FATHOM TUTORIAL

FATHOM TUTORIAL FOR MATHS 181 PREPARED BY Mohammed A. Halim, Instructor Dept. of Mathematical Sciences And Rebecca L. Pierce, Assoc. Professor Dept. of Mathematical Sciences MAH & RLP Page 2 of 24 Contents Part IABCDEFPart IIABCDPart III- DATA, VARIABLES, AND FATHOM Entering Data 3 Changing Name of an Attribute 3 Moving and Resizing Case Tables and Graphs Renaming a Collection Box 4 Entering Formula 4 Data Sorting 4 3 DISPLAYING AND DESCRIBING DATA DISTRIBUTIONS Producing Graphs 5-9 Two-way Tables and Conditional Distributions 10 Graphical Displays of Association- Regression and Correlation Measures of Center and Spread of Distributions 12-13 COLLECTING DATA Random Sampling 14-16 RANDOMNESS IN DATA Simulation and Probability Calculation 17-18 Calculation of Normal Probability 18-19 Sampling Distributions of Sample Proportions 19-21 Sampling Distributions of Sample Means 21-23 INFERENCE FROM DATA Confidence Interval for Population Proportion Confidence Interval for Population Mean 24 11-12 Part IVABCDPart VAB- 23-24 MAH & RLP PART I- DATA, VARIABLES, and FATHOM Page 3 of 24 Download Fathom data files and In-Class Activities: 1. Create two desktop folders- Fathom Data Files and In-Class Activities. 2. Go to www.keycollege.com/ws and click on your book picture. This opens the resource page. 3. Click on Downloadable Files. Download Scrabble.ftm to BabyMatch24.ftm files and save those in your Fathom Data Files folder. 4. Go back to the resource page. Click on “Answers to In-Class Activities.” Download and save. 5. After working on a Fathom Data File, always check “no” when asked about saving changes. A. ENTERING DATA 1. Start FATHOM from the desktop or from „All Programs‟. 2. Click on Insert on the menu bar and choose Case Table. 3. Refer to Activity 2-1: Scrabble Names, page 21 of the textbook. Click on in the blank Case Table. Type Name and enter. This is the first attribute (variable) in the Case Table. Enter all the names in this column. A Collection Box appears near the Case Table. It will contain all the data. 4. Click on in the next column. Type Letters and enter the data in the column. Do the same in the third column with the attribute Points. The case table is seen below. collection 1 Nam e 1 = 2 3 4 5 6 Nightingale Tukey Fisher Blackw ell Neym an Gossett Norw ood Le tte r s 11 5 6 9 6 6 7 Points 16 12 12 20 11 7 11 Ratio 1.5 2.4 2 2.2 1.8 1.2 1.6 B. CHANGING THE NAME OF AN ATTRIBUTE (VARIABLE) 8 P earson 7 9 1.3 Double click the name you want to change and type the new name and enter. 9 Dem ing 6 10 1.7 MOVING AND RESIZING Case Tables and Graphs. Click on the Table to highlight the borders. Grab the top border and move it. To resize, place the cursor showing double arrow on the right or bottom border and move. Note: To work on a Case Table or on a Graph, click on it to highlight the frame (border). RENAMING THE COLLECTION BOX Double click on Collection 1 in the upper left corner of the Case Table. A text box appears. Type something meaningful, like StatNames in this case. The name should be one word. 10 Galton 6 7 1.2 7 C. D. MAH & RLP E. Page 4 of 24 ENTERING FORMULA Calculate the ratios, Points/Letters, in the Case Table on page 3. 1. Enter the attribute name Ratio in the Case Table and right click on it. 2. Select Edit Formula on the drop down menu. A Formula Editor appears. 3. Click on the plus sign of + Attributes and double click on Points in the list. Attribute Points appears on the formula area. 4. Click on ÷ sign on the formula calculator and double click on Letters in the Attributes list. Click OK and the calculated ratios will appear under the attribute Ratio in the Case Table. If you make a mistake, edit the formula by double clicking on it. 5. Edit the decimal place of data of an attribute: Select the attribute. Click on Display on the menu bar. Choose Number Format. Change the significant digits to 2, for instance. 