MATH 221 Statistics for Decision Making Week 6 iLab Name: Melissa Frawley Statistical Concepts: Data Simulation Confidence Intervals Normal Probabilities Short Answer Writing Assignment All answers should be complete sentences. We need to find the confidence interval for the SLEEP variable. To do this, we need to find the mean and then find the maximum error. Then we can use a calculator to find the interval, (x - E, x + E). First, find the mean. Under that column, in cell E37, type =AVERAGE(E2:E36). Under that in cell E38, type =STDEV(E2:E36). Now we can find the maximum error of the confidence interval. To find the maximum error, we use the \"confidence\" formula. In cell E39, type =CONFIDENCE.NORM(0.05,E38,35). The 0.05 is based on the confidence level of 95%, the E38 is the standard deviation, and 35 is the number in our sample. You then need to calculate the confidence interval by using a calculator to subtract the maximum error from the mean (x-E) and add it to the mean (x+E). 1. Give and interpret the 95% confidence interval for the hours of sleep a student gets. (5 points) 6.73 and 7.67 are the confidence intervals for the hours of sleep a student gets. Which means that 95% of the students get between 6.73 hours of sleep and 7.67 hours of sleep. Then, you can go down to cell E40 and type =CONFIDENCE.NORM(0.01,E38,35) to find the maximum error for a 99% confidence interval. Again, you would need to use a calculator to subtract this and add this to the mean to find the actual confidence interval. 2. Give and interpret the 99% confidence interval for the hours of sleep a student gets. (5 points) 6.59 and 7.81 are the confidence intervals. Which then would mean that 99% of students get between 6.59 hours of sleep and 7.81 hours of sleep 3. Compare the 95% and 99% confidence intervals for the hours of sleep a student gets. Explain the difference between these intervals and why this difference occurs. (5 points) In comparing the two, 99% of the students have a wider range of sleep hours, while the 95% of the students have a narrower range of sleep hours. To have more confidence that your interval contains the true population mean, it has to be widened. In the week 2 lab, you found the mean and the standard deviation for the HEIGHT variable for both males and females. Use those values for follow these directions to calculate the numbers again. (From week 2 lab: Calculate descriptive statistics for the variable Height by Gender. Click on Insert and then Pivot Table. Click in the top box and select all the data (including labels) from Height through Gender. Also click on \"new worksheet\" and then OK. On the right of the new sheet, click on Height and Gender, making sure that Gender is in the Rows box and Height is in the Values box. Click on the down arrow next to Height in the Values box and select Value Field Settings. In the pop up box, click Average then OK. Write these down. Then click on the down arrow next to Height in the Values box again and select Value Field Settings. In the pop up box, click on StdDev then OK. Write these values down.) You will also need the number of males and the number of females in the dataset. You can either use the same pivot table created above by selecting Count in the Value Field Settings, or you can actually count in the dataset. Then in Excel (somewhere on the data file or in a blank worksheet), calculate the maximum error for the females and the maximum error for the males. To find the maximum error for the females, type =CONFIDENCE.T(0.05,stdev,#), using the females' height standard deviation for \"stdev\" in the formula and the number of females in your sample for the \"#\". Then you can use a calculator to add and subtract this maximum error from the average female height for the 95% confidence interval. Do this again with 0.01 as the alpha in the beginning of the formula to find the 99% confidence interval. Find these same two intervals for the male data by using the same formula, but using the males' standard deviation for \"stdev\" and the number of males in your sample for the \"#\". 4. Give and interpret the 95% confidence intervals for males and females on the HEIGHT variable. Which is wider and why? (7 points) The males confidence intervals for their heights are 68.02 and 71.31. While the females confidence intervals for their heights are 65.46 and 68.66. The males range is wider because the standard deviation is larger than the females. 