1) Problem #PRAJBX2 "PRAJBX2 - 239932 - &n..." |
A)
The above figure is a density curve that represents the distribution of random variable X. What is the height of the density curve? Round answer to the nearest hundredth. © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
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Solve for Height |
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B)
What is the probability that X is between 7.8 and 8.6? Round answer to the nearest hundredth. |
Algebraic Expression:
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Hints: |
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LENGTH = 8.6 - 7.8= 0.8 HEIGHT = 0.5 |
Remember, area equals probability, thus the probability equals 0.4. |
C)
What is the probability that X=7.4? Round answer to the nearest hundredth. |
Algebraic Expression:
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D)
What is the probability that X is between 8.5 and 8.9 ? Round answer to the nearest hundredth. |
Algebraic Expression:
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Hints: |
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LENGTH = 8.9-8.5=0.4 HEIGHT = 0.5 |
Remember, area equals probability, thus the probability equals 0.20. |
E)
What is the probability that X is between 8.2 and 9.2? Round answer to the nearest hundredth |
Algebraic Expression:
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Hints: |
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LENGTH = 8.2-9.0=0.8 HEIGHT = 0.5 |
Remember, area equals probability, thus the probability equals 0.40. |
2) Problem #PRAJBX3 "PRAJBX3 - 239933 - X~N(300,25) Y~N(4..." |
A)
X~N(300,25) Y~N(400,34) What is the mean of Z? © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
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E(Y) = 400 E(Z) = 300+400 |
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B)
What is the standard deviation of Z? Round answer to the nearest hundreth. |
Algebraic Expression:
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C)
X~N (300, 25) Y~N (400, 34) Z=X + Y What is P(Z<629)? Round answer to the nearest hundreth. Link to z-table. |
Algebraic Expression:
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D)
What is P(Z > 786)? Round answer to the nearest hundreth. |
Algebraic Expression:
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Therefore, P(Z>786) = 0.02 (ROUNDED) |
E)
What is the probability Z is less than 638 or greater than 752? Round answer to the nearest hundreth. |
Algebraic Expression:
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Hints: |
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P(Z<638 or Z>752) = P(Z<638) + P(Z>752) - P(Z<638 & Z>752) |
P(Z<638 or Z>752) = 0.07 + 0.11 - 0
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3) Problem #PRAJBX4 "PRAJBX4 - 239934 - Red Sox investors..." |
Red Sox investors profit $76 when the team wins and lose $19 when the team loses. The probability the Red Sox win is 0.58. What is the expected value of the net profit? © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
P (lose) = 0.42 |
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4) Problem #PRAJBX5 "PRAJBX5 - 71-7-A" |
A)
A research project was done in the state of Rhode Island to find out the amount of dogs per household.
Find the mean amount of dogs per household. Round to the nearest hundreth. © STATS4STEM.ORG |
Algebraic Expression:
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B)
What is the standard deviation of the amount of dogs, according to the chart? Round to the nearest hundreth. |
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C)
What is the probability of a household having more than two dogs? Round answer to the nearest hundreth. |
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D)
What is the probability a house hold has at most three dogs? Round to the nearest hundreth. |
Algebraic Expression:
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5) Problem #PRAJBX6 "PRAJBX6 - 71-7-A" |
Variance is the ________________ squared. © STATS4STEM.ORG |
Multiple Choice:
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6) Problem #PRAJBX7 "PRAJBX7 - 71-7-A" |
Discrete variables have countable outcomes. © STATS4STEM.ORG |
Multiple Choice:
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7) Problem #PRAJHMT "PRAJHMT - 245381 - The area of a cir..." |
The area of a circle is ∏*r2. The semicircle below is a density curve. Calculate the value of r. Round answer to the nearest hundredth. © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
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Solve for r. |
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Answer = 0.80 (ROUNDED) |
8) Problem #PRAJBV9 "PRAJBV9 - 239877 - A survey wa..." |
A)
A survey was done to see the average amount of money a wizard spends at Hogsmeade per day. Results were:
A.) Calculate the mean amount of money spent. © STATS4STEM.ORG |
Algebraic Expression:
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B)
B.) Calculate the standard deviation of the money spent. |
Algebraic Expression:
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Hints: |
To find the standard deviation take the square root of the calculated variance |
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9) Problem #PRAJBX8 "PRAJBX8 - 71-7-A" |
What is the probability that x will be at most 2? round to the nearest hundreth. © STATS4STEM.ORG |
Algebraic Expression:
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10) Problem #PRAJBWZ "PRAJBWZ - 239900 - Below is a table ..." |
A)
Below is a table of test scores of Mr. Simoneau's AP Statistics students.
