Approximately 85.87 mg of Bismuth-210 would remain after 8 days.
Explanation:To calculate the amount of Bismuth-210 remaining after 8 days, we need to apply the concept of radioactive decay. Bismuth-210 decays by about 13% each day, meaning that 13% of the remaining Bismuth-210 transforms into another atom (Polonium-210) each day.
Let's calculate the amount remaining:
Start with 233 mg of Bismuth-210.After the first day, 13% of the remaining Bismuth-210 will decay, leaving 87% of the original amount: 0.87 * 233 mg = 202.71 mg.Repeat this process for each subsequent day.After 8 days, the amount remaining would be: (0.87)^8 * 233 mg = 85.87 mg.Therefore, after 8 days, approximately 85.87 mg of Bismuth-210 would remain.
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After 8 days, approximately 76.69 mg of Bismuth-210 remains out of the initial 233 mg.
To calculate the amount of Bismuth-210 remaining after 8 days of radioactive decay, follow these steps:
Step 1:
Understand the decay rate.
Each day, 13% of the remaining Bismuth-210 decays into Polonium-210. This means that 87% of the Bismuth-210 remains each day.
Step 2:
Calculate the remaining amount each day.
Start with the initial amount of 233 mg of Bismuth-210.
After the first day: [tex]\( 233 \text{ mg} \times 0.87 = 202.71 \text{ mg} \)[/tex]
After the second day: [tex]\( 202.71 \text{ mg} \times 0.87 = 176.43 \text{ mg} \)[/tex]
Continue this process for 8 days.
Step 3:
Perform the calculations for 8 days.
[tex]\[ \text{Day 1: } 233 \text{ mg} \times 0.87 = 202.71 \text{ mg} \][/tex]
[tex]\[ \text{Day 2: } 202.71 \text{ mg} \times 0.87 = 176.43 \text{ mg} \][/tex]
[tex]\[ \text{Day 3: } 176.43 \text{ mg} \times 0.87 = 153.62 \text{ mg} \][/tex]
[tex]\[ \text{Day 4: } 153.62 \text{ mg} \times 0.87 = 133.67 \text{ mg} \][/tex]
[tex]\[ \text{Day 5: } 133.67 \text{ mg} \times 0.87 = 116.33 \text{ mg} \][/tex]
[tex]\[ \text{Day 6: } 116.33 \text{ mg} \times 0.87 = 101.28 \text{ mg} \][/tex]
[tex]\[ \text{Day 7: } 101.28 \text{ mg} \times 0.87 = 88.10 \text{ mg} \][/tex]
[tex]\[ \text{Day 8: } 88.10 \text{ mg} \times 0.87 = 76.69 \text{ mg} \][/tex]
Step 4:
Interpret the result.
After 8 days, approximately 76.69 mg of Bismuth-210 remains.
So, after 8 days, approximately 76.69 mg of Bismuth-210 remains.
Find the following standard normal probability: p(-1.23 szs +1.23) OA..7814 B. .8907 O C..1093 D. .2186
In the formula used to convert "real-world" data values into z numbers, the standard deviation of the data is never considered. O True O False
Answer: a) 0.7814
b) False
Step-by-step explanation:
To find : The p-value of z , where
[tex]-1.23<z< +1.23[/tex]
[tex]P(-1.23<z< +1.23)=1-P(z<-1.23)\\\\=1-0.1093=0.7814[/tex]
Hence, the standard normal probability [tex]P(-1.23<z< +1.23)=0.7814[/tex]
In the formula used to convert "real-world" data values into z numbers, the standard deviation of the data is considered.
The formula to calculate the z score is [tex]z=\dfrac{x-\mu}{\sigma}[/tex].
If sample size (n) is given then , the [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Hence, the statement is false.
A soft drink machine outputs a mean of 28 ounces per cup. The machine's output is normally distributed with a standard deviation of 2 ounces. What is the probability of filling a cup between 30 and 31 ounces? Round your answer to four decimal places.
To find the probability of the soft drink machine filling a cup between 30 and 31 ounces, calculate the z scores for 30 and 31 ounces, use them to find the cumulative probabilities from a standard normal distribution table, then subtract the two probabilities. The result is 0.0919 or 9.19%.
Explanation:This is a question about probability in a normal distribution. In this case, we want to find the probability of the output being between 30 and 31 ounces, given a mean of 28 ounces and a standard deviation of 2 ounces.
First, we find the z-scores for 30 and 31 ounces. The z-score is calculated by subtracting the mean from the value and dividing the result by the standard deviation. For 30 ounces, the z-score is (30-28)/2 = 1. For 31 ounces, the z-score is (31-28)/2 = 1.5.
Next, we use these z-scores to find the cumulative probabilities from a standard normal distribution table. The cumulative probability for a z-score of 1 is 0.8413 and for 1.5, it's 0.9332.