6. Entering “if. then” formula. Suppose you want to classify the names in the above table as long or short if the number of letters is above or below 6. Enter a new attribute NameLength and right click on it for Formula Editor and enter the formula: if (Letters > 6) Long { " Short" " " StatNames Nam e 1 = 2 3 4 5 6 7 8 Nightingale Tukey Fisher Blackw ell Neyman Gossett Norw ood Pearson Le tte rs 11 5 6 9 6 6 7 7 Points 16 12 12 20 11 7 11 9 Ratio 1.5 2.4 2 2.2 1.8 1.2 1.6 1.3 Nam e Le ... Long Short Short Long Short Short Long Long F. 9 Deming 6 10 1.7 Short DATA SORTING 6 7 1.2 Click on 10 attribute you want to sort in the Case Table. Short the Galton Click on Data on the menu bar and choose Sort ascending or descending. MAH & RLP PART II- DISPLAYING AND DESCRIBING DATA DISTRIBUTIONS A. Page 5 of 24 PRODUCING GRAPHS 1. Click on Insert on the menu bar and choose Graph. An empty graph box appears. 2. Go to the Case Table, grab the attribute Letters & drop it on the horizontal axis of the graph box. The graph shows dot plot. Make a dot plot of the attribute Points. Click on a dot. It turns red and highlights the name of the Statistician. 3. Again, insert a graph. Drag & drop the attribute NameLength on the horizontal axis to make a Bar Chart. See the graphs below. Note: Any change in the Case Table will automatically change the graph. Collection 1 Dot Plot StatNames Bar Chart 6 Count 5 6 7 8 9 10 11 12 4 2 Letters Long Short NameLength 4. Open Geyser.ftm file from Fathom Data Files and insert a graph. Drag the count   InterEruptionTime attribute from the Case Table to the horizontal axis of the graph to produce a Dot Plot. Click on the Dot Plot at the upper right corner of the graph and select Histogram to create a histogram and then click on Histogram and select Box Plot to make a boxplot. You should get the graphs as shown on the next page. OldFaithf ul Day 1 = 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 Inte rEru... 78 74 68 76 80 84 50 93 Duration 4.4 3.9 4 4 3.5 4.1 2.3 4.7 MAH & RLP Page 6 of 24 OldFaithf ul Dot Plot 40 50 60 70 80 90 100 InterEruptionTime OldFaithf ul 50 40 Histogram Count 30 20 10 30 OldFaithf ul Box Plot 50 70 90 110 InterEruptionTime 40 50 60 70 80 90 100 Note: You can rescale the histogram by double clicking on the horizontal axis and editing the pop-up information or by moving the cursor on the bars until a double arrow appears and dragging it to change the bin width. InterEruptionTime 5. Splitting a numerical distribution by categories on the same graph Open Cars99.ftm file from Fathom Data Files and show whether the distribution of CityMPG differs among different types of cars by splitting the graph of CityMpg by types of car. 1. Insert a blank graph. Drag & drop CityMPG on the horizontal axis. 2. Click on Dot Plot at the upper right corner of the graph and select Histogram. 3. Drag & drop the attribute Type on the vertical axis and you have histograms for the types of cars on the same graph. 4. Now, click on Histogram at the upper right corner of the graph and select Box Plot to create box plots for Types of cars. See the graphs on the next page. MAH & RLP Page 7 of 24 Cars99 M ode l 1 = 2 3 4 5 6 7 8 Buick Century Buick Regal CityM PG 20 19 Hw yM pg 29 30 29 29 30 27 28 27 We ight 3350 3325 3465 3350 3040 3170 3185 3170 Type f amily f amily f amily f amily f amily f amily f amily f amily Cars99 Chevrolet Im... 20 Chevrolet L... Chevrolet M... Chrysler Cir... Daew oo La... Dodge Status 20 22 19 20 19 upscale 5 0 sports 5 0 Count Type small 5 0 luxury 5 0 large 5 0 f amily 5 Cars99 upscale sports Box P lot Type small luxury large f amily 16 18 20 22 24 