Chapter 7 Assignment due Wednesday, May 24

Transcription

1 due Wednesday, May 24 Calculating Probabilities for Normal Distributions Overview What you re going to do in this assignment is use an online applet to calculate: probabilities associated with given -scores or data values, and -scores or data values associated with given probabilities. You will do these calculations for both: the standard normal distribution, and a non-standard normal distribution. How to Use the Applet Here is the website for the applet (I will include the link on the Assignments page also): I have currently tested the applet with Internet Explorer 11. If you have troubles getting it to work, then try a different browser, such as Moilla Firefox or Google Chrome, both of which are free downloads, and I just tested them both. Or, give me a call if you have a problem running it, and I will try to help. The help may involve you trying it on another computer. Actually I have never had any complaints from students in the past about this particular applet not working.

2 So, once you get to the website, wait for a few (or several) seconds while the applet loads. Once it loads, the page should hopefully look like this. Use this button to calculate a probability from a -score or data value. Input the mean, standard deviation, and the -score(s) or data value(s), hit correct button. The output is the area, and the graph will show where the area is (according to your input). The top button (Area from a value) is what you will use to calculate the probabilities when you are given a data value or a -score. The value shown in the yellow box at the bottom of the first distribution: Area (probability) is the answer for this portion; remember that probability = area. In other words, the area above a -score of 1.96 =

3 If you hit the second button (Value from an area), the input and output regions will change like this: Use this button to calculate a - score or data value from a given area. Input the mean, standard deviation, and the area. Choose the correct button, and the -scores(s) or data value(s), and the graph will show where the area is (according to your input). The second button (Value from an area) is what you will use if you are given a probability (area), and want to calculate the associated data value or -score. If you click that button, the yellow box will change and give you four options to enter the area. Once you make that selection the answer will appear immediately to the right of the option (Above, Below, ). The answer(s) will be the -scores or data values. So again: use top button on applet to go from -scores/values to probabilities use bottom button on applet to go from probabilities/areas to - scores/values.

4 Looking at either part of the applet, you can see that the default is for the standard normal distribution, since the mean = 0 and the standard deviation = 1. You will use this for problems 1-3, and for problem 4 you will input a different mean and standard deviation. Make sure that you hit the Recalculate button every time you change an input to the applet! Other than this, I m NOT going to give you a detailed step-by-step procedure on how to use the applet, because I think it s pretty self-explanatory, and you should be able to just play around with it and figure it out. I will include some specific pointers in the problems below. One suggestion would be to try reproducing the problems that we did in class, to verify that you are using the applet correctly and getting the same answers. Since you are also going to do the problems by hand (using Table V), I would suggest the following: Draw the figures by hand first, and calculate your probabilities or data values THEN verify by using the applet if the figure does not at all match what you drew, then you re either using the applet incorrectly, or you drew it wrong to begin with. However, I will give the following CAUTION for using the applet: ONLY input the values that are given in the problem statements. Do NOT calculate something by hand, or look it up in the table, and then use that value as input to the applet. The Table V result and the Applet Result should be completely independent of each other!

5 Problems: Note that you are going to do all of these problems using two methods: 1. by hand, using Table V look-up method (make SURE you know how to do this, it s how you ll have to do it on the test!) 2. with the applet Therefore, you have the perfect opportunity to check your own answers, since the results from both methods should be approximately the same. If there is a large discrepancy, then go back and check because you have done one (or both) incorrectly. Note: DO NOT round off your answers from the Applet. 1. Determine the total area under the standard normal curve: a. that lies to the left of = 1.38 b. that lies to the right of = 1.22 c. that lies between = 0.89 and = 0.57 d. to the left of = 0.96 or to the right of = 2.22 (note: you can use the Outside button for this one) 2. Find the -score such that the area under the standard normal curve to its left is In other words, find the 17 th percentile, P17, which is the - score separating the bottom 17% of the -scores from the top 83%. So in other words, you want to use the bottom button on the applet, to find the value from an area. 3. Find the -scores (two of them) that separate the middle 47% of the distribution from the area in the tails of the standard normal distribution. Again, you want to use the bottom button on the applet, to find the value from an area. You can use either the Between or the Outside button, but be careful that you put in the correct area that corresponds to that option. Check that your results match your hand results from Table V.