5. Give and interpret the 99% confidence intervals for males and females on the HEIGHT variable. Which is wider and why? (7 points) The males confidence intervals for their heights are 67.41 and 71.93. While the females confidence intervals for their heights are 64.85 and 69.26. The males range is again wider because the standard deviation is larger than those of the females. 6. Find the mean and standard deviation of the DRIVE variable by using =AVERAGE(A2:A36) and =STDEV(A2:A36). Assuming that this variable is normally distributed, what percentage of data would you predict would be less than 40 miles? This would be based on the calculated probability. Use the formula =NORM.DIST(40, mean, stdev,TRUE). Now determine the percentage of data points in the dataset that fall within this range. To find the actual percentage in the dataset, sort the DRIVE variable and count how many of the data points are less than 40 out of the total 35 data points. That is the actual percentage. How does this compare with your prediction? (10 points) Mean ____51.66__________ Standard deviation _______25.8_____________ Predicted percentage ______32.57%________________________ Actual percentage ________40%_____________________ Comparison _The actual percentage is larger by 7.43% The predicted percentage is smaller and would mean that 32.57% of the students would drive less than 40 miles when in actuality 40% drive less than 40 miles. __________________________________________________ ______________________________________________________________ 7. What percentage of data would you predict would be between 40 and 70 and what percentage would you predict would be more than 70 miles? Subtract the probabilities found through =NORM.DIST(70, mean, stdev, TRUE) and =NORM.DIST(40, mean, stdev, TRUE) for the \"between\" probability. To get the probability of over 70, use the same =NORM.DIST(70, mean, stdev, TRUE) and then subtract the result from 1 to get \"more than\". Now determine the percentage of data points in the dataset that fall within this range, using same strategy as above for counting data points in the data set. How do each of these compare with your prediction and why is there a difference? (11 points) Predicted percentage between 40 and 70 __43.57%____________________________ Actual percentage _________22.9%____________________________________ Predicted percentage more than 70 miles ________23.86%________________________ Actual percentage __________37.1%_________________________________ Comparison _About 5% of the between 40 and 70 case and 2% on the greater than 70 case. They are still pretty close. ___________________________________________________ _______________________________________________________________ Why? _We only had 35 data points, so as the amount of data increases, the actual percentage should be getting closer to the predicted percentage. _________________________________________________________ ________________________________________________________________ 1 MATH 221 Statistics for Decision Making Week 2 iLab Name: Melissa Frawley Statistical Concepts that you will learn after completing this iLab: Using Excel for Statistics Graphics Shapes of Distributions Descriptive Statistics Empirical Rule Week 2 iLab Instructions-BEGIN Data have already been formatted and entered into an Excel worksheet. Obtain the data file for this lab from your instructor. The names of each variable from the survey are in the first row of the Worksheet. This row has a background color of gray to identify it as the variable names. All other rows of the Worksheet represent a certain students' answers to the survey questions. Therefore, the rows are called observations and the columns are called variables. On page 6 of this lab, you will find a code sheet that identifies the correspondence between the variable names and the survey questions. Follow the directions below and then paste the graphs from Excel in the grey areas for question 1 through 3. Type your answers to questions 4 through 11 where noted in the grey areas. When asked for explanations, please give thorough, multi-sentence or paragraph length explanations. PLEASE NOTE that various