The probability of filling a cup between 30 and 31 ounces is the difference between the cumulative probabilities of the two z-scores. So, the answer is 0.9332 - 0.8413 = 0.0919.
Therefore, the probability of the soft drink machine filling a cup between 30 to 31 ounces is 0.0919 or 9.19% when rounded to four decimal places.
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The probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]
The probability of filling a cup between 30 and 31 ounces when the mean output is 28 ounces and the standard deviation is 2 ounces can be found using the Z-score formula for a normal distribution.
First, we calculate the Z-score for 30 ounces:
[tex]\[ Z_{30} = \frac{X - \mu}{\sigma} = \frac{30 - 28}{2} = \frac{2}{2} = 1 \][/tex]
Next, we calculate the Z-score for 31 ounces:
[tex]\[ Z_{31} = \frac{X - \mu}{\sigma} = \frac{31 - 28}{2} = \frac{3}{2} = 1.5 \][/tex]
Now, we look up the probabilities corresponding to these Z-scores in the standard normal distribution table or use a calculator.
The probability of getting a value less than or equal to [tex]\( Z_{30} \)[/tex]is:
[tex]\[ P(Z \leq 1) \approx 0.8413 \][/tex]
The probability of getting a value less than or equal to is:
[tex]\[ P(Z \leq 1.5) \approx 0.9332 \][/tex]
To find the probability of filling a cup between 30 and 31 ounces, we subtract the probability of filling up to 30 ounces from the probability of filling up to 31 ounces:
[tex]\[ P(30 < X < 31) = P(Z \leq 1.5) - P(Z \leq 1) \][/tex]
[tex]\[ P(30 < X < 31) \approx 0.0919 \][/tex]
Rounded to four decimal places, the probability is 0.0919.
Therefore, the probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]
The Cooper family, consisting of 6 children and 2 adults, goes to the movies and pays a total of $86. The Griffen birthday party, consisting of 10 children and 3 adults, pays a total of $139 for tickets. Let x = the cost of a child ticket and y = the cost of an adult ticket. How much does a child's ticket cost and how much does an adult ticket cost?
Answer: The cost of child's ticket = $10
The cost of adult ticket = $ 13
Step-by-step explanation:
Let x be the cost of a child ticket and y be the cost of an adult ticket.
Then According to the question, we have
[tex]6x+2y=86..........................(1)\\\\10x+3y=139.......................(2)[/tex]
Multiply equation (1) by 3 and equation (2) by 2, then we have
[tex]18x+6y=258.......................(1)\\\\20x+6y=278...........................(2)[/tex]
Subtract equation (1) from equation (2), we have
[tex]2x=20\\\\\Rightarrow\ x=10[/tex]
Substitute the value of x in equation (1), we get
[tex]60+2y=86\\\\\Rightarrow\ 2y=26\\\\\Rightarrow\ y=13[/tex]
Hence, the cost of child's ticket = $10
The cost of adult ticket = $ 13
Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 192 daysμ=192 days and standard deviation sigma equals 12 daysσ=12 days.What is the probability that a randomly selected pregnancy lasts less than 188188 days? The probability that a randomly selected pregnancy lasts less than 188188 days is approximately
Answer: 0.5237
Step-by-step explanation:
Mean : [tex]\mu=192\text{ days}[/tex]
Standard deviation : [tex]\sigma = 12\text{ days}[/tex]
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = 188 days ,
[tex]z=\dfrac{188-192}{12}\approx-0.33[/tex]
For x = 107 miles per day ,
[tex]z=\dfrac{107-92}{12}=1.25[/tex]
The P-value =[tex]P(-0.33<z<1.25)=P(z<1.25)-P(z<-0.33)[/tex]
[tex]0.8943502-0.3707=0.5236502\approx0.5237[/tex]
Hence, The probability that a randomly selected pregnancy lasts less than 188 days is approximately 0.5237.
In a clinical trial of a cholesterol drug, 374 subjects were given a placebo, and 21% of them developed headaches. For such randomly selected groups of 374 subjects given a placebo, identify the values of n, p, and q that would be used for finding the mean and standard deviation for the number of subjects who develop headaches. The value of n is __________. (Do not round.) The value of p is __________. (Type an integer or a decimal. Do not round.) The value of q is __________. (Type an integer or a decimal. Do not round.)
Answer: The value of n is 374. The value of p is 0.21. The value of q is 0.79.
Step-by-step explanation:
Given : In a clinical trial of a cholesterol drug, 374 subjects were given a placebo, and 21% of them developed headaches.
∴ Sample size : [tex]n=374[/tex]
The probability that cholesterol drug developed headaches :[tex]p=0.21[/tex]
Then , the probability that cholesterol drug did not develop headaches :[tex]q=1-p=1-0.21=0.79[/tex]
Hence, The value of n is 374.
The value of p is 0.21.
The value of q is 0.79.
