FINAL NOTEBOOK

1. FINAL NOTEBOOK Amy Hartoon

2. Preface This semester I will be using Yogurt Mountain. Yogurt mountain has 45 total flavors, but for simplicity we will be using the 16 flavors, since each store has 16 flavors that rotate. Yogurt mountain also has 50+ toppings.

3. Final Notebook – Section 3 • Let’s say there are 6 stores in the surrounding Memphis area, and we decided to count the amount of customers that came into each store in a 2 hour span, and the resulting numbers are: 58, 59, 71,71, 76, and 80. Mean: (58+59+71+71+76+80)/6 = 69.167 So, the average amount of people that visited the stores within a 2 hour span is about 69 people. Median: 71 The mid range number of people is 71 people, which is a little bit over the average amount Mode:71 the number that most occurred in the data is 71 people, which is also a little bit over the average. Quartiles: Q1: (25/100)*6=1.5 rounded up to 2, 2nd position number is 59 Q2: (50/100)*6=3, 3rd position number is 71 Q3: (75/100)*6=4.5 rounded up to 5, 5th position number is 76 Interquartile range is equal to Q3-Q1, this eliminates the extreme highs and extreme lows from the data, and focuses more on the middle 50% So, 5-2 = 3, and the number in the 3rd position is 71. It seems that 4 of the stores rest in the middle and upper quartiles for customer traffic, which means they are bringing in the median amount or more. The other 2 stores rest in the 25th percentile, which means they are bringing in less than the mean, so they probably want to figure out something to bring more customers in.

4. Section 3 Cont’d • Standard deviation: the average distance from the points in my set to the mean. • Mean=69 • Sum of (xi - µ)²=399 • s²=399/6-1=79.8 • SD= Square root of 79.8=8.93 • CV=(SD/mean)*100=8.93/69=12.9% • This is a large CV since it is over the 10% we agreed was excessive.

5. Notebook - Section 4 • Yogurt Mountain has 8 machines, each containing 2 flavors. So, there are 1 possible slots for different flavors. What are the possible combinations of 4 of them being chocolate? • C 416 = 16! = 16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 = 16*15*14*13 = 4*5*7*13 = 1820 4! (16-4)! 4*3*2*1*12*11*10*9*8*7*6*5*4*3*2*1 4*3*2*1 1*1*1*1 • There are 1820 possible different combinations of 4 of the slots being chocolate. This is good since there are 5 different flavors of chocolate to choose from: Chocolate, Triple Chocolate, Smores, No Sugar Added Chocolate, and Frozen Hot Chocolate. • The implications that could come with 4 of the flavors being chocolate involve sales. Chocolate is usually one of the better selling flavors, so having 4 different kinds could increase sales dramatically. On the other hand, if chocolate sales are decreasing, it could decrease sales dramatically.

6. Notebook Section 5(binomial) There are 20 people in the line at Yogurt Mountain. The probability of a customer getting a drink is .24, and you are looking for 9 successes. This is important to the business because it shows the probability of a drink being bought in addition to yogurt. If there is a low percent chance, maybe the company needs to rethink providing drinks, because they may be losing money supplying the drinks if there is not enough demand. N! * px *(1-p)(n-x) x!(n-x)! • 20! * .249 * (1-.24)11 • 9!(11)!) 20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 (9*8*7*6*5*4*3*2*1*11*10*9*8*7*6*5*4*3*2*1 4*19*3*17*4*5*14*13*6 *.249 * (.76)11 9*8*7*1*1*1*1*1*1 84651840 * 0.000002642 * 0.048859556 504 167960 * .000002642 * 0.048859556 =0.021681444 = 2.2% chance there will be 9 successes.

7. Section 5 Binomial • Expected Value= E(x) = µ=np • (20)(.24)=4.8 • Variance=np(1-p) • Variance=4.8(.76)=3.648 • Standard deviation = the square root of 3.648 = 1.909 • CV- Standard deviation= 1.9099 = .397 mean 4.8 This is a large standard deviation, because it is a lot larger than 10%. • If the probability was larger, the company would probably need to consider offering more drink choices. If it were smaller, obviously they would need to think about what drinks on which to cut back. If the Standard Deviation were larger, it would be harder to concisely see how many drinks were sold in addition to yogurt, and if it were smaller, that means there would be less of a gap between drink sales.

8. Notebook section 5 Poisson Customers arrive at the average rate of 30 per hour on any given weekday. What is the probability of 10 arrivals in 10 minutes on any given weekday? This is important for the business because they will be able to see the busier times of the day, and make sure employees are more friendly during this time/promote buying drinks, really anything that will earn more money. • 30/hr=5/10 minutes • F(10)=510 (2.71828)-5 10! F(10)=9765625(0.00673797) 3628800 F(10)=65800.48858 3628800 F(10)=0.018132 There is a 1.81% chance there will be 10 arrivals within 10 minutes If the number were bigger, this indicates that more customers will come in in the goal time slot, and if it were smaller maybe they need to think about doing something else to promote the store and get its name out in the community.