Determine the mean. © STATS4STEM.ORG |
Algebraic Expression:
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B)
Determine the standard deviation. |
Algebraic Expression:
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C)
Since everyone failed the test, Mr. Simoneau, the good guy that he is, decides to scale the test score. The actual grades will becaome: grade= 3*Score-10. Determine the new mean. |
Algebraic Expression:
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D)
Determine the new standard deviation. |
Algebraic Expression:
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11) Problem #PRAJBXV "PRAJBXV - 239927 - The following is ..." |
A)
The following is the distribution of MCAS test scores at Boston High School:
a) What is the expected MCAS score at Boston High? Round answer to the nearest hundredth. © STATS4STEM.ORG |
Algebraic Expression:
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B)
b) What is the variance of the scores? Round answer to the nearest hundredth. |
Algebraic Expression:
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Hints: |
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(1-2.55)2 (0.25) + (2-2.55)2(0.10) + (3-2.55)2(0.5) + (4-2.55)2(0.15) |
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C)
c) What is the standard deviation of the scores? Round answer to the nearest hundredth. |
Multiple Choice:
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Hints: |
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D)
d) The teachers at Boston High, in fear of having their school questioned due to low test scores, decide to scale by using this formula: New Score=2*Score-2.5 What is the mean of the new score? Round answer to the nearest hundredth. |
Algebraic Expression:
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Recall, E(X) = 2.55 μx = 2*2.55 - 2.5 |
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E)
e) What is the standard deviation of the adjusted scores? Round answer to the nearest hundredth. |
Algebraic Expression:
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12) Problem #PRAJBXT "PRAJBXT - 239925 - Rosy Cheeks Corpo..." |
A)
Rosy Cheeks Corporation for makeup is planning a major investment. The amount of profit X is uncertain but a probabilistic estimate gives the following distribution (in millions of dollars): Profit: 3 5.5 7 11 Probability: 0.1 0.4 0.3 0.2
Find the mean of the profit. Round answer to the nearest hundredth.
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Algebraic Expression:
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B)
Calculate the standard deviation of the profit. |
Algebraic Expression:
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Answer = 2.38 (ROUNDED) |
C)
B. Rosy Cheeks owes its source of capital a fee of $500,000, plus 10% of the profits X. Find the value of retained profit Y after covering the expenses of the capital fee and percentage. The answer is in millions of dollars. Round to the nearest hundredth. |
Algebraic Expression:
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13) Problem #PRAJBXM "PRAJBXM - 239919 - The Boston Latin ..." |
The Boston Latin School Science Department gives a survey to all of its students. The distribution of answers:
Calculate the mean of the distribution. © STATS4STEM.ORG |
Algebraic Expression:
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14) Problem #PRAJBWN "PRAJBWN - 238090 - #5" |
A)
The probability you will complete a certain level in video game X is shown below:
a) Is this a discrete or continuous distribution? © STATS4STEM.ORG |
Multiple Choice:
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B)
b) P (X ≤ 3) |
Number:
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Hints: |
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C)
c) P(X ≥ 5) |
Number:
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15) Problem #PRAERFK "PRAERFK - Rules for Means and Variances" |
Copy down the following rules for means and variances in your notebook. Assume X and Y are random variables, and that a and b are constants. 1) E(a + b*X) = a + b*E(X) 2) VAR(a + b*X) = b2VAR(X) 3) E(X+Y) = E(X) + E(Y) If X and Y are independent: VAR(X + Y) = VAR(X) + VAR(Y) OR VAR(X-Y) = VAR(X) + VAR(Y) VAR(constant number) = 0 For example, VAR(5) = 0 © STATS4STEM.ORG |
Multiple Choice:
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16) Problem #PRAERMN "PRAERMN - Rules for means & variance - HP example" |
A)
Team Harry Potter enters a relay race. There are two runners, Harry and Ron. Harry's run times for his given course are normally distributed with a mean time of 20 minutes and a standard deviation of 1 minute. Ron's run times for his given course are normally distributed with a mean time of 30 minutes and a standard deviation of 2 minute. Assuming the two run times are independent, answer the following questions. The mean time for Harry and Ron to complete the relay race is: © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
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E(R) = 30 |
B)
The variance for Harry and Ron to complete the relay race is: |
Algebraic Expression:
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Hints: |
VAR(H+R) = VAR(H) + VAR(R) |
VAR(R) = 4 |
C)
What is the standard deviation for the time it takes Harry and Ron to complete the relay race? Round answer to the nearest hundredth. |
Algebraic Expression:
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D)
Find the probability that it takes Harry and Ron longer than 51 minutes to finish the relay race. Round answer to the nearest hundredth. Link to z-table |
Algebraic Expression:
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Hints: |
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P(X<51) = 0.6736 Therefore, P(X>51) = 1 - 0.6736 = 0.33 (ROUNDED) |