Graph the solution set of the system of inequalities or indicate that the system has no solution.
y ≥ 2x – 4
x + 2y ≤ 7
y ≥ -2
x ≤ 1
Answer:
the graph is shown below
Step-by-step explanation:
The solution region is that quadruply-shaded area above the line y=-2 and to the left of the line x=1. It is further bounded above by the line y = -1/2x +3.5.
Consider the following equation. f(x, y) = sin(2x + 5y), P(−5, 2), u = 1 2 3 i − j (a) Find the gradient of f. ∇f(x, y) = (b) Evaluate the gradient at the point P. ∇f(−5, 2) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(−5, 2) =
The gradient of function f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)). The gradient at the point P(-5, 2), is ∇f(-5, 2) = (2cos(-20), 5cos(-20)). Rate of change of f at P in the direction of the vector u is (-1/2, 3). Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).
(a) To find the gradient of the function f(x, y) = sin(2x + 5y), we need to compute the partial derivatives with respect to x and y:
Gradient of f(x, y) = (∂f/∂x, ∂f/∂y).
Taking the partial derivative with respect to x:
∂f/∂x = ∂/∂x(sin(2x + 5y)) = 2cos(2x + 5y).
Taking the partial derivative with respect to y:
∂f/∂y = ∂/∂y(sin(2x + 5y)) = 5cos(2x + 5y).
So, the gradient of f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)).
(b) To evaluate the gradient at point P(-5, 2), we substitute these values into the gradient expression:
∇f(-5, 2) = (2cos(2(-5) + 5(2)), 5cos(2(-5) + 5(2))).
Calculating these values gives the gradient at P.
The gradient at point P is ∇f(-5, 2) = (2cos(-20), 5cos(-20)).
(c) To find the rate of change of f at point P(-5, 2) in the direction of the vector u = (1/2, 3) - (1, 0) = (-1/2, 3), we use the dot product:
Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).
Calculate this dot product to find the rate of change of f in the direction of u at point P.
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What Number is 5% of 220 ???
Answer: 11
Step-by-step explanation:
5% × 220 =
(5 ÷ 100) × 220 =
(5 × 220) ÷ 100 =
1,100 ÷ 100 =
11;
5% of 220 = 11
Answer:
44% is your answer.
Step-by-step explanation:
So, 220/5= 44/1= 44%
Because if you divide 220 by 5 you'll get 44 and 5 divided by 5 ofcourse is 1.
(I'm doing these also in my class, so hopefully i helped you.)
Suppose a variable has a normal distribution with mean 67 and standard deviation 4. What percentage of the distribution is less than 75? (Use z-score.)
Answer:
The percentage of the distribution is less than 75 is 97.72%.
Step-by-step explanation:
Given,
Mean of the distribution,
[tex]\mu=67[/tex]
Standard deviation,
[tex]\sigma = 4[/tex]
Thus, the z-score of the score 75,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]=\frac{75-67}{4}[/tex]
[tex]=\frac{8}{4}[/tex]
[tex]=2[/tex]
With the help of z-score table,
[tex]P(x<75)=0.9772=97.72\%[/tex]
Hence, the percentage of the distribution is less than 75 is 97.72%.
It took 2 men 5 hours to build an airstrip. Working at the same rate, how many additional men could have been hired in order for the job to have taken 1/hour less? (A) Two (B) Three (C) Four (D) Six
The problem is to calculate the number of additional men needed to build an airstrip in less time given two men already took five hours. The additional man required is one after solving the equation, but this is not an option provided, indicating a potential error in the question.
Explanation:The question involves calculating the number of additional men required to build an airstrip in less time. If it took 2 men 5 hours to build an airstrip, we can say they have a combined work rate of 1 airstrip per 5 hours, or (1/5) airstrip per hour. To finish the job in 4 hours, which is 1 hour less than the original time, we would need a work rate of 1 airstrip per 4 hours.
So, if we let the number of additional men be X, we can set up the equation as follows:
(2 + X) men × 4 hours = 2 men × 5 hours(2 + X) × 4 = 102 + X = 10 / 4X = 2.5 - 2X = 0.5Since we cannot hire half a person, we round up to the nearest whole number. Hence, one additional man would be sufficient to complete the work in 1 hour less time. However, none of the options given (A) Two, (B) Three, (C) Four, or (D) Six, are correct. Therefore, the answer is not provided in the given options and this represents a possible error in the question itself.
If a dart was thrown randomly at the dart board shown below, what is the probability that it would land between the outer circle and the middle circle? The radius of the bulls eye is 2 cm, the radius of the middle circle is 8 cm, and the radius of the outer circle is 14 cm
A.68%
B.67%
C.14%
D.75%
Answer: B) 67%
Step-by-step explanation:
Find the Area of the Bullseye and Middle ring
A = π r²
A (inside) = π(8)² = 64π
Find the Area of the entire Target
A (target) = π (14)² = 196π
Find the Area of the Outer ring
A (outer ring) = A (target) - A(inside)
= 196 π - 64π
= 132 π
The last step is to find the probability of landing on the outer ring:
[tex]P=\dfrac{success (area\ of\ outer\ ring)}{total\ possible\ outcomes(area\ of\ target)}=\dfrac{132\pi}{196\pi}=0.673=\large\boxed{67\%}[/tex]
Answer:
67%
Step-by-step explanation:
Part A: Sam rented a boat at $225 for 2 days. If he rents the same boat for 5 days, he has to pay a total rent of $480.