26 28 30 32 CityMP G More examples: Open AgeChild.ftm file. Insert a blank graph. Drag & drop ParentAge on the horizontal axis and Parent on the vertical axis. Click on Dot Plot and choose Histogram. Again insert a blank graph. Drag ParentAge to the horizontal axis and Parent to the vertical axis. Click on Dot Plot and choose Box Plot. You should get the graphs as shown on the next page. MAH & RLP Page 8 of 24 AgeChild Par e ntAge 1 = 2 3 4 5 6 7 8 13 13 13 13 14 14 15 15 Par e nt f ather f ather f ather f ather f ather f ather f ather f ather Fathe r Age 13 13 13 13 14 14 15 15 M othe r Age 13 14 14 14 14 14 15 15 Fa 24 26 31 30 21 22 22 24 A 6. Splitting a Bar Graph and making a Ribbon Chart of two categorical attributes. Open Senate99.ftm file. Insert a graph. 1. Drag Party to the horizontal axis and you will see a bar graph for Party. 2. Drag & drop the Sex attribute in the middle of the Bar graph. Black borders will be seen around the graph for a moment and the bars will be split for male and female. 3. Click on the Bart Chart at the upper right corner of the split-Bar graph and choose Ribbon Chart. You should get the graphs as shown on the next page. father mother MAH & RLP Page 9 of 24 Senate99 ID 1 = 2 3 4 5 6 1 2 3 4 5 6 Nam e Abraham Akaka Allard Ashcrof t Baucus Bayh m m m m m m Se x Par ty Rep Dem Rep Rep Dem Dem State Michigan Haw aii Colorado Missouri Montana Indiana Senate99 60 Bar Chart Count 40 20 Dem P arty Senate99 60 40 20 Rep Bar Chart count   Count Dem P arty Se x f count   m Rep MAH & RLP B. Page 10 of 24 TWO-WAY TABLES AND CONDITIONAL DISTRIBUTIONS Open Senate99.ftm file. Insert a Summary Table. Grab the Party attribute and drop it in the empty space in front of the Right arrow (). Now, grab the Sex attribute and drop it in the empty space below the down arrow (). The result is a Two-way Table showing the distribution of Party and Sex attributes as shown below. Conditional distributions of Party by Sex: Right click on the Summary Table and select Add Formula. Click on the plus sign of + Special and double click on ColumnProportion in the Formula Editor. Click OK. The Table will show the column proportions as seen in the Summary Table below. You can get the row proportions by double clicking on RowProportion in the Formula Editor. Conditional distribution of Party by Sex (column proportions) Conditional distributions of Sex by Party (row proportions) MAH & RLP C. Page 11 of 24 GRAPHICAL DISPLAY OF ASSOCIATION 1. Regression Open MarriageAge24.ftm and insert a blank graph. Drag & drop Wife on the horizontal axis and Husband on the vertical axis. Click on Graph on menu bar and select Movable Line. You have a scatterplot with a movable line at a 450 angle. Grab the line at the top end, or at the bottom end, or at the middle area and move it to see the effect on its slope. M arriageA ge24 70 Husband 50 30 10 10 20 30 Now, click on Graph on the menu bar and select Least Squares Line. The graph Husband shows 2 the regression line with proportion of variability, r = 0.89 and a slope of + 1.01 showing a positive association between Wife‟s age and Husband‟s age. = 0.961Wif e MarriageA ge24 70 Open Cars99.ftm and insert a Graph. Drag and drop Weight on the horizontal axis and 50 HwyMpg on the vertical axis. You have a regression line with a slope of –0.00533 showing a negative association between Weight and HwyMpg of the cars. 30 . Scatter P lot Husband 10 10 20 30 40 50 60 70 80 Wif e Husband = 1.01Wif e + 1.4; r^2 = 0.89 MAH & RLP Page 12 of 24 2. Correlation For Cars99.ftm, insert a Summary Table. Drag and drop Weight on the right  (the arrow becomes dark) and then drop HwyMpg on the same right arrow. Now, right click on the Summary Table and select Add Formula. Click on the plus signs of + Functions, + Statistical, + Two