E)
Find the probability that it takes Harry and Ron between 48 and 52 minutes to finish the relay race. Round answer to the nearest hundredth. Link to z-table |
Algebraic Expression:
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Hints: |
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z48 = (48-50)/2.24 = -0.89 |
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17) Problem #PRAERE9 "PRAERE9 - Variance of two independent R.V.'s" |
A)
If X and Y are independent random variables with variances, VAR(X)=1 and VAR(Y)=2, find the variance of the random variable Z=6X-3Y. © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
VAR(X+Y) = VAR(X) + VAR(Y) |
VAR(Z) = VAR(6X-3Y) = VAR(6X) + VAR(-3Y) |
Therefore, VAR(6X) + VAR(-3Y) = 36*VAR(X) + 9*VAR(Y) |
VAR(Z) = 36 + 18 = 54 |
B)
If X and Y are independent random variables with variances, VAR(X)=1 and VAR(Y)=2, find the variance of the random variable Z=4X+2Y+4 |
Algebraic Expression:
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Hints: |
2nd - Remember, VAR(aX + bY) = a2Var(X) + b2VAR(Y) |
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VAR(4X+2Y+4) = 16*1 + 4*2 = 16 + 8 = 24 Answer = 24 |
18) Problem #PRAENY4 "PRAENY4 - Discrete R.V." |
A)
A random sample of graduating seniors was surveyed just before graduation. One question that was asked is: How many times did you change majors? The results are displayed in a probability distribution. Using this probability distribution we can answer probability questions such as: What is the probability that a randomly selected senior has changed majors more than once? This can be written as P(X > 1). Round answer to the nearest hundredth. © STATS4STEM.ORG |
Algebraic Expression:
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B)
As you just saw/noticed in this example, we need to pay attention to the wording of the probability question. The key words that told us which values to use for X are more than. The following will clarify and reinforce the key words and their meaning. Key Words Let's begin with some everyday situations using at least and at most. Suppose someone said to you, "I need at least 10 pages for a term paper." What does this mean? It means that 10 pages is the smallest amount you are going to write. In other words, you will use 10 or more pages in the term paper. This would also be the same as saying, not less than 10 pages. So, for example, writing 9 pages would be unacceptable. On the other hand, suppose you are considering the number of children you will have. You want at most 3 children. This means that 3 children is the most that you wish to have. In other words, you will have 3 or fewer children. This would also be the same as saying, not more than 3 children. So, for example, you would not want to have 4 children. What is the probability that a random student changed their major at least twice. |
Algebraic Expression:
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C)
What is the probability that a random student changed their major at most twice. |
Algebraic Expression:
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19) Problem #PRAENZZ "PRAENZZ - Expected Value & Variance of RV - ILL ELEM ST" |
A)
A public speaker computes the probabilities for the number of speeches she gives each week. The distribution is shown below:
Compute the expected value. Round answer to the nearest tenths. © STATS4STEM.ORG |
Algebraic Expression:
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B)
Calculate the variance. Round your answer to the nearest tenths. |
Algebraic Expression:
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C)
Calculate the standard deviation. Round your answer to the nearest tenths. |
Algebraic Expression:
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D)
If she receives $100 per speech, about how much will she earn per week? Round answer to the nearest whole number. |
Algebraic Expression:
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20) Problem #PRAJBV8 "PRAJBV8 - 239876 - Honeydukes Sweets..." |
A)
Honeydukes Sweetshop begins to sell their Chocolate Frogs for half price. Ambrosius Flume, the owner, calculates her weekly profits per box of Chocolate Frogs will have the following distribution:
A.)Find the mean profit per box of Chocolate Frogs. © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
Formula for calculating mean is E(X) = Σ(xi)(Pi) |
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B)
B.) Calculate the standard deviation of the profit per box. Round answer to the nearest hundredth. |
Algebraic Expression:
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To find standard deviation take the square root of the variance |
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C)
C.) Find the mean of a new random variable Y by using the equation: Y=3X+1 |
Algebraic Expression:
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Where: E(X) = 3.2 |
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D)
D.) Calculate the standard deviation of variable Y using the same equation: Y=3x+1 Round answer to the nearest hundredth. |
Algebraic Expression:
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21) Problem #PRAJBXD "PRAJBXD - 239912 - Z=X+Y X~N(500,50)..." |
A)
Z=X+Y X~N(500,50) Y~N(280,30) Find the mean: © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
Answer=780
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B)
Find the standard deviation:(round to the nearest 10th) |
Algebraic Expression:
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Hints: |
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Answer=58.3
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22) Problem #PRAJBXE "PRAJBXE - 239913 - Below are the mea..." |
A)