Write an equation in the standard form to represent the total rent (y) that Sam has to pay for renting the boat for x days. (4 points)
Answer:
y = 85x + 55
Step-by-step explanation:
If we use the given prices as coordinates we will get the coordinates (2, 225) and (5, 480) and if we use the rise over run which in this case is 255/3 then we get the slope which is $85 and now we plug what we know into y = mx + b. we now have the equation 225 = (85)2 + b which can be simplified to 55 = b and now we can create the equation y = 85x + 55.
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using induction, verify the inequality. (1 + x)" ? 1 + nx, for x ?-1 and ? 1
Answer with explanation:
The Inequality should be:
[tex](1+x)^n\geq 1+n x[/tex]
Where, n and x are any integers.
⇒For, x= -1
L HS
[tex]=[1+(-1)]^n\\\\=(0)^n\\\\=0[/tex]
R HS
→1+n × (-1)
=1-n
If, n is any Integer, then for, n=1
1-1=0
For, n=2
1-2= -1
....
So, [tex](1+x)^n\geq 1+n x[/tex]
for, x=-1.
⇒For, x=1
L HS
[tex]=[1+(1)]^n\\\\=(2)^n[/tex]
For, n=1
L H S=1
For, =2
L H S=4
R HS
→1+n × (1)
=1+n
If, n is any Integer, then for, n=1
1+1=2
For, n=2
1+2= 3
....
So, [tex](1+x)^n\geq 1+n x[/tex]
for, x=1.
Find the sum of the geometric sequence 3, 15, 75, 375, … when there are 9 terms and select the correct answer below.
a. -976,563
b. 976,563
c. 1,464,843
d. 976,562
Answer:
c. 1,464,843
Step-by-step explanation:
The sum of n terms of a geometric sequence with first term a1 and common ratio r is given by ...
sn = a1(r^n -1)/(r -1)
Filling in the values a1=3, r=5, n=9, we get ...
s9 = 3(5^9 -1)/(5 -1) = 1,464,843
Answer: -976,563 PLEASE READ DESCRIPTION
Step-by-step explanation:
My question had the same exact numbers but some of them were negative! Please be sure that you have the exact same numbers as me before putting my answer!!
"Find the sum of the geometric sequence −3, 15, −75, 375, ... when there are 9 terms and select the correct answer below"
-3 X -5 = 15 X -5 = -75 X -5 = 375.
Ratio = -5
Use the formula [tex]s_{n} = \frac{a_{1 - a_1 (r)^{n} }}{1 - r}[/tex].
Our [tex]a_1[/tex] (first term) = -3, r (ratio)= -5, and n (number of terms) = 9. Knowing this, plug them into the equation.
[tex]s_9 = \frac{-3 - (-3)(-5)^9}{1-(-5)}[/tex].
First, simplify the exponent. -5 to the ninth power = -1,953,125. Multiply this by the nearest number in exponents (-3). -1,953,125 X -3 = 5,859,375. Continue simplifying your numerator. -3 - (5,859,375) = -5,859,378. Now, simplify your denominator. 1 - (-5) = 6.
Divide.
[tex]s_9 = \frac{-5,859,372}{6}[/tex] = -976,563
I tried to make sure there weren't any typos, but please comment if there's something wrong!
Use induction to prove: For every integer n > 1, the number n5 - n is a multiple of 5.
Answer:
we need to prove : for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
1) check divisibility for n=1, [tex]f(1)=(1)^{5}-1=0[/tex] (divisible)
2) Assume that [tex]f(k)[/tex] is divisible by 5, [tex]f(k)=(k)^{5}-k[/tex]
3) Induction,
[tex]f(k+1)=(k+1)^{5}-(k+1)[/tex]
[tex]=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1[/tex]
[tex]=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k[/tex]
Now, [tex]f(k+1)-f(k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)[/tex]
[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k[/tex]
[tex]f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k[/tex]
Take out the common factor,
[tex]f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)[/tex] (divisible by 5)
add both the sides by f(k)
[tex]f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)[/tex]
We have proved that difference between [tex]f(k+1)[/tex] and [tex]f(k)[/tex] is divisible by 5.
so, our assumption in step 2 is correct.
Since [tex]f(k)[/tex] is divisible by 5, then [tex]f(k+1)[/tex] must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.
Therefore, for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.
Final answer:
The statement is true for all integers greater than 1.
Explanation:
The n5 - n is a multiple of 5 for every integer n > 1, we will use proof by induction.