Attributes on the Formula Editor and double click on Correlation. Type (Weight, HwyMpg) inside the parenthesis on the formula editor. Click Ok. You should get a summary table as shown below. Cars99 Summary Table Weight 3185.5046 -0.79947357 Hw yMpg 28.735849 -0.79947357 S1 = mean   S2 = correlation  Weight Hw yMpg  D. MEASURES OF CENTER AND SPREAD OF DISTRIBUTIONS 1. Mean and Median Open Cars99.ftm and insert a Summary Table. Drag and drop CityMPG on the right arrow. Fathom automatically calculates the mean for the attribute. Now, right click on the Summary Table and choose Add Formula. On the formula editor, click on the + sign of Functions, Statistical, One Attribute and double click on Median. Click OK. Fathom calculates the median. You don‟t have to type CityMPG within the parenthesis. Do the same for HwyMpg to calculate its mean and the median. Cars99 Model 1 = 2 3 4 5 6 7 8 PageNum CityMPG 20 19 20 20 22 19 20 19 Hw yMpg FuelCap... 29 30 29 29 30 27 28 27 17.5 17.5 17 16.6 15 16 15.8 16 Buick Century 74 Buick Regal 77 Chevrolet Im... 88 Chevrolet L... 89 Chevrolet M... 90 Chrysler Cir... 101 Daew oo La... 108 Dodge Status 118 Cars99 Summary Table CityMPG 20.962264 20.5 S1 = mean   S2 = median  CityMPG  MAH & RLP Page 13 of 24 2. Calculate: First quartile (Q1), Third quartile (Q3) and Interquqrtile Range (Q3 – Q1) Open AnnualTemp.ftm and insert a Summary Table. Drag the Raleigh attribute from the case table and drop it on the right arrow in the Summary Table. Fathom calculates the mean. Right click on the Summary Table and click on Add Formula. Type Q1 ( ) on the formula editor. Click OK. Fathom calculates the Q1. Right click on the Summary Table and click on Add Formula. Type Q3 ( ) on the formula editor. Click OK. Fathom calculates the Q3. Right click on the Summary Table and click on Add Formula. Type iqr ( ) on the formula editor. Click OK. Fathom calculates iqr. AnnualTemp Month 1 = 2 3 4 5 6 7 8 Jan Feb Mar Apr May Jun Jul Aug MonthNo 1 2 3 4 5 6 7 8 Raleigh 39 42 50 59 67 74 78 77 SanFran... 49 52 53 56 58 62 63 64 AnnualTemp Summary Table Raleigh 59.25 46.5 72.5 26 S1 = S2 = S3 = S4 = mean   Q1   Q3   iqr   3. Calculate standard deviation for the data for Raleigh temperature in the example above. Insert a Summary Table. Drag and drop Raleigh on the right arrow in the Summary Table. Right click on Summary Table and click on Add Formula. On the Formula Editor, click on the + sign of Functions, Statistical, One Attribute, and double click on stdDev. Click OK. OR simply Type stdDev ( ) on the formula editor and click OK. AnnualTemp Summary Table Raleigh 59.25 14.168627 S1 = mean   S2 = stdDev   MAH & RLP PART III- COLLECTING DATA Page 14 of 24 RANDOM SAMPLING 1. Open Senate99.ftm. Right click on Senate99 Collection Box and select Sample Cases. Fathom collects a sample of 10 Senators and names the collection as Sample of Senate99. 2. Double click on Sample of Senate99. Uncheck “With replacement” box & “Animation on” box. 3. Change the sample size from 10 to 25 and click on “Sample More Cases”. Sample of Senate99 Collection Box will now have a sample of 25 Senators. 4. Insert a Case Table. The Case Table appears with all the data for the 25 Senators. Sample of Senate99 17 = 18 19 20 21 92 10 11 65 22 ID Nam e Stevens Bond Boxer Lieberman Collins Shelby Akaka Hatch Collins 5. Insert a Summary Table. Drag and drop the Party attribute from the new Case Table on the right arrow and the Sex attribute on the bottom arrow to see the Party22 Sex and 87 23 2 distribution in the sample. 