Below are the mean and standard deviations nationally on both sections of the SATs. SAT VERBAL: μx = 625 σx = 80 SAT MATH: μy = 600 σy = 70 Find the mean overall score on the SATs. © STATS4STEM.ORG |
Algebraic Expression:
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B)
Find the standard deviation of the overall SAT scores. |
Algebraic Expression:
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Hints: |
Y~N(625,70) |
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23) Problem #PRAJBXF "PRAJBXF - 239914 - Sullivan's at Cas..." |
Sullivan's at Castle Island loses $20 on days when it rains and gains $180 on sunny days. If the probability of rain is 0.3, what is the expected value of Sullivan's net profit? © STATS4STEM.ORG |
Algebraic Expression:
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24) Problem #PRAJBXG "PRAJBXG - 239915 - x  ..." |
A)
Does this distribution fit the requirements to be a discrete probablility distribution? © STATS4STEM.ORG |
Algebraic Expression:
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B)
Find the probability that x is less than 3. |
Algebraic Expression:
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25) Problem #PRAJBXK "PRAJBXK - 239918 - A cook stacks thr..." |
A)
A cook stacks three pots on top of each other for storage. The height of the first pot is normally distributed with a mean of 5 inches and a standard deviation of .3 inches. The height of the second pot is also normally distributed, and has a mean of 6 inches and a standard deviation of .5 inches. The height of the final pot is normally distributed as well with a mean of 5 inches and a standard deviation of .4 inches. Find the expected value of the height of the stack of pots. © STATS4STEM.ORG |
Number:
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Hints: |
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E(X+Y+Z)=5+6+5 = 16
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B)
Find the standard deviation of the height of the stack of pots. |
Number:
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Hints: |
Note on notation: VAR(X)=σ2 |
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C)
The cook decides to get a new set of pots. The height of the new pots stacked compare to the old stacked pots height in the following way: New = 2 * Old - 3 Find the new mean of the height of the pots. Remember, with the original pots, STACKED HEIGHT~N(16, 0.71). |
Number:
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Hints: |
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E(a+bX)= (-3) + (16)(2)
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D)
Which distribution would best represent the stack of pots? |
Multiple Choice:
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26) Problem #PRAKBZJ "PRAKBZJ - 269770 - Using the followi..." |
A)
Using the following R output, calculate A. Round answer to the nearest hundredth. © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
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B)
Calculate B. Round answer to the nearest hundredth. |
Algebraic Expression:
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Therefore, σ = √13 = 3.606 or 3.61 (ROUNDED) |
27) Problem #PRAJBWH "PRAJBWH - 237897 - #1" |
The distribution above can be labeled as: © STATS4STEM.ORG |
Multiple Choice:
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28) Problem #PRAJBWQ "PRAJBWQ - 238717 - #7" |
Test scores on a given test have a mean of 70 with a standard deviation of 2. If the teacher decides to scale everyone’s test by 5 points, what is the resulting mean and standard deviation? © STATS4STEM.ORG |
Check All That Apply:
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29) Problem #PRAKPMQ "PRAKPMQ - 279974 - Ricki goes to the..." |
A)
Ricki goes to the toy store with 3 dollars. He wants to buy a bag of marbles. He randomly chooses 15 beautiful marbles. The marbles cost 1 penny per gram. If the weight of each marble is normally distributed with a mean of 19.5 grams and a standard deviation of 1.5 grams, what is the mean weight for the weight of bags containing 15 marbles? Assume that the weight of the bags holding the marbles is negligible. Round answer to the nearest hundredth. © STATS4STEM.ORG |
Number:
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Hints: |
Answer = 292.5 |
B)
What is the standard deviation for the weight of bags containing 15 marbles? Assume that the weight of the bag holding the marbles to be negligable. Round answer to the nearest hundredth. |
Number:
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Hints: |
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Answer = 22.5 |
C)
Ricki brings his bag of 15 marbles to the checkout counter to be weighed. What is the probability that he doesn't have enough money to buy the marbles? Round answer to the nearest hundredth. Link to z-table. |
Number:
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P(Bag Weight > 300 g) = 1 - 0.6293 = 0.37 (ROUNDED) |
30) Problem #PRAP6GU "PRAP6GU - Problem #383642" |
A)
Traveling Travis travels to work. His travel time to work is normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. He learns that a new bridge is being built that will save him time. The time saved is normally distributed with a mean of 6 minutes and a standard deviation of 1 minute. After the bridge is built, Traveling Travis' travel time will have what mean? © STATS4STEM.ORG |
Algebraic Expression:
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Hints: |
Answer = 19 |
B)
After the bridge is built, Traveling Travis' travel time will have what standard deviation? Round answer to the nearest hundredth. |
Algebraic Expression:
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Hints: |
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Therefore, the standard deviation is √5 = 2.236 or 2.24 (ROUNDED) ANSWER = 2.24 |