Base Case: For n = 2,
n5 - n = 25 - 2 = 32 - 2 = 30,
which is clearly a multiple of 5. Hence, our base case holds true.
Inductive Step: Assume that for some integer k > 1, the statement holds true, i.e.,
k5 - k is a multiple of 5. We need to show that k5 - k + 5(k4 + k3 + k2 + k + 1) is also a multiple of 5.
If k5 - k is a multiple of 5, then adding a number which is a multiple of 5 (5 times a sum of powers of k) to it will also result in a multiple of 5. This means that (k + 1)5 - (k + 1) will be a multiple of 5, hence the statement holds for k + 1. By the principle of mathematical induction, the statement holds true for all integers n > 1.
Part A: Solve −np − 80 < 60 for n. Show your work. (4 points)
Part B: Solve 2a − 5d = 30 for d. Show your work. (6 points)
Answer:
Part A ⇒ n>-140/p and p≠0
Part B. ⇒ d=-30-2a/5
Step-by-step explanation:
Part A. -np-80<60
First, add by 80 both sides of equation.
-np-80+80<60+80
Simplify.
60+80=140
-np<140
Then, multiply by -1 both sides of equation.
(-np)(-1)>140(-1)
Simplify.
np>-140
Divide by p both sides of equation.
np/p>-140/p; p≠0
Simplify to find the answer.
n>-140/p; p≠0 is the correct answer from part a.
___________________________________
Part B. 2a-5d=30
First add by 2a from both sides of equation.
2a-5d+2a=30+2a
Then, simplify.
-5d=30-2a
Divide by -5 from both sides of equation.
-5d/-5=30/-5-2a/5
Simplify, to find the answer.
d=-30-2a/5 is the correct answer from part b.
Part A:
For this case we have the following inequality:
[tex]-np-80 <60[/tex]
We add 80 to both sides of the inequality:
[tex]-np <60+80\\-np <140[/tex]
Dividing between -p on both sides, having to change the inequality sign:
[tex]n> - \frac {140} {p}[/tex]
Part B:
For this case we have the following equation:
[tex]2a-5d = 30[/tex]
Subtracting 2a on both sides:
[tex]-5d = 30-2a[/tex]
Dividing between -5 on both sides:
[tex]d = \frac {30-2a} {- 5}\\d = \frac {-30+2a} {5}\\d = - \frac {30} {5} +\frac {2a} {5}\\d = -6+ \frac {2a} {5}[/tex]
Answer:
[tex]n> - \frac {140} {p}\\d = -6+\frac {2a} {5}[/tex]
Stephanie cuts a five-meter-long piece of rope into five equal parts. What is the length of each piece of rope?
Final answer:
Stephanie cuts the rope into five 1-meter-long pieces by dividing the total length (5 meters) by the number of pieces (5).
Explanation:
If Stephanie cuts a five-meter-long piece of rope into five equal parts, we would divide the total length of the rope by the number of parts to find the length of each piece. In this case, the calculation would be:
Total length of rope = 5 metersNumber of parts = 5Length of each part = Total length of rope ÷ Number of partsSo, the length of each part would be 5 meters ÷ 5, which equals 1 meter. Therefore, each piece of rope would be 1 meter long.
Q1. Ten percent of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. What is the standard error of the distribution of the sample proportions? Round your answer to three decimal places.
Answer: 0.030
Step-by-step explanation:
Given: The percent of the items produced by a machine are defective :[tex]P=10\%=0.10[/tex]
The percent of the items produced by machine which are not defective:[tex]Q=1-0.1=0.9[/tex]
Sample size : n = 100
Now, the standard error of the proportion is given by :-
[tex]\text{S.E.}=\sqrt{\dfrac{0.1\times0.9}{100}}\\\\\Rightarrow\text{S.E.}=0.030[/tex]
Hence, the the standard error of the distribution of the sample proportions=0.030
The standard error for the distribution of sample proportions of a machine producing 10% defective items determined from a random sample of 100 items is ±0.030.
Explanation:Ten percent of the items produced by a machine are defective. From a random sample of 100 items, the standard error of the sample proportions is calculated using the formula for the standard error of a proportion, which is √(pq/n), where p is the proportion of successes, q is the proportion of failures (1-p), and n is the sample size.
In this case, p is 0.10 (the proportion defective), q is 0.90 (1-p, representing the proportion not defective), and n is 100 (the sample size). Substituting these values into the formula gives: √[(0.10)(0.90) / 100] which equals ±0.030 (rounded to three decimal places).
Therefore, the standard error for the distribution of sample proportions is ±0.030.
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The half-life of a certain material is 0.004 second. Approximately how much of a 20-gram sample will be left after 0.016 second?
A. 5 grams
B. 16 grams
C. 0.32 gram
D. 1.25 grams
Answer:
D. 1.25 grams
Step-by-step explanation:
Half-life is: 0.004 sec
Time spent : 0.016 sec
Quantity = 20 gram
In order to find the material after 0.016 sec, we have to calculate how many number of half-lives have been passed
No. of half-lives passed = 0.016/0.004
= 4
The number of lives passed will be raised to the power of 0.5.
