Answer:100 people
Step-by-step explanation:
140 people x 5 ladle/1 person = 700 ladles full
700 ladles x 1 person/7 ladles = 100 people fed
What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success p? Choose the correct forumla below.
A. Upper E (Upper X )equalsStartRoot np (1 minus p )EndRoot
B. Upper E (Upper X )equalsp Superscript n
C. Upper E (Upper X )equals (1 minus p )Superscript n(1 minus p )Superscript n
D. Upper E (Upper X )equalsnp
Answer:
[tex](D)E[ X ] =np.[/tex]
Step-by-step explanation:
Given a binomial experiment with n trials and probability of success p,
[tex]f(x)=\left(\begin{array}{c}n\\k\end{array}\right)p^x(1-p)^{n-x}, 0\leq x\leq n[/tex]
[tex]E(X)=\sum_{x=0}^{n}xf(x)= \sum_{x=0}^{n}x\left(\begin{array}{c}n\\k\end{array}\right)p^x(1-p)^{n-x}[/tex]
Since each term of the summation is multiplied by x, the value of the term corresponding to x = 0 will be 0. Therefore the expected value becomes:
[tex]E(X)=\sum_{x=1}^{n}x\left(\begin{array}{c}n\\x\end{array}\right)p^x(1-p)^{n-x}[/tex]
Now,
[tex]x\left(\begin{array}{c}n\\x\end{array}\right)= \frac{xn!}{x!(n-x)!}=\frac{n!}{(x-)!(n-x)!}=\frac{n(n-1)!}{(x-1)!((n-1)-(x-1))!}=n\left(\begin{array}{c}n-1\\x-1\end{array}\right)[/tex]
Substituting,
[tex]E(X)=\sum_{x=1}^{n}n\left(\begin{array}{c}n-1\\x-1\end{array}\right)p^x(1-p)^{n-x}[/tex]
Factoring out the n and one p from the above expression:
[tex]E(X)=np\sum_{x=1}^{n}n\left(\begin{array}{c}n-1\\x-1\end{array}\right)p^{x-1}(1-p)^{(n-1)-(x-1)}[/tex]
Representing k=x-1 in the above gives us:
[tex]E(X)=np\sum_{k=0}^{n}n\left(\begin{array}{c}n-1\\k\end{array}\right)p^{k}(1-p)^{(n-1)-k}[/tex]
This can then be written by the Binomial Formula as:
[tex]E[ X ] = (np) (p +(1 - p))^{n -1 }= np.[/tex]
The correct formula for the expected number of successes in a binomial experiment with n trials and probability of success p is μ = np (Option D), which means that the expected number of successes is calculated by multiplying the total number of trials by the probability of success.
Explanation:The formula for the expected number of successes in a binomial experiment with n trials and probability of success p is given by μ = np. In this formula, 'n' represents the number of trials, and 'p' is the probability of success on each trial. Therefore, the correct option, in this case, is D. Upper E (Upper X ) equals np.
This essentially means that the expected number of successes is obtained by multiplying the total number of trials by the probability of success. For instance, if you were to flip a coin (where the probability of getting heads is 0.5) 10 times, the expected number of times you'd get heads would be 0.5 * 10 = 5.
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Auto pistons at Wemming Chung's plant in Shanghai are produced in a forging process, and the diameter is a critical factor that must be controlled. From sample sizes of 10 pistons produced each day, the mean and the range of this diameter have been as follows:
Day Mean Bold x overbar (mm) Range R (mm)
1 158.9 4.0
2 151.2 4.6
3 155.6 4.3
4 155.5 5.0
5 154.6 4.3
a) What is the value of x ?
Missing part of the question
b)What is the value of R?
Answer:
a. X = 155.2
b. R = 4.4
Step-by-step explanation:
a. Calculating the value of x.
The value of x is the mean of the mean observation; i.e. the mean column
Mean is calculated as: the summation of all observation divided by the number of observations.
Summation of Observation = 158.9 + 151.2 + 155.6 + 155.5 + 154.6
Summation of Observation = 775.8
Number of Observations = 5
X = Mean = 775.8/5
X = 155.16
X = 155.2 ----- Approximated to 1 decimal place
b. Calculating the value of R.