versions of Excel may have slightly different formula commands. For example, some versions use =STDEV.S while other versions would use =STDEVS (without the dot before the last \"S\"). The completed iLab Word Document with your responses to the 11 questions will be the ONE and only document submitted to the dropbox. When saving and submitting the document, you are required to use the following format: Last Name_ First Name_Week2iLab. Week 2 iLab Instructions-END 2 Creating Graphs 1. Create a pie chart for the variable Car Color: Select the column with the Car variable, including the title of Car Color. Click on Insert, and then Recommended Charts. It should show a clustered column and click OK. Once the chart is shown, right click on the chart (main area) and select Change Chart Type. Select Pie and OK. Click on the pie slices, right click Add Data Labels, and select Add Data Callouts. Add an appropriate title. Copy and paste the chart here. (4 points) Count of Car colors white; 3% black; 20% silver; 20% red; 11% orange; 3% blue; 26% black blue dark blue green orange red silver white green;dark 14%blue; 3% 2. Create a histogram for the variable Height. You need to create a frequency distribution for the data by hand. Use 5 classes, find the class width, and then create the classes. Once you have the classes, count how many data points fall within each class. It may be helpful to sort the data based on the Height variable first. Create a new worksheet in Excel by clicking on the + along the bottom of the screen and type in the categories and the frequency for each category. Then 3 select the frequency table, click on Insert, then Recommended Charts and choose the column chart shown and click OK. Right click on one of the bars and select Format Data Series. In the pop up box, change the Gap Width to 0. Add an appropriate title and axis label. Copy and paste the graph here. (4 points) Height Histogram 13 8 Frequency 5 4 6 1 -6 3 6 4 -6 6 5 6 7 -6 9 7 0 -7 2 7 3 -7 5 Height 3. Type up a stem-and-leaf plot chart in the box below for the variable Money, with a space between the stems and the group of leaves in each line. Use the tens value as the stem and the ones value for the leaves. It may be helpful to sort the data based on the Money variable first. An example of a stem-and-leaf plot would look like this: 1 2 3 4 5 6 9 3 5 6 3 6 9 2 The stem-and-leaf plot shown above would be for data 4, 5, 6, 9, 3, 15, 16, 13, 16, 29, and 22. (4 points) stem-and-leaf plot for Money 4 0 1 2 3 4 5 13445556666777789 03556 01369 124 347 23 Calculating Descriptive Statistics 4. Calculate descriptive statistics for the variable Height by Gender. Click on Insert and then Pivot Table. Click in the top box and select all the data (including labels) from Height through Gender. Also click on \"new worksheet\" and then OK. On the right of the new sheet, click on Height and Gender, making sure that Gender is in the Rows box and Height is in the Values box. Click on the down arrow next to Height in the Values box and select Value Field Settings. In the pop up box, click Average then OK. Type in the averages below. Then click on the down arrow next to Height in the Values box again and select Value Field Settings. In the pop up box, click on StdDev then OK. Type the standard deviations below. (3 points) Females Males Mean 67.0588 69.6667 Standard deviation 3.1119 3.3077 Short Answer Writing Assignment All answers should be complete sentences. 5. What is the most common color of car for students who participated in this survey? Explain how you arrived at your answer. (5 points) Blue is most common color of cars in survey. As in pie chart blue is of maximum area of pie with 26% area. 5 6. What is seen in the histogram created for the heights of students in this class (include the shape)? Explain your answer. (5 points) Height looks like approximately symmetric in shape. First height increases to maximum and then decreases. 7. What is seen in the stem and leaf plot for the money variable (include the shape)? Explain your answer. (5 points) According to stem and leaf plot money variable is right skewed. We can clearly see that frequency decreases from maximum to minimum. 