6 4. For this problem, you are going to use the Body Data data set which is preloaded into STATDISK, 13 th edition. HOWEVER! the data file in STATDISK is a mixture of male and female data. I sorted it out according to gender, and saved the data into two different Excel files. Pick the file that corresponds to YOU! Therefore, ladies will be using one file and guys will be using a different file. Ladies: Body_data_women_pulse (147 data values) Guys: Body_data_men_pulse (153 data values) The variable that you will be looking at is the pulse rate, in beats per minute. Therefore, take your own resting pulse rate for one minute and record that on the Answer sheet. The Excel files contain the pulse rate data for a sample of 153 men and 147 women in the respective files, and the original source is from the National Center for Health Statistics. Copy the column of pulse rate data to STATDISK, and make a histogram to verify that the data is approximately normal. Make sure to add a title and x-axis label with measurement units, and print out the histogram. Use STATDISK to find the Descriptive Statistics for the data set. Note: make sure that you have copied all 153 (men) or 147 (women) data values over. Because the data is approximately normal, we can use the mean and the standard deviation of the data to model the variable pulse rate as a normal distribution. Do NOT round off the mean or the standard deviation, use all of the digits from the STATDISK calculation for both your hand calculations and as input to the applet. Note: answer each of the questions below with respect to the data set YOU are using either the Men s or the Women s. You don t need to do both! a. Using the Empirical Rule, calculate the minimum and maximum usual values for the pulse rates, in bpm. Along the x-axis of your histogram, mark and label the approximate locations of the mean, the minimum and maximum usual values, and also your own pulse rate.

7 b. Men: Based on the result in part a, would a man with a pulse rate of 95 bpm be considered unusual? Explain why or why not. Women: Based on the result in part a, would a woman with a pulse rate of 95 bpm be considered unusual? Explain why or why not. c. Using the normal model, what proportion of men or women have a pulse rate greater than 85 bpm? d. What proportion of men s or women s pulse rates are between 60 and 90 bpm? e. Find the fourth decile of the pulse rates, in other words the pulse rate (in bpm) that separates the bottom 40% of pulse rates from the top 60%. f. Using the normal model, determine the interval of pulse rates that make up the middle 80% of the pulse rates. g. Based on your pulse rate measurement, what percentile are you in? (round to the nearest percent) h. Is your pulse rate unusual? Explain why or why not. What You Need to Turn In: Print out the following table, fill it out, and turn it in (5 pages total). Histogram from STATDISK, including a title and x-axis label, and with the values marked and labeled as noted in problem 4a. For the Table V column, SHOW ALL OF YOUR HAND CALCULATIONS (for many problems, these will be minimal)! For the Applet result, do not round off the answers! For each Figure on the Answer Sheet, just make a rough sketch showing the appropriate shaded region for the problem. Include any applicable measurement units with your answers (this only applies to question 4).

8 Prob. # 1a. Figure (just a rough sketch) Table V Result (and any work if applicable) Applet Result (do not round off) 1b. 1c.

9 Prob. # 1d. Figure (just a rough sketch) Table V Result (and any work if applicable) Applet Result (do not round off) 2. 3.

10 4. Does the STATDISK histogram indicate that a normal distribution could be used as a model for the variable? (yes or no) List the mean (do not round off): List the standard deviation (do not round off): (bpm) (bpm) List your pulse rate: (bpm) 4a. Minimum usual value = (bpm) Maximum usual value = (bpm) 4b. Men: Based on the result in part a, would a man with a pulse rate of 95 bpm be considered unusual? Explain why or why not. Women: Based on the result in part a, would a woman with a pulse rate of 95 bpm be considered unusual? Explain why or why not. Prob. # 4c. Figure (just a rough sketch) Table V Result (and any work if applicable) Applet Result (do not round off)

11 Prob. # 4d. Figure (just a rough sketch) Table V Result (and any work if applicable) Applet Result (do not round off) 4e. 4f.

12 Prob. # 4g. Figure (just a rough sketch) Table V Result (and any work if applicable) Applet Result (do not round off) 4h. Unusual or not? Explain.