The value of R is the mean/average of the Range
Mean is calculated as: the summation of all observation divided by the number of observations.
Summation of Observation = 4.0 + 4.6 + 4.3 + 5.0 + 4.3
Summation of Observation = 22.2
Number of Observations = 5
R = Mean = 22.2/5
R = 4.44
R = 4.4 ----- Approximated to 1 decimal place
The average diameter [tex](\( \overline{x} \))[/tex] of the pistons from the provided data is 155.16 mm, calculated by summing the daily means and dividing by the number of days.
In the context provided, [tex]\( \overline{x} \)[/tex] refers to the average of the piston diameters measured across different days at Wemming Chung's plant.
To find this value, we need to calculate the mean of the daily means [tex](\( \overline{x} \))[/tex] which are 158.9, 151.2, 155.6, 155.5, and 154.6 mm. This is done by summing these values and dividing by the number of days (5 in this case).
[tex]\[ \overline{x} = \frac{158.9 + 151.2 + 155.6 + 155.5 + 154.6}{5} \][/tex]
[tex]\[ \overline{x} = \frac{775.8}{5} \][/tex]
[tex]\[ \overline{x} = 155.16 \text{ mm} \][/tex]
Hence, the value of [tex]\( \overline{x} \)[/tex] is 155.16 mm.
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 12 vinegar. The chef wants to make 390 milliliters of a dressing that is 11 vinegar. How much of each brand should she use
Answer:
78 ml of 7% vinegar and 312 ml of 12% vinegar.
Step-by-step explanation:
Let x represent ml of 7% vinegar brand and y represent ml of 12% vinegar brand.
We have been given that chef wants to make 390 milliliters of the dressing. We can represent this information in an equation as:
[tex]x+y=390...(1)[/tex]
[tex]y=390-x...(1)[/tex]
We are also told that 1st brand 7% vinegar, so amount of vinegar in x ml would be [tex]0.07x[/tex].
The second brand contains 12 vinegar, so amount of vinegar in y ml would be [tex]0.12y[/tex].
We are also told that the chef wants to make 390 milliliters of a dressing that is 11% vinegar. We can represent this information in an equation as:
[tex]0.07x+0.12y=390(0.11)...(2)[/tex]
Upon substituting equation (1) in equation (2), we will get:
[tex]0.07x+0.12(390-x)=390(0.11)[/tex]
[tex]0.07x+46.8-0.12x=42.9[/tex]
[tex]-0.05x+46.8=42.9[/tex]
[tex]-0.05x+46.8-46.8=42.9-46.8[/tex]
[tex]-0.05x=-3.9[/tex]
[tex]\frac{-0.05x}{-0.05}=\frac{-3.9}{-0.05}[/tex]
[tex]x=78[/tex]
Therefore, the chef should use 78 ml of the brand that contains 7% vinegar.
Upon substituting [tex]x=78[/tex] in equation (1), we will get:
[tex]y=390-78[/tex]
[tex]y=312[/tex]
Therefore, the chef should use 312 ml of the brand that contains 12% vinegar.
Marcela took out a $600 discount loan with a 4% annual interest rate over a period of 8 months. How much money does marcela get from the bank when she receives the loan? Round to the nearest dollar.
Answer:
Step-by-step explanation:
4 % of 600 over 8 months is 16
600-16=584
The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value 32 mm and standard deviation 7.3 mm. (a) What is the probability that defect length is at most 20 mm? Less than 20 mm?
Answer:
The probability that defect length is at most 20 mm (or less than 20 mm) is 0.0505
Step-by-step explanation:
Mean defect length = u = 32
Standard deviation = [tex]\sigma[/tex] = 7.3
The distribution is normal and we have the value of population standard deviation so we will use the concept of z-score and probability from z-table to find the said probability.
We have to find the probability that the defect length is at most 20 mm. At most 20 mm means equal to or lesser than 20 mm. Since the normal distribution is a continuous distribution, the probability of at most is approximately equal to probability of lesser than.
If X represents the distribution of defect lengths, then we can write:
P(X ≤ 20) ≅ P(X < 20)
We can find the probability of defect length being lesser than 20 mm by converting it to z-score and find the corresponding probability from the z-table.