24 25 47 22 6. Insert a second Summary Table and Drag the Years attribute to the right arrow to calculate the mean number of years of service (13.08) for the Senators in your sample. MAH & RLP Page 15 of 24 7. Insert another Summary Table and drag the Party attribute from the new Case Table to the right arrow. The summary shows the party breakdown by Dem and Rep. What percentage of the 25 sampled Senators are Democrats? Right click on the Summary Table and select Add Formula. On the formula editor, click on the + sign of Special and double click on rowProportion. Click OK. Proportion of Democrat and Republican Senators in the sample: Proportion of male and female Senators in the sample: 8. Sampling results for ten samples of size 25 each: 1. Double click on Sample of Senate99 collection box. 2. Click on the “Sample More Cases” tab in the inspection box. 3. You will see that the proportion of Dem and Rep Senators, and the mean number of service of the sampled Senators are changing. Record the results as shown below. Note: your results may be different because of sampling variability. Sample PropDem MeanYrs 1 44% 13.08 2 40% 12.84 3 60% 14.6 4 24% 7.6 5 56% 10.88 6 52% 12.08 7 56% 13.56 8 40% 14.24 9 36% 9.68 10 48% 8.2 9. The data that you have collected are called measures of the sample. Fathom can collect it automatically by creating a Collection of Measures. i) Double click on Sample of Senate99. Click on the measures tab in the inspector. ii) Under the Measure column, click on and enter PropDem. In the Formula column, Double click in the blank cell. Select Add Formula. Type the formula: Proportion (Party=”Dem”) on the formula editor. Click OK. iii) Go back to the Measure column. Click on and enter MeanYrs. In the formula column, Double click in the blank cell. Select Add Formula. Type the formula: Mean (Years) on the formula editor and click OK. MAH & RLP Page 16 of 24 iv) Click once on Sample of Senate99. Click on Analyze on the menu bar and choose Collect Measures. A collection box “Measures from Sample of Senate99” appears. Click it once and insert a case table to view the contents of this collection. By default, it shows the results of five samples. v) Double click on Measures from Sample of Senate99 collection. On the Inspector, change the number from 5 to 25. Click on the Collect More Measures button. The Measures from Sample of Senate99 will now have data for the Democrats from 25 samples. Insert a Case Table to see the results. Measures from Sample of Senate99 PropDem 15 = 16 17 18 19 20 21 22 23 24 25 0.36 0.6 0.4 0.28 0.4 0.44 0.44 0.44 0.52 0.44 0.48 MeanYrs 11.32 9.12 12.08 12.52 13.44 10.2 10.72 14.2 14.16 10.4 6.68 vi) Make a dot plot and a histogram for PropDem. Dot Measures from Sample of Senate99 Plot 0.2 0.3 0.4 0.5 PropDem 0.6 Measures from Sample of Senate99 6 Histogram Count 4 2 0.2 0.3 0.4 0.5 PropDem 0.6 0.7 MAH & RLP PART IV- RANDOMNESS IN DATA A. SIMULATION AND PROBABILITY CALCUALTIONS Refer to Activity 14-4 in the book. Page 17 of 24 Use Fathom to simulate the gender breakdown for 1000 families with four children and then for 1000 families with ten children. 1. Insert an empty Collection and click on Data on the menu bar. Select New Cases and enter 1000 in the pop-up box. Click OK. 3. Double click on the collection box. Click on on the Inspector, type Girls4 and enter. 4. Click on in the next row, type Girls10 and enter. 5. Double click in the Formula section for Girls4. Type the formula: randomBinomial (4, 0.5) on the formula Editor. Click OK. 6. Now, double click in the Formula section for Girls10 and type the formula: randomBinomial (10, 0.5) on the Editor and click OK. 7. Click on Collection 1 box and insert a Case Table. It shows data for 1000 families on Girls4 and Girls10 as shown below. 8. Again, click on Collection 1 box. Insert a Summary Table. Holding down the Shift key, drag Girls4 from the Case Table to the down arrow in the Summary Table. You have a frequency distribution of Girls4. To see the histogram, click on the Case Table and insert a graph. Drag and drop Girls4 on the horizontal axis. Click on Dot Plot and select Histogram. 