0.5 ^ 4 = 0.0625
The answer will be multiplied with the quantity we started with.
Remaining material is:
20*0.0625 = 1.25 grams
Hence, Option D is correct ..
How do you simplify this sum?
Answer:
Step-by-step explanation:
You can't simplify the sum, but you can factor it.
t^2 + 2t - 3
(t + 3)(t - 1 )
That's about all you can do.
Answer:
It is already in simplest form.
Step-by-step explanation:
It cannot be further simplified because it does not have any like terms.
#5 Points possible: 3. Total attempts: 5
Using your calculator, find the range and standard deviation, round to two decimals places:
The table below gives the number of hours spent watching TV last week by a sample of 24 children.
76
57
89
73
88
42
31
46
80
42
38
57
49
50
89
36
69
82
27
88
39
89
95
18
Range =
Standard Deviation =
Answer:
honesty you cant realy use a calculator because you need to meadian mode and range them first
Step-by-step explanation:
An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the 10 problems, what is the probability that he will answer correctly (a) all 5 problems? (b) at least 4 of the problems?
(a) The probability of answering all 5 problems correctly is 0.08.
(b) The probability of answering at least 4 problems correctly is 0.5.
There are a total of 10 problems, and the student has figured out how to solve 7 of them which means there are 3 problems that the student hasn't figured out how to solve.
(a) To find the probability that the student answers correctly to all 5 problems, we need to consider that the student must select 5 out of the 7 problems they know how to solve and 0 out of the 3 problems they don't know how to solve.
The probability of selecting 5 specific problems out of 7 is given by:
(7 choose 5) / (10 choose 5) [tex]=\frac{^7C_5}{^{10}C_5}[/tex]
=21/252
=0.083
(b) To find the probability that the student answers at least 4 problems correctly, we need to consider two cases: when the student answers 4 problems correctly and when the student answers all 5 problems correctly.
Case 1: Student answers 4 problems correctly and 1 problem incorrectly:
P(4 correct and 1 incorrect) = (7 choose 4) × (3 choose 1) / (10 choose 5)
Case 2: Student answers all 5 problems correctly:
P(5 correct out of 5) = (7 choose 5) / (10 choose 5)
Now, add the probabilities of these two cases to get the probability of answering at least 4 problems correctly:
P(at least 4 correct) = P(4 correct and 1 incorrect) + P(5 correct out of 5)
Calculate each part using combinations:
P(4 correct and 1 incorrect) = (35 × 3) / 252
= 0.4167
P(5 correct out of 5) = 0.08 (as calculated in part a)
Now, add these probabilities:
P(at least 4 correct) = 0.4167 + 0.08
= 0.5
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The probability that the student will correctly answer all 5 questions is 0.083 (8.3%) and the probability of correctly answering at least 4 questions is 0.5 (50%).
Explanation:This is a probability question related to the field of combinatorics. To answer the student's question, we have to calculate the probability of correctly answering the questions out of the known ones.
The total ways the instructor can select 5 problems out of 10 is represented by the combination formula C(10,5). This equates to 252.
The student can answer 7 questions, so a) the number of ways of getting all 5 correctly is represented by C(7,5) which equals 21. Therefore, the probability of answering all 5 correctly is 21/252 = 0.083 or 8.3%.
For b), the student aims to answer at least 4 correctly. This means that we calculate the probability for getting 4 and 5 problems correct. For 4 problems it's C(7,4)*C(3,1) = 105. Adding the ways to get 5 problems correct, we get 105+21=126. So, the chance of answering at least 4 correct is 126/252 = 0.5 or 50%.
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Problem Page
A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 7% vinegar, and the second brand contains 15% vinegar. The chef wants to make 240 milliliters of a dressing that is 13% vinegar. How much of each brand should she use?
Let [tex]x[/tex] be the amount (in mL) of the first brand (7% vinegar) and [tex]y[/tex] the amount of the second brand (15% vinegar). She wants to end up with a mixture with volume 240 mL, so that
[tex]x+y=240[/tex]
and she wants it contain 13% vinegar. Each mL of the first brand contributes 0.07 mL (i.e. 7% of 1 mL) vinegar, while each mL of the second brand contributes 0.15 mL (i.e. 15% of 1 mL). The final mixture needs to contribute 0.13 mL (i.e. 13% of 1 mL) for each mL of dressing, so that
[tex]0.07x+0.15y=0.13(x+y)=31.2[/tex]
Now solve:
[tex]x+y=240\implies y=240-x[/tex]
[tex]0.07x+0.15y=31.2\implies0.07x+0.15(240-x)=31.2[/tex]
[tex]\implies-0.08x+36=31.2[/tex]
[tex]\implies4.8=0.08x[/tex]
[tex]\implies\boxed{x=60}[/tex]
[tex]y=240-x\implies\boxed{y=180}[/tex]
The chef needs to use 60 mL of the first brand and 180 mL of the second brand.