The formula to calculate the z-score is:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Substituting the values, we get:
[tex]z=\frac{20-32}{7.3}=-1.64[/tex]
Therefore, P(X < 20) is equivalent to P(z < -1.64)
From z-table we can find the probability of z being lesser than - 1.64, which comes out to be:
P(z < -1.64) = 0.0505
Therefore, we can conclude that:
The probability that defect length is at most 20 mm (or less than 20 mm) is 0.0505
The volume of a cube is given by the expression s to the power of 3, and its surface area is given by the expression 6s to the power of 2, where s is the length of the cube’s side. What is the volume of a cube with a side length of 5 inches?
Answer:
The volume of the cube would be 125in³
Step-by-step explanation:
The volume of a cube is s³ (side). 5 x 5 x 5 = 125.
Layla and Sam are both dog sitters. Layla charges $2 per day plus a sign-up fee of $3. Sam charges a flat rate of $3 per day. The system of linear equations below represents y, the total amount earned in dollars for x days of dog sitting.
A. Write the equation to represent Layla’s fees.
B. Write the equation to represent Sam’s fees.
C. After how many days do Layla and Sam earn the same amount for dog sitting?
What is that amount?
Answer:
a) [tex]L(t) = 3 + 2\cdot t[/tex], b) [tex]S (t) = 3\cdot t[/tex], c) Third day.
Step-by-step explanation:
a) The equation to represent Layla's fees is:
[tex]L(t) = 3 + 2\cdot t[/tex]
b) The equation to represent Sam's fees is:
[tex]S (t) = 3\cdot t[/tex]
c) Both earn the same amount for dog sitting at the third day.
45 POINTS! NEED HELP QUESTION IS A IMAGE.
A portion of the Quadratic Formula proof is shown. Fill in the missing statement.
Statements Reasons
x squared plus b over a times x plus the quantity b over 2 times a squared equals negative 4 times a times c all over 4 times a squared plus b squared over 4 a squared Find a common denominator on the right side of the equation
x squared plus b over a times x plus the quantity b over 2 times a squared equals b squared minus 4 times a times c all over 4 times a squared Add the fractions together on the right side of the equation
the quantity x plus b over 2 times a squared equals b squared minus 4 times a times c all over 4 times a squared Rewrite the perfect square trinomial on the left side of the equation as a binomial squared
? Take the square root of both sides of the equation
x plus b over 2 times a equals plus or minus the square root of the quantity b squared minus 4 times a times c end quantity, all over 4 times a squared
x plus b over 2 times a all squared equals plus or minus b squared minus 4 times a times c, all over 4 times a squared all squared
the square root of x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 4 times a squared
x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 4 times a squared
Answer:
Take the square root of each side
D (x+b/2a) =±sqrt( (b^2-4ac)/ 4a^2)
Step-by-step explanation:
We have
(x+b/2a) ^2 = (b^2-4ac)/ 4a^2
We need to take the square root of each side to continue to isolate x
(x+b/2a) =±sqrt( (b^2-4ac)/ 4a^2)
Answer:
Option 3
Step-by-step explanation:
[x + b/2a]² = (b² - 4ac)/4a²
Taking square root both sides, you get this:
x + b/2a = +/- sqrt(b² - 4ac)/2a
Remember we're trying to make x the subject,
So does get rid of the square by applying square root both sides
Two candidates ran for class president. The candidate that won received 80% of the 290 total votes. How many votes did the winning candidate receive?
Answer:
232 total votes
Step-by-step explanation:
change 80% to a decimal.... .8 and then multiply that by 290
Answer:
232
Step-by-step explanation:
80% of 290 is 232
290-232 is 58
20% of 290 is 58
The population of a certain species of fish has a growth rate of 2.2% per year. It is estimated that the the current population is 350,000. Estimate the number of years it will take the fish population to reach 1,000,000. Round your answer to the nearest tenth.
We have been given that the population of a certain species of fish has a growth rate of 2.2% per year. It is estimated that the the current population is 350,000.
We are asked to find the time it will take the fish population to reach 1,000,000.
We will use exponential growth formula to solve our given problem.