9. Insert another Summary Table and do the same for Girls10. Collection 1 Girls4 993 = 994 995 996 997 998 999 3 3 1 3 1 3 1 7 5 4 5 6 4 6 6 Girls10 1000 2 Shows the frequency distribution of Girls4 Collection 1 Summary Table 0 1 Girls4 2 3 4 S1 = count   65 226 377 248 84 Column Summary 1000 MAH & RLP Shows the frequency distribution of Girls10 Collection 1 Summary Table Page 18 of 24 1 2 3 4 Girls10 5 6 7 8 9 10 S1 = count   14 42 129 204 239 207 120 37 7 1 Column Summary 1000 Histogram for Girls4 Collection 1 400 350 300 Histogram Count 250 200 150 100 50 -1 0 1 2 3 Girls4 4 5 6 B. CALCULATION OF NORMAL PROBABILITY Refer to Activity 15-2: Birth Weights. Assume normal distribution of weights of babies born in the United States with mean,  =3250 grams, and standard deviation, = 550 grams Find the probability that a baby will have a birth weight less than 2500 grams. 1. Open Fathom. Insert a Summary Table and right click on it. Choose Add Formula. Click on the + signs of Functions, Distributions, normal, and double click on normalCumulative and within the parenthesis type the numbers 2500, 3250, 550. Click OK. The probability is 0.08634 no data Summary Table 0.086341021 S1 = normalCumulative        MAH & RLP Page 19 of 24 2. Find the probability that a baby has a birth weight more than 4536 grams. Open on Fathom. Insert a Summary Table and right click on it. Choose Add Formula. Click on the + signs on Functions, Distributions, normal, and double click on normalCumulative and within the parenthesis type the numbers 4536, 3250, 550. Click OK. P(Baby weight > 4536 ) = 1 - P(Baby weight < 4536) = 1- 0.9903 =0.0097 Gives area from -∞ to (4536 – 3250)/550 no data Summary Table 0.99031109 S1 = normalCumulative        C. SAMPLING DISTRIBUTION OF SAMPLE PROPORTIONS Refer to Activity 16-2: Colors of Reese’s Pieces Candies Take a sample of size 25 and find the distribution of colors. 1. Open SimReeses.ftm. Right click on ReesesPieces collection and choose Sample Cases. 2. Double click on “Sample of Reeses Pieces” collection. Uncheck “With Replacement” box and “Animation On” box. Change sample size to 25 and click “Sample More Cases”. 3. Insert a Case Table and a Summary Table. Drag the Color attribute to the right arrow on the Summary Table. 4. Right click on the Summary Table. Select Add Formula. Click on the + signs of Functions, and Statistical. Select One Attribute and double click on Proportion. Enter: count (Color/25) within the parenthesis on the formula editor. Click OK. The Summary Table shows the proportions of colors in the sample of 25. Note: Your result may be different due to sampling variability. Sample of ReesesPieces 21 = 22 23 24 25 Color Brow n Orange Brow n Brow n Orange Sample of ReesesPieces Color Summary Table Row Brow n Orange Yellow Summary 7 8 10 25 0.28 S1 = count   S2 = proportion  count  Color   0.32 0.4 1 MAH & RLP Page 20 of 24 5. Note that in the summary table above the proportion of orange color in the sample is 0.32. Now, assume that 45% of all Reese‟s Pieces candies (population) is orange, that is  = 0.45. Simulate 100 samples of size 25 each from this population to see the distribution of the sample proportions. 1. Click once on Sample of ReesesPieces box and insert a Collection. 2. Right click on Collection 1. Click New Cases and enter 100 in place of 1. Click OK. 3. Double click on the Collection 1 Box to bring up the Inspector. Click on and type Count and enter. Double click in the Formula section of Count and type the formula: randomBinomial (25, 0.45) on the formula editor. Click OK. 4. Click , type Phat and double click in the Formula section of Phat and type the formula: Count/25. Click OK. 5. Insert a Summary Table. Holding down the Shift Key, drag the Phat attribute from the Inspector to the down arrow in the Summary Table. The Summary Table will show the distribution of 100 Phats. 