Answer:
First brand: 60 milliliters
Second brand: 180 milliliters
Step-by-step explanation:
Let's call m the amount of the first dressing mark that contains 7% vinegar
Let's call n the amount of the second dressing mark that contains 15% vinegar
The resulting mixture should have 13% vinegar and 240 milliliters.
Then we know that the total amount of mixture will be:
[tex]m + n = 240[/tex]
Then the total amount of vinegar in the mixture will be:
[tex]0.07m + 0.15n = 0.13 * 240[/tex]
[tex]0.07m + 0.15n = 31.2[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.15 and add it to the second equation:
[tex]-0.15m -0.15n = 240 * (- 0.15)[/tex]
[tex]-0.15m -0.15n = -36[/tex]
[tex]-0.15m -0.15n = -36[/tex]
+
[tex]0.07m + 0.15n = 31.2[/tex]
--------------------------------------
[tex]-0.08m = -4.8[/tex]
[tex]m = \frac{-4.8}{-0.08}[/tex]
[tex]m = 60\ milliliters[/tex]
We substitute the value of m into one of the two equations and solve for n.
[tex]m + n = 240[/tex]
[tex]60 + n = 240[/tex]
[tex]n = 180\ milliliters[/tex]
force of 400 N stretches a spring 2 m. A mass of 50 kg is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. Find the equation of motion?
Answer:
x = -5 sin (2t)
Step-by-step explanation:
k is the spring stiffness. The unstretched length of the spring is L.
When the mass is added, the spring stretches to an equilibrium position of L+s, where mg = ks. When the mass is displaced a distance x (where x is positive if the displacement is down and negative if it's up), the spring is stretched a total distance s + x.
There are two forces on the mass: weight and force from the spring. Sum of the forces in the downward direction:
∑F = ma
mg − k(s + x) = ma
mg − ks − kx = ma
Since mg = ks:
-kx = ma
Acceleration is second derivative of position, so:
-kx = m d²x/dt²
Let's find k:
F = kx
400 = 2k
k = 200
We know that m = 50. Substituting:
-200x = 50 d²x/dt²
-4x = d²x/dt²
d²x/dt² + 4x = 0
This is a linear second order differential equation of the form:
x" + ω² x = 0
The solution to this is:
x = A cos (ωt) + B sin (ωt)
Here, ω² = 4, so ω = 2.
x = A cos (2t) + B sin (2t)
We're given initial conditions that x(0) = 0 and x'(0) = -10 (remember that down is positive and up is negative).
Finding x'(t):
x' = -2A sin (2t) + 2B cos (2t)
Plugging in the initial conditions:
0 = A
-10 = 2B
Therefore:
x = -5 sin (2t)
The response to a question has three alternatives: A, B, and C. A sample of 120 responses provides 60 A, 23 B, and 37 C. Show the frequency and relative frequency distributions (use nearest whole number for the frequency column and 2 decimal for the relative frequency column).
We know that in a frequency table we write the frequency corresponding to each of the data.
The relative frequency is the ratio of the frequency to the total frequency corresponding to each entry.
Here we have a total sample as: 120
Also, the frequency corresponding to A is: 60
Corresponding to B is: 23
and corresponding to C is: 37
Hence, the Frequency table is as follows:
Data Frequency
A 60
B 23
C 37
The relative frequency table is given by:
Data Relative Frequency
A 60/120=0.5
B 23/120=0.19
C 37/120=0.31
Hence, we get:
Data Relative Frequency
A 0.5
B 0.19
C 0.31
The frequency distribution for the responses A, B, and C are 60, 23, and 37 respectively. The relative frequency distribution for the responses A, B, and C are 0.50, 0.19, and 0.31 respectively. Sum of the relative frequencies equals to 1, indicating all portions of the sample have been accounted for.
Explanation:Frequency and Relative Frequency DistributionsTo create these distributions for the given response data, we'll organize our observations into a table.
Frequency distribution: The frequency shows the number of times each response (A, B, or C) occurs in the sample.
- Response A: 60
- Response B: 23
- Response C: 37
Adding these frequencies would equal the total sample size, 120.
Relative frequency distribution: The relative frequency is calculated as the frequency of each response divided by the total sample size, expressed as a decimal.
- Response A: 60/120 = 0.50
- Response B: 23/120 = 0.19
- Response C: 37/120 = 0.31
Note that if we add up the relative frequencies, we should get a sum of 1, indicating that we've accounted for all portions of the sample.
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Papa John wants to know if he should rename an old favorite pizza for his restaurant. He gives 60 patrons a pepperoni pizza and asks them to rate both the names ‘pepperoni deluxe’ and ‘venti extra cheesy pepperoni on toasted pan bread with tomato zest pesto’. He takes the ratings and wants to perform a statistical test. What would be the best statistical test to be used?