An exponential growth function is in form [tex]y=a\cdot (1+r)^x[/tex], where,
y = Final amount,
a = Initial amount,
r = Growth rate in decimal form,
x = Time
[tex]2.2\%=\frac{2.2}{100}=0.022[/tex]
[tex]y=350,000\cdot (1+0.022)^x[/tex]
To find the time for the fish population to reach 1,000,000, we will substitute [tex]x=1,000,000[/tex] in our equation as:
[tex]1,000,000=350,000\cdot (1+0.022)^x[/tex]
[tex]1,000,000=350,000\cdot (1.022)^x[/tex]
[tex]\frac{1,000,000}{350,000}=\frac{350,000\cdot (1.022)^x}{350,000}[/tex]
[tex]2.8571428571428571=(1.022)^x[/tex]
Now we will take natural log on both sides:
[tex]\text{ln}(2.8571428571428571)=\text{ln}((1.022)^x)[/tex]
[tex]\text{ln}(2.8571428571428571)=x\cdot \text{ln}(1.022)[/tex]
[tex]x=\frac{\text{ln}(2.8571428571428571)}{\text{ln}(1.022)}[/tex]
[tex]x=\frac{1.0498221244986776733}{0.0217614917815127}[/tex]
[tex]x=48.2421947[/tex]
Upon rounding to nearest tenth, we will get:
[tex]x\approx 48.2[/tex]
Therefore, it will take approximately 48.2 years for the fish population to reach 1,000,000.
Answer:
47.7
Step-by-step explanation:
right answer
On Saturday, Cannor drives 62 1/4 miles each hour. If he travels 4 hours, how many miles does he travel altogether?
Final answer:
To find the total distance Cannor travels, multiply his hourly distance of 62 1/4 miles by the total travel time of 4 hours, resulting in 249 miles.
Explanation:
The question asks about calculating the total distance traveled by Cannor on Saturday. Since Cannor drives 62 1/4 miles each hour and travels for 4 hours, we can find the total distance by multiplying the number of miles driven in an hour by the number of hours traveled.
To calculate this, convert 62 1/4 to an improper fraction which is 249/4 and multiply it by 4 (the number of hours):
Total Distance = (249/4 miles/hour) × 4 hours
When we do the multiplication, the hours unit cancels out and we get:
Total Distance = 249 miles
This calculation shows how far Cannor will travel altogether if he maintains his speed for the entire duration of his trip.
Does the equation specify a function with independent variable x? If so, find the domain of the function. If not, find a value of x to which there corresponds more than one value of y. 6 x plus 19 yequals7
Yes, the equation specifies a function with independent variable x.
Equation: 6x + 19y = 7
This can be put in y = mx + b form,
6x + 19y = 7
19y = -6x + 7
y = -[tex]\frac{6}{19}[/tex]x + [tex]\frac{7}{19}[/tex]
Since the equation can be put in the form of a linear function, the domain is all read numbers (-∞,∞) or -∞ < x < ∞ .
The equation 6x + 19y = 7 does not specify a function in terms of x because for each value of x, there exist multiple possible values of y. A specific example is shown when x = 1.
Explanation:The equation "6x + 19y = 7" does not specify a function with the independent variable x. In a function, each input (x) corresponds to exactly one output (y). However, in this equation, each value of x corresponds to multiple values of y, thus it does not represent a function.
In order to find a value of x for which there are multiple corresponding values of y, we can set x to any arbitrary value. For example, if we set x = 1, we get "6(1) + 19y = 7", which simplifies to "19y = 1". This implies that there are infinite solutions for y when x = 1, confirming that the equation does not represent a function.
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ANSWER ONLY IF YK IT IM GIVING CORRECT BRAINLIEST
Factor the expression completely : -15x^3+5
Answer:
-5(3x^3-1)
Step-by-step explanation:
-15x^3+5
Factor out -5
-5(3x^3-1)
how do I find the answer to 3km 6hm 7dam 5m ÷ 15?
Final answer:
To solve 3km 6hm 7dam 5m ÷ 15, first convert the distance to meters (3675 m) and then divide by 15 to get 245 meters.
Explanation:
The question given by the student involves performing a division operation with a composite distance measurement in different units. First, we need to convert the entire distance into a single unit (meters) before we divide it by the given number (15).
