimate the probability density function (p.d.f.)

In your spreadsheet, go to the bottom of Column B (“Time-Gaps”). In an empty cell a row or two below the bottom of Column B, calculate the average of all your observed time gaps by entering a formula similar to: “ =average(B2:B103)” (In your formula, substitute the numbers of the top and bottom data cells in your column B.)

Record the result for Lab Report 5, #5

We can use this average as an estimate for the mean μ of our random variable X.

Do Lab Report 5, #6, #7, #8

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download