Answer:
gragh
Step-by-step explanation:
its one of the few ways to analyze data and compare it to find the best choice
Prove that for all integers n, if n^(2)-6n is odd then n is odd.
Step-by-step explanation:
o - odd number
e - even number
n × e = e, n is either odd or even...rule 1
n - e = o, n must be odd...rule 2
n - e = e, n must be even...rule 3
n^2 = o, n must be odd...rule 4
n^2 = e, n must be even...rule 5
6n is even, no matter if n is odd or even following rule 1
if n^2 - 6n = o, n must be odd following rule 2
if n^2 = o, n must be odd following rule 4
Based on sample data, newborn males have weights with a mean of 3277.9 g and a standard deviation of 571.6 g. Newborn females have weights with a mean of 3091.6 g and a standard deviation of 625.7 g. Who has the weight that is more extreme relative to the group from which they came: a male who weighs 1700 g or a female who weighs 1700 g?
To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.
Explanation:To determine who has the weight that is more extreme relative to their group, we need to calculate the z-scores for both the male and female who weigh 1700 g.
The z-score formula is: z = (x - mean) / standard deviation.
For the male who weighs 1700 g:
z = (1700 - 3277.9) / 571.6 = -1.956.
For the female who weighs 1700 g:
z = (1700 - 3091.6) / 625.7 = -1.394.
Since the absolute value of -1.956 is greater than the absolute value of -1.394, the male with a weight of 1700 g is more extreme relative to their group.
(CO 3) A survey indicates that shoppers spend an average of 22 minutes with a standard deviation of 8 minutes in your store and that these times are normally distributed. Find the probability that a randomly selected shopper will spend less than 20 minutes in the store. 0.40 0.50 0.60 0.22
Final answer:
To calculate the probability of a shopper spending less than 20 minutes in the store, the Z-score is found using the formula Z = (X - μ) / σ, resulting in a Z-score of -0.25, which corresponds to a probability of approximately 0.40.
Explanation:
The question asks for the probability that a randomly selected shopper will spend less than 20 minutes in a store, given that the average time spent is 22 minutes with a standard deviation of 8 minutes, and that these times are normally distributed. To find this probability, we use the Z-score formula:
Z = (X - μ) / σ
Where X is the value we are checking (20 minutes), μ is the mean (22 minutes), and σ is the standard deviation (8 minutes). Plugging in the numbers, we get:
Z = (20 - 22) / 8 = -0.25
Next, we look up the Z-score in a standard normal distribution table, or use a calculator with normal distribution functions, to find the probability that a Z-score is less than -0.25. This probability is approximately 0.40.
Find Each Measure.
m1, m2, m3
Answer:
The second choice is correct.
Step-by-step explanation:
These are triangles, so the interior angles have to add up to 180 degrees. Since 45 degrees and m<1 are vertical angles, they will have the same measure. So right off the back we know that m<1 =45. To find the measure<2 all you need to do is add 45 and 59, then subtract that answer from 180. M<2= 76. That leaves only one option. The second one.
The measurements of the angles are Angle 1 = 45 degrees, Angle 2 = 76 degrees, Angle 3 = 80 degrees. The correct option is b) 1 = 45, 2 = 76, 3 = 80.
Let's solve this step by step to find the measurements of angles 1, 2, and 3.
We named the two triangles as OAB and OCD, with a common vertex O. angle O makes vertically opposite angle 45 degree in triangle OAB and angle 1 in OCD, angle A makes angle 2, angle B makes angle 59 degree, angle C makes 55 degree and angle D makes angle 3.Given the information:
In triangle OAB:
Angle OAB (angle 2) = 59 degrees
Angle O = 45 degrees (vertically opposite to angle 1)
In triangle OCD:
Angle O = 45 degrees (again, vertically opposite to angle 1)
Angle OCD (angle 3) = ?
First, let's find angle 2 in triangle OAB:
In triangle OAB, the sum of angles in a triangle is 180 degrees.
So, angle 2 + angle O + angle OAB = 180 degrees.
59 + 45 + angle OAB = 180 degrees.
Now, solve for angle OAB:
angle OAB = 180 degrees - 59 degrees - 45 degrees
angle OAB = 76 degrees
Now, let's find angle 3 in triangle OCD:
In triangle OCD, the sum of angles in a triangle is also 180 degrees.
So, angle O + angle OCD + angle C = 180 degrees.
45 + angle 3 + 55 = 180 degrees.
Now, solve for angle 3:
angle 3 = 180 degrees - 45 degrees - 55 degrees
angle 3 = 80 degrees
So, the measurements of angles 1, 2, and 3 are as follows:
Angle 1 = 45 degrees
Angle 2 = 76 degrees
Angle 3 = 80 degrees
The correct option is:
b) 1 = 45, 2 = 76, 3 = 80
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