To convert the given distance to meters, we use the metric system unit conversion:
1 kilometer (km) = 1000 meters (m), 1 hectometer (hm) = 100 meters (m), and 1 decameter (dam) = 10 meters (m). Therefore, 3 km is 3000 m, 6 hm is 600 m, 7 dam is 70 m, and adding the given 5 m to these conversions, we get a total distance of 3675 meters (3000 m + 600 m + 70 m + 5 m = 3675 m).
To find the final answer, divide 3675 meters by 15: which yields 245 meters. Thus, 3km 6hm 7dam 5m divided by 15 equals 245 meters.
I've been trying this for days I can't get my head round it any help would be appreciated.
The one line of symmetry is vertical, so we could fold the hexagon in half in such a way that the vertices A and B would meet at the same point, and the same goes for the pairs C,F and D,E. Because of this symmetry, we know angle AFE is congruent to BCD, and angle FED is congruent ot CDE.
Let x be the measure of angle CDE.
In any convex polygon with n sides, the interior angles sum to (n - 2)*180º in measure. ABCDEF is a hexagon, so n = 6.
We have 2 angles of measure 123º, 2 of measure x, and 2 of measure 2x. So
2(123º + x + 2x ) = (6 - 2)*180º
246º + 2x + 4x = 720º
6x = 474º
x = 79º
Angle AFE is congruent to angle BCD, which is twice the measure of CDE, so angle AFE has measure 2*79º = 158º.
Rearrange the equation so u is the independent variable.
4u+8w=−3u+2w
Answer:
w = [tex]-\frac{7}{6}u[/tex]
Step-by-step explanation:
If u will be the independent variable, then w will have to be made the subject of the equation so that w will depend on the changes in u
4u + 8w = −3u + 2w
First collect all like terms
8w − 2w = −3u − 4u
6w = −7u
divide both sides by 6
∴ w = [tex]-\frac{7}{6}u[/tex]
In the equation w = [tex]-\frac{7}{6}u[/tex] , U is independent variable.
Please help with this question!!?
Answer:
6 seconds
Step-by-step explanation:
The ball will hit the ground when its height is 0, so you can plug this into the equation to find your answer.
-16t^2+96t=0
Factoring out -16t, you get:
-16t(t-6), meaning that the solutions are 0 and 6 seconds. Since 0 is the starting point, the answer is 6 seconds. Hope this helps!
Students arrive randomly at the help desk of the computer lab. There is only one service agent, and the time required for inquiry varies from student to student. Arrival rates have been found to follow the Poisson distribution, and the service times follow the negative exponential distribution. The average arrival rate is 12 students per hour, and the average service rate is 20 students per hour. On average, how long does it take to service each student?
Considering Poissoniated student arrivals and exponential service times in a queueing situation, with an average service rate of 20 students per hour, it takes on average 3 minutes to service each student.
Explanation:
The student's question pertains to the field of queueing theory, more specifically dealing with Poisson arrivals and exponential service time. In this context, the given arrival rate (λ) is 12 students per hour and the service rate (μ) is 20 students per hour.
Since the service rate is provided in terms of how many students can be serviced per hour (20 students per hour), to find the average service time for each student, we take the reciprocal of the service rate.
Therefore, average service time = 1/μ = 1/20 = 0.05 hours per student.
Note that 0.05 hours is equivalent to 3 minutes, hence on average, it takes 3 minutes to service each student.
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A 95% confidence interval for the proportion of students achieving a reading achievement score that is above the standard set by the teachers for a population of third grade students is (0.43, 0.49).
The margin of error of this interval is:
Group of answer choices
0.05
0.03
0.06
None of the above
Miss Perkins want to rent a car for a day. It will cost a daily fee of $75 plus $.55 per mile driven. M equals the number of miles Miss Perkins drive for the day. Write an expression that shows the amount she will pay for the car. Evaluate the expression you wrote to find them out Miss Perkins will pay if she drive 300 miles.
Answer:
.55M + 75 = the total cost
.55(300) + 75 = $240
Step-by-step explanation:
What is 10x5??? Help
Answer:
The actual Answer is 10x5=50
Answer: 50
Step-by-step explanation: it’s like when you get 10 and add it up 5 times you get 50
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Which inequality is represented by this graph?