6. Now, insert a graph and drag the gray box from the Summary Table to the horizontal axis. Click on Dot plot and select Histogram. See the table and the graph below. Note: Your result may be different due to sampling variability. Distribution of 100 phats from samples of size 25 Collection 1 Sum a m 0.24 0.28 1 4 9 13 18 20 12 8 5 6 3 1 Histogram of the above data 0.32 0.36 0.4 0.44 Phat 0.48 0.52 0.56 0.6 0.64 0.68 Colum Sum ary 100 n m S1 = count   MAH & RLP Page 21 of 24 Calculate the mean and the standard deviation of these 100 phats as shown in the Summary Table below. 1. Insert a Summary Table. Drag the Phat attribute from the gray area in the Summary Table to the right arrow. Fathom calculates the mean of the 100 Phats as seen in the table below. 2. Now, right click this summary table and select Add Formula. Click on the + signs of Functions, Statistical, and One Attribute and double click on stdDev. Click ok. OR simply type stdDev ( ) and click OK. Fathom calculates the standard deviation of the 100 phats as seen in the table below. Calculate how many and what proportion of the 100 Phats are within  .10 of .45 1. Double click on Collection 1. Click on Measures tab. Click on and type Within10. Double click in the formula section and enter the formula: count (inRange (Phat, 0.35, 0.55 )) on the formula editor. Click OK. The Value column of the Inspector will show how many of the 100 Phats are within this range. 2. Now, click on and type Proportion. Double click in the formula section and enter the formula: proportion (inRange (Phat, 0.35, 0.55 )). Click ok. The Value column of the Inspector will show what proportion of the 100 Phats are within this range. D. SAMPLING DISTRIBUTION OF SAMPLE MEANS Refer to Activity 17-2: Coin Ages Take a sample of size 25 of the pennies and find the find the mean age, x . 1. Open 1000Pennies.ftm and Right Click on 1000Pennies Box. Choose Sample Cases. 2. Right click on “Sample of 1000Pennies” box. Choose Inspect Collection. 3. Uncheck “With Replacement” box and “Animation On” box. Change sample size to 25 and click “Sample More Cases”. 4. Click on “Sample of 1000Pennies” collection. Insert a Case Table. Insert a Summary Table and drag the Age attribute to the right arrow on the Summary Table. It will show the mean age of the 25 Pennies in the sample. See the tables on the next page. Note: Your result may be different due to sampling variability. MAH & RLP A Sample of 25 pennies Sample of 1000Pennies 17 = 18 19 20 21 22 23 24 25 85 Year 72 84 77 83 96 86 87 97 14 Age 27 15 22 16 3 13 12 2 Page 22 of 24 Sample mean x of 25 pennies Summary Table Age 13.6 Sample of 1000Pennies S1 = mean   Simulate 100 samples of size 25 each from the population of 1000 pennies and see the distribution of the sample means. 1. Double click on “Sample of 1000Pennies” box. Click on Measure tab on the Inspector, click on and type XBAR. Double click in the Formula section of Xbar. Type the Formula: mean (age) on the Formula Editor. Click ok. 2. Right click on “Sample of 1000Pennies” box. Click on Collect Measures. By default, Fathom will generate 5 samples of 25 each in a collection box called “Measures from Sample of 1000Pennies”. 