1) The document discusses milk production data from CowMilk Boss's farm and uses it to answer several questions. 2) For question A, the data is determined to approximate a normal distribution based on calculating the mean and standard deviation. 3) For question B, it is determined that 119 cows will be used based on killing cows producing less than 50% of the normal milk and obtaining milk from cows producing over 80% of the normal amount. 4) For question C, it is calculated that if CowMilk uses the $7,470 monthly from milk sales to pay the loan payments, he can pay it off in around 4 years and save $13,749.60 compared to the full payment

1. The Cows By TN

2. CowMilk Boss of Milk-Cow Farm All of the calculations base on TI-84 Plus

3. After came back from the statistic class of Mr.K He started gather the data about the milk of his cows in a month, and he made this table

4. a/ Is the data approximates the normal distribution? Using the probabilities of the normal distribution. b/ Mr.K makes a small party in his class. CowMilk decided that he will kill all of the cows which give him less than 50% of the normal milk to make the steaks and get milk from the cows which give him greater than 80% of the normal. What is the number of cows he will use? C/ After the party, Cow Milk decided that he will buy a car which total cost 300 000$.He’s going to rent the money from the bank which give him 7.5% annually interest in 5 years, compound monthly. CowMilk is also going to, How much he can save in the total if CowMilk uses all the money from selling milk of the cows he has to pay the interest otherwise he pay exactly the number of payment monthly the bank will give him?

5. Oki… lets start ….. Question a. “ Is the data approximates the normal distribution? Using the probabilities of the normal distribution .” 1 σ 2 σ 3 σ Remember Standard Normal Distribution curve! To determine . We need to imagine that the number of cow should lay on x-axis and the number of milk should lay on the y-axis Number of Cow Number of Milk So….

6. We need to calculate the mean( μ ) and standard deviation ( σ ) base on the data of Milk and Cows ! To calculate mean and standard deviation, we can use the calculator: List 2 List 1 We put the number of cow to List 2 and the number of Milk to List 1 Use 1-Var stats to calculate σ and μ : [stat] => CALC => 1-Var Stats => [ENTER] And set up the screen like this …

7. 1-Var Stats L1 L2 The number of Milk Frequency ( the number of Cow) And…. ENTER We will see on the main screen of the calculator: σ μ For convenient, we can save the result of μ and σ by use : [VARS] => Statistics… => =>[STO] => [ALPHA] => [M]( the divide key) => [ENTER] ,

8. And now, we have mean and standard deviation, Lets calculate 1 σ , 2 σ , and 3 σ 1-Standard deviation : Max: + =20.7 Min: - =17.12 So the Cows which give the milk between 17.12L and 20.7L .. uhmm lets count …There are 153 Cows 2-Standard deviation : Max: + 2 =20.7 Min: - 2 =17.12 3-Standard deviation : Max: + 3 =20.7 Min: - 3 =17.12 There are 193 Cows There are 209 Cows

9. After that, we change the number of Cows to %, because we want to compare with “68-95-99 rule” 1-Standard deviation : 2-Standard deviation : 3-Standard deviation : 153 /209=73.2% We know total of the Cows : 209 Cows 193 /209=92.34% 209 /209=100% Base on “68-95-99 rule” the data approximates the normal distribution To be..continue =>

10. …… Time to Party ^_^! And now…. B /Mr.K makes a small party in his class. CowMilk decided that he will kill all of the cows which give him less than 50% of the normal milk to make the steaks and get milk from the cows which give him greater than 80% of the normal. What is the number of cows he will use?

11. When solve any kind of this question, we usually need a GRAPH Because we know that the data approximates the normal distribution (question a/) . We can make a graph “… he will kill all of the cows which give him less than 50% of the normal milk… “ “… get milk from the cows which give him greater than 80% of the normal…” The question give us the %, we need to change it to the z-score to put on the graph !

12. To convert from % to Z-score, we use the calculator to convert : [2 ND ] => [VARS] => invNorm( => [ENTER] And then put 0.80 (80%) => [ENTER] The calculator will give us z-score of 0.80(80%) , this is 0.84162 0.84162 So on for 50% REMEMBER put invNorm(0.5) in the calculator We will have z-score of 0.50(50%) = 0 0

13. 0.84162 0 “… get milk from the cows which give him greater than 80% of the normal…” “… he will kill all of the cows which give him less than 50% of the normal milk… “ Look at the question So, look at the graph, we will see the cows he will use The Cows he will use

14. Use this formula, we can calculate the volume of milk (X) at 2 points on the graph mean Z-score Volume of milk We all know Z , , mean => we calculate X X 50 =18.913 X 80 =20.41 And the pull out the table and…count => The number of the cows he use : 8+11+21+45+18+8+8 =119 Cows

16. Can I borrow 300 000$ to buy that Car ? Uhmmm…we offer the loan with 7.5% annually interest for 5 years, compound monthly Ok !! Supercar

17. if I use all the money from selling milk of the cows I have, to pay the interest otherwise pay exactly the number of payment monthly the bank will offer. Assume The amount of milk every month doesn’t change and 1L Milk =3$ HOW MUCH CAN I SAVE ?

18. At first, he need to calculate the total amount of milk he have in 1 month : 48*19 + 42*20 + 18*21 + 8*22 + 8*23 = 2490 L (The rest were killed to make steak). => Total $/month = 2490L * 3$/L = $7470

19. After that, he take his calculate and use TVM solve to calculate the monthly payment which can be offered by the bank: TVM solve : [APPS] => Finance…=> TVM solve => payment/month =$ 6011.38 Total payment : $6011.38 * 60 = $360682.8

20. If he use all of the money he have from selling the milk to pay for monthly payment , he must find how long it will take … … .so that, he also use TVM solve with PMT = $7410 Total amount : 46.82 * $7410 = $346936.2

21. Consider 2 results, we can know the difference : $360682.8 - 346936.2 =$ 13749.6 SO…if he use all the money from selling the milk, he can make 46.8 ~ 47 payments ( 47/12 ~4 years) and save $13749.6 !!! He should do it.. I mean.. I should do it ^^