8. Compare the mean for the heights of males and the mean for the heights of females in these data. Compare the values and explain what can be concluded based on the numbers. (5 points) Mean of height of females are slightly less than mean of height of males. So we can conclude that males are normally taller than females. 9. Compare the standard deviation for the heights of males and the standard deviation for the heights of females in the class. Compare the values and explain what can be concluded based on the numbers. (5 points) Standard deviation of height of males are larger than height of females. So height of females are less variant than height of males. 6 10. Using the empirical rule, 95% of female heights should be between what two values? Either show work or explain how your answer was calculated. (5 points) 95% female heights will lie within (67.0588 - 2*3.1119, 67.0588 + 2*3.1119) = (60.835, 73.2826) 11. Using the empirical rule, 68% of male heights should be between what two values? Either show work or explain how your answer was calculated. (5 points) 68% male heights will lie within ( 69.6667 - 3.3077, 69.6667 + 3.3077) = (66.359 , 72.9744) 7 Code Sheet Do NOT answer these questions. The Code Sheet just lists the variables name and the question used by the researchers on the survey instrument that produced the data that are included in the data file. This is just information. The first question for the lab is after the code sheet. Variable Name Drive State Temp Rank Height Shoe Sleep Gender Race Car TV Money Coin Die1 Die2 Die3 Die4 Die5 Die6 Die7 Die8 Die9 Die10 Question Question 1 - How long does it take you to drive to the school on average (to the nearest minute)? Question 2 - What state/country were you born? Question 3 - What is the temperature outside right now? Question 4 - Rank all of the courses you are currently taking. The class you look most forward to taking will be ranked one, next two, and so on. What is the rank assigned to this class? Question 5 - What is your height to the nearest inch? Question 6 - What is your shoe size? Question 7 - How many hours did you sleep last night? Question 8 - What is your gender? Question 9 - What is your race? Question 10 - What color of car do you drive? Question 11 - How long (on average) do you spend a day watching TV? Question 12 - How much money do you have with you right now? Question 13 - Flip a coin 10 times. How many times did you get tails? Question 14 - Roll a six-sided die 10 times and record the results. Drive (miles) 20 28 54 76 36 88 6 25 40 42 55 71 76 78 4 20 25 29 33 36 36 36 63 73 73 76 80 33 54 63 76 80 80 94 6 State OR NY NV PA TX SC MI CA IL IL OH MI NY CA NY IL GA TX TX MI IL OR MI FL SC NY FL SC FL PA PA NV SC KY OR Shoe Size 7 7 11 11 7 6 10 8 10 8 9 11 9 11 9 5 9 11 9 10 12 9 5 12 11 9 9 11 9 10 11 12 10 11 13 Height (inches) 65 67 65 63 66 63 66 69 68 65 67 64 63 63 69 62 63 70 75 61 61 73 69 70 68 65 66 68 70 66 70 69 63 74 75 Sleep (hours) 4 4 4 4 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 10 Gender Car Color F green F black F silver M silver M orange F black M black F silver M silver F red F dark blue F silver M silver M blue F blue F black M silver M silver M silver M blue M red M blue F orange F green F red F red F black M blue F green F silver F blue M white F black M red M silver TV (hours) 1 1 5 5 3 3 5 4 5 4 4 2 4 2 5 3 3 2 5 6 4 5 3 4 2 6 5 1 2 3 3 4 2 3 6 Money (dollars) 46.00 25.00 54.00 13.00 7.00 43.00 34.00 30.00 16.00 4.00 27.00 1.00 38.00 47.00 21.00 10.00 53.00 28.00 54.00 5.00 14.00 23.00 9.00 15.00 18.00 7.00 22.00 9.00 5.00 5.00 5.00 3.00 7.00 20.00 7.00 Coin 4 4 7 5 3 6 3 4 5 4 2 2 2 2 5 2 5 3 7 2 3 5 2 3 5 2 2 4 3 7 4 4 7 4 4 Die1 Die2 5 6 6 4 2 6 4 3 2 3 2 4 3 3 4 2 2 3 4 6 2 2 6 1 4 1 6 5 5 3 4 2 4 3 2 Die3 5 2 2 5 4 2 4 5 2 4 6 3 5 4 2 4 4 3 6 6 5 5 2 6 2 6 4 1 4 1 4 3 4 5 1 Die4 1 6 3 2 4 4 5 3 4 1 5 5 2 4 1 5 4 2 2 4 3 1 4 4 3 6 4 5 3 2 5 2 1 3 2 Die5 5 5 6 3 1 2 1 6 2 3 5 4 1 5 5 4 2 6 6 5 6 4 4 6 1 5 2 6 3 1 2 5 6 2 6 Die6 2 5 1 3 6 3 4 1 1 2 5 3 2 2 6 5 5 6 4 4 4 5 6 2 6 1 2 2 3 6 4 6 3 6 4 1 4 5 5 3 5 4 3 3 5 3 1 3 5 5 5 2 6 3 4 3 1 2 3 4 5 4 5 3 5 4 4 2 4 5 Die7 Die8 5 3 3 3 1 3 5 1 2 5 5 2 4 5 2 6 3 6 1 6 2 1 2 2 3 2 1 2 6 4 1 6 4 5 2 Die9 1 1 6 6 6 1 5 4 4 5 2 3 5 4 1 3 4 5 4 4 4 1 5 2 3 5 1 1 3 2 3 3 6 3 2 Die10 6 3 5 2 4 4 5 1 4 6 3 6 5 6 6 4 3 5 3 6 3 2 3 4 3 1 2 3 4 1 3 2 5 4 3 3 1 1 1 4 2 1 5 2 6 3 1 3 5 2 1 3 1 5 6 3 1 1 6 2 6 1 2 2 5 1 6 1 4 5