3. Right click on “Measures from Sample of 1000Pennies” and click on Inspect Collection. Uncheck the “Animation On” box and change the number of samples from 5 to 100. Click on Collect More Measures. This will generate 100 samples of size 25 each. 4. Insert a Case Table to see the distribution of 100 XBars. Create a dotplot and a histogram for this data. See the table below and the histogram on the next page. Note: Your result may be different due to sampling variability. Data of XBar from 100 samples of size 25 each Measures f rom Sample of 1000Pennies xBar 92 = 93 94 95 96 97 98 99 100 12.64 9.64 11.6 12.28 12.68 14.12 11.2 10.64 13.6 MAH & RLP Histogram of the XBars Measures from Sample of 1000Pennies 20 Histogram Page 23 of 24 Count 16 12 8 4 6 8 10 12 14 xBar 16 18 20 PART VA. INFERENCE FROM DATA CONFIDENCE INTERVAL FOR POPULATION PROPORTION, θ. Refer to Activity 19-5: Reese’s Pieces Open SimReeses.ftm. Assume the population proportion of Orange candies, θ = 0.45. Estimate the 95% confidence interval for the proportion of orange color candies based on a sample of size 75. 1. Click on Analyze on the menu bar and choose Estimate Parameters. 2. Click on Empty Estimate. Choose Estimate Proportion. 3. Drag the color attribute from ReesesPieces and drop it on “Attribute (Categorical): 4. Click on Brown and choose Orange. 5. Record the sample proportion and the 95% confidence interval. 6. Simulate 200 such intervals 1. Keep the estimation Box selected. Click on Analyze on the menu bar and choose Collect Measures. Shows a new collection “Mesures from Estimates of ReesesPieces”. 2. Right click on it and choose Inspect Collection. Change number from 5 to 200 and uncheck the “Animation On” box. 3. Click on “Collect More Measures”. This will simulate confidence intervals for orange color candies with lowerConfidenceBound and upperConfidenceBound for 200 samples. 4. Insert a Case Table to see the intervals. Click on “CountlnCat..” attribute on the case table. Click on Data on the menu bar and Choose Sort Ascending. See next page. 6. Check how many intervals do not contain θ = 0.45 from the sorted table and check their sample proportion. Your result may be different due to sampling variability. Estimate of ReesesPieces Attribute (categorical): Color Interval estimate f or population proportion of Orange in Color In the sample 33 out of 75, or 0.44, are Orange . Based on the sample, the 95 % conf idence interval f or the population proportion of Orange in Color is f rom 0.3277 to 0.5523. If the sampling process w ere perf ormed repeatedly, the conf idence intervals generated w ould capture the population proportion 95 % of the time. Estimate Proportion MAH & RLP Page 24 of 24 Measures from Estimate of ReesesPieces countInCat... 195 = 196 197 198 199 200 43 43 43 44 45 46 low erC... sam ple... upperC... 0.461398 0.461398 0.461398 0.475221 0.489128 0.50312 0.573333 0.573333 0.573333 0.586667 0.6 0.613333 0.685268 0.685268 0.685268 0.698112 0.710872 0.723547 category Orange Orange Orange Orange Orange Orange B. CONFIDENCE INTERVAL FOR POPULATION MEAN, µ Refer to Activity 20-18: Coins Ages 1. Open 1000Pennies.ftm and right click on the Collection box. Choose Sample Cases. 2. Right click on “Sample of 1000Pennies” and choose Inspect Collection. 3. Uncheck “With Replacement” box and “Animation On” box and change cases from 10 to 25. Click on “Sample More Cases”. 4. Click on “Sample of 1000Pennies” box and Insert a case table. It will show ages of 25 coins. 5. Now, click on Analyze on the menu bar and choose Estimate Parameters. 6. Click on “Empty Estimate” on the estimation box and choose Estimate Mean. 7. Drag the Age attribute from the “Sample of 1000Pennies” case table and drop it on Attribute (Continuous): . This produces a 95% confidence interval for the population mean, µ. Record the sample mean and the confidence interval. Your result may be different due to sampling variability. Estimate of Sampl Attribute (continu Interval estimate f o Sample count: 25 Sample mean: 11.6 Standard deviation Standard error: 1.8 Based on the sam of Age is 11.68 plu If the sampling pro intervals generate