A. x>-53
B. x_<- 53
C. X<-53
D. x_>-53
Answer:
x ≥ -53
Step-by-step explanation:
There is a closed circle at -53 which means equal to
The line goes to the right which means greater than
x ≥ -53
Answer:
D) x>=-53
Step-by-step explanation:
A closed dot means -53 is included in the inequality as a solution, and since it's going right, it's greater than or equal to -53.
Angeline bought s yards of satin fabric priced at $8.09 per yard, and c yards of cotton fabric priced at $3.79 per yard. Use an expression to determine the amount that Angeline would spend in all if she bought 5 yards of satin fabric and 8 yards of cotton fabric. *
Answer:
$70.77
Step-by-step explanation:
A=8.09s+3.79c
A=8.09(5)+3.79(8)
A=40.45+30.32
A=70.77
Answer:
$70.77
Step-by-step explanation:
5s+8c=?
5(8.09) + 8(3.79) = ?
40.45 + 30.32 = ?
$70.77
Finish the following proof for Theorem 1.4.12. Assume B is a countable set. Thus, there exists f : N -+ B, which is 1-1 and onto. Let A ~ B be an infinite subset of B. We must show that A is countable. Let nI = min{n EN: f(n) E A}. As a start to a definition of g: N -+ A, set g(l) = f(nI). Show how to inductively continue this process to produce a 1-1 function 9 from N onto A.
Answer:
A is a countable set
Step-by-step explanation:
Assume B is a countable set, Go through the attached file for a detailed step by step explanation of the proof that A is a countable set.
a whole number is squared. the result is between 200 and 260. the number is between?
Final answer:
The whole number squared resulting in a number between 200 and 260 must have the original whole number between 15 and 16.
Explanation:
If a whole number is squared and the result is between 200 and 260, we must determine which whole numbers will yield a square within that range. We find the square roots of both 200 and 260 to identify the range of whole number bases. The square root of 200 is approximately 14.14, and the square root of 260 is approximately 16.12. Therefore, the number must be greater than 14 but less than or equal to 16 because squaring whole numbers greater than 16 gives results exceeding 260. Hence, the number is between 15 and 16.
Jonas earns $77 in 7 hours. At this rate, how many dollars will he earn in 20 hours?
Answer:
$220?
Step-by-step explanation:
Answer$220
Step-by-step explanation:
77/7=$11 so $11x20=220
How do I find the area of a circle with the diameter of 4
Answer:
use [tex]a = 1/4\pi d^2\\[/tex]
all you have to do is plug 4 into d and then square it, and multiply everything together.
Step-by-step explanation:
Help me! ? Its math and I I bad at it!!!!!!!
Answer:
C. 31
Step-by-step explanation:
(570.7 - n) ÷ 16
(570.7 - 74.7) ÷ 16
496 ÷ 16
31
Michaela says that |3| is bigger than |-3|.
Is Michaela’s statement true or false?
Answer:
False
Step-by-step explanation:
Because with the lines on the outside automatically changes the number to a positive number so, therefor the numbers are equal hope this help.
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Michaela's statement that |3| is bigger than |-3| is true.
Explanation:Michaela's assertion that |3| is greater than |-3| is accurate. The absolute value of a number represents its distance from zero on the number line. In this case, |3| signifies a 3-unit distance from zero, and |-3| also corresponds to a 3-unit distance from zero. Since both values are equidistant from zero, their magnitudes are equal. Consequently, |3| and |-3| both have a value of 3, emphasizing that the absolute value of a number signifies its distance from zero, regardless of its sign, and in this case, both distances are identical at 3 units.
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The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. Step 2 of 2 : Suppose a sample of 1291 tenth graders is drawn. Of the students sampled, 1098 read above the eighth grade level. Using the data, construct the 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level. Round your answers to three decimal places.
Answer:
The 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is (0.13, 0.168).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
1291 tenth graders, 1098 read above the eighth grade level.
1291 - 1098 = 193 read at or below this level.
We want the 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level.
So [tex]n = 1291, \pi = \frac{193}{1291} = 0.149[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.149 - 1.96\sqrt{\frac{0.149*0.851}{1291}} = 0.13[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.149 - 1.96\sqrt{\frac{0.149*0.851}{1291}} = 0.168[/tex]
The 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is (0.13, 0.168).