9. Notebooks – Section 6 Normal • Identify and complete a Normal application • Buying yogurt is based completely on the weight of everything you get. Yogurt mountain offers a 16 oz cup and a 32 oz cup, but the usual weight of one person’s cup is 10 oz. What is the probability of buying more than 15 oz? The Standard deviation is 2 • It’s helpful to the business because they can see the likelihood of getting a larger amount of money from one cup of yogurt. The more yogurt they sell, the more money they make. • (15-10)/2 = 5/2 = 2.50 • The z value for 2.50 is .4938 • .50-.4938=.0062 • There is a .62% chance of buying more than 15 oz of yogurt. • If the number was even smaller than .62, they might want to consider making a smaller cup to save on cost of the cup. If no one ever fills the cup up, there’s no use for that big of a cup. Conversely, if the number was bigger, they might want to consider making the smaller of the tw cups a 20 oz cup, because the people that fill the 16 oz cups up might fill the 20 oz cups up, resulting in a higher profit for the company.

10. Notebook – Sections 7 • Section 7: Create and complete a sample means problem • There are 44 yogurt mountain stores in the US. Were looking at one store’s pre-pack yogurt sales over 30 days. From previous records, the average amounts of pre-packs sold are 20 a day. In 30 days of watching the store, the average amount of pre-packs bought was 18, with a standard deviation of 7. • This is helpful information for the business because they can see if it’s profitable to keep the pre-packs, or to just let people exclusively buy soft serve yogurt. • Z = x-µ z=18-20 z= -2 z= -2(5.47) z= -1.56 o77 7 Square root of N square root of 30 5.47 • Z=.4406 • These results lie outside of the established 40% parameters, which shows that they might want to consider having less available in pre-pack form, or to consider getting rid of it altogether. • If the results had been in the 40% parameters, there would be no question of getting rid of the pre-packs. The only consideration is whether or not to have more available. If the results had been farther away from the 40% parameters, like 60%, the decision of getting rid of the pre-packs becomes more urgent, because they are losing a lot more money.

11. Notebook - Section 8 • Historically, Yogurt Mountain sells 26 sodas in a day. We’ll be watching one store over 25 days, and the standard deviation is 13. We will also be using the 40% standard. • This is helpful for the business because they will be able to see the very outer limits of soda sales that are acceptable. • T1= -z(σ) +µ T1= (-1.28)(13) +26 Square root of n 5 T1= 16.64 +26 T1= 22.672 5 T2= (1.28)(13) +26 T2 =29.328 5 These results show that the very outer limits are around 4 sodas within the mean. If the results were farther apart, there would be a wider margin before anyone got worried over soda sales. Now that we know our T values, lets suppose we went to one store and the average was 27 sodas a day. Z= 27-26 z= 1 X 5 z= .3846 13 13 5 With z being within the 40% range, soda sales at this location are just as good as all of the locations. The company doesn’t need to worry about soda sales at this specific location at this time. If the results would have been outside of the 40% range, then the store would need to be examined to see if this period of observation was a fluke, or if soda sales are normally like this, and if they are, they need to think of a way to push soda sales or just get rid of them altogether. If the sales had been even more inside of the 40% range, they might have had to consider raising the price of the soda, or maybe even having more soda available at one time.

12. Notebook – Section 9 • There are 16 flavors available to a customer at any given time. You are allowing yourself to have 4 flavors of yogurt, so what’s the probability you’ll get 2 pink colored yogurts? • This is good for the business to know so they can see how some of their yogurt flavors hold up to other flavors. Yogurt mountain has 8 pink flavored yogurts: Cherry Amaretto, Cherry Limeade, No sugar added strawberry, pomegranate energy tart, Raspberry tart, strawberry, strawberry cheesecake, and watermelon Sorbet. If we can see how some flavors get favored over others, we could consider getting rid of some of the more unpopular flavors, and even consider replacing them with other flavors. • What’s in the Bucket: what’s in your ‘hand’ N=16 n=4 r= 8 x=2 N-r=8 n-x=2 8! * 8! = 8*7 * 8*7 = 56 * 56 784 * 24 = .4307 2!(6)! 2!(6!)2*1 2*122 43680 16!16*15*14*1343680 4!(12)! 4*3*2*1 24 There’s a 43.1% chance of getting 2 pink colored yogurts when 4 flavors of yogurt are chosen. That’s almost a 50% chance. The company might want to consider getting more pink colored yogurts. If the result would have been lower, say, 20%, then the company might have to consider actually getting rid of some of the pink colored yogurts offered. If the percentage were higher, say 80%, then they might want to start thinking of new pink colored yogurt flavors and start tesing and implementing them, now.

13. Notebook – Section 10 • In contemplating which concept is the most useful, I’ve just come to realize that they’re all useful in their different ways. Some help figure out if other products should be sold in addition to yogurt, some help figure out what sort of flavors would yield the most profit, some help figure out customer traffic in certain times of the day. I think all of these concepts need to be used to help the business run more efficiently. • What was the most difficult concept to apply? Why? • The most difficult concept to apply was the very beginning mean, median mode, etc. It’s one thing to talk about customer traffic in different stores, but to put in in writing and trying to figure out quartiles and the standard deviation was hard work. • What was the least difficult concept to apply? Why? • The easiest concept to grasp all semester for me was the Standard Normal problem. It might be because the math behind the problem is so simple, but in figuring out the likelihood of a customer getting a cup over the standard 10 oz weight was really easy for me to think through.