Use differentiation rules to find the values of a and b that make the function f(x) = ( x 2 if x ≤ 2, ax3 + bx if x > 2 differentiable at x = 2.
To make the function differentiable at x = 2, we must ensure continuity by matching function values and differentiability by equating derivatives from both sides at x = 2. Solving the system of equations obtained from these conditions will give the values of a and b.
Explanation:To find the values of a and b that make the function f(x) = { x² if x ≤ 2, ax³ + bx if x > 2 } differentiable at x = 2, we need to ensure both continuity and differentiability of f(x) at this point. First, continuity at x = 2 requires that the limits from the left and the right are the same, meaning f(2) = 2² = 4 should equal a(2)³ + b(2). Second, for differentiability, the derivatives from the left and right at x = 2 must also be equal. The derivative of x² is 2x, so f'(2) = 4. Differentiating ax³ + bx gives 3ax² + b, so f'(2) = 12a + b must also equal 4. Solving these equations:
4 = 8a + 2b4 = 12a + bgives us a system of equations that when solved, will provide the exact values of a and b required.
PLEASE HELP!!!!
WILL GIVE BRAINLY
Which statements are true?
Select each correct answer.
9g3+12=3(3g3+4)
35g5−25g2=5g2(7g3−5)
24g4+18g2=6g2(4g2+3g)
4g2−g=g2(4−g)
Rate data often follow a lognormal distribution. Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parameters μ = 4 and σ = 2. What is the probability that the company uses more than 270 dB during any particular hour?
The probability that the company uses more than 270 dB during any particular hour is approximately 0.8963 or 89.63%.
To find the probability that the company uses more than 270 dB during any particular hour, we need to use the properties of the lognormal distribution.
The lognormal distribution is characterized by two parameters: μ (mean of the logarithm of the data) and σ (standard deviation of the logarithm of the data).
μ = 4
σ = 2
To find the probability of the company using more than 270 dB, we need to convert this value to the logarithmic scale and then calculate the corresponding probability.
Convert 270 dB to the logarithmic scale:
Let X be the random variable following a lognormal distribution with parameters μ and σ. The logarithm of X is normally distributed with mean μ and standard deviation σ.
Using the lognormal properties, we can convert 270 dB to the logarithmic scale:
ln(270) = 5.5984 (approximately)
Calculate the probability of X being greater than ln(270):
We now need to find the probability of X being greater than ln(270) in the lognormal distribution with parameters μ = 4 and σ = 2.
Using statistical software, a lognormal distribution table, or a calculator, we find the probability P(X > ln(270)) to be approximately 0.8963.
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To find the probability that the company uses more than 270 dB during any particular hour, we can use the properties of the lognormal distribution. The lognormal distribution is characterized by its parameters μ and σ, where μ is the mean and σ is the standard deviation of the natural logarithm of the variable. In this case, μ = 4 and σ = 2. The probability is approximately 0.227.
Explanation:To find the probability that the company uses more than 270 dB during any particular hour, we can use the properties of the lognormal distribution. The lognormal distribution is characterized by its parameters μ and σ, where μ is the mean and σ is the standard deviation of the natural logarithm of the variable. In this case, μ = 4 and σ = 2.
To find the probability, we can convert the value of 270 dB to its corresponding value on the lognormal distribution. Let's call this value x. Using the formula x = e^μ + σ^2/2, we have x = e^4 + 2^2/2 = 54.598.
Now, we can use the cumulative distribution function (CDF) of the lognormal distribution to find the probability that the company uses more than 270 dB. The CDF gives us the probability that the variable is less than or equal to a given value.
Since we want the probability that the variable is greater than 270 dB, we can subtract the CDF value from 1. Using a calculator or a statistical software, we can find that the CDF of the lognormal distribution at x = 54.598 is approximately 0.773. Therefore, the probability that the company uses more than 270 dB during any particular hour is 1 - 0.773 = 0.227.
A blueprint shows a house with two fences. Fence A is 1 4/5 inches long on the blueprint and is to be 1 1/2 feet long. How long is Fence B on the blueprint?
The problem is about finding a length on a blueprint using scale factor. A ratio is established between blueprint measurements and real-life measurements. However, information about Fence B's actual size is needed to calculate its blueprint length.
Explanation:The subject of this question is mathematics, more specifically proportionality and scale factor. To solve this problem, you need to establish the ratio or scale represented by the blueprint to the actual size. In the given problem, 1 4/5 inches on the blueprint represents 1 1/2 feet in real life. First, convert the measures to improper fractions to easily manage the computations. Hence, 1 4/5 becomes 9/5 inches and 1 1/2 feet becomes 3/2 feet. The scale becomes 9/5 inches:3/2 feet on blueprint:real life. Now, we need additional information about the actual size of Fence B to calculate its length on the blueprint.
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Marc is decorating 60 cupcakes for a school fund-raiser. He starts working at 1:00 P.M. In the afternoon, first setting up his supplies and then starting to decorate. At 1:05 P.M. He has 5 cupcakes decorated. At 1:08 P.M. He has 10 cupcakes decorated. If he decorates cupcakes at a constant rate, at what time that afternoon will he finish decorating the 60 cupcakes?
Answer:
1.38 pm.
Step-by-step explanation:
After 1.05 PM the rate at which he decorates the cakes is 10 - 5 = 5 cakes per 3 minutes = 5/3 cakes per minute. At 1.08 pm he has made 10 cupcakes.
At 1.08 he has to decorate another 50 cakes so the time he will take to do these is 50 / 5/3 = 50 * 3/5 = 30 minutes.
So the time when he finishes 60 cupcakes is 1.08 + 30
= 1.38 pm.
The 60 cupcakes will be decorated at 1.38 pm.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
After 1.05 PM the rate at which he decorates the cakes is 10 - 5 = 5 cakes per 3 minutes = 5/3 cakes per minute. At 1.08 pm he made 10 cupcakes.
At 1.08 he has to decorate another 50 cakes so the time he will take to do these is 50 / 5/3 = 50 * 3/5 = 30 minutes.
The time when he finishes 60 cupcakes is:-
1.08 + 30 = 1.38 pm.
Therefore, 60 cupcakes will be decorated at 1.38 pm.
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4. Corrupt professor Z has a class of 50 students. He needs to give exactly 10 A's. However five students already have a special deal (they are professor Z's nephews and nieces) and will get A's for sure. How many ways can the 10 A's be distributed?
Answer:
Step-by-step explanation:
Given that the professor Z has a class of 50 students is corrupt. However five students already have a special deal (they are professor Z's nephews and nieces) and will get A's for sure.
Thus out of 10 students 5A's are reserved
Remaining 5 can be distributed in
I 5 to any one of the 45, II to any one of the 44....
i.e. 45P5 ways
no of ways = 45P5 == 146611080
Please Help! I don't know where to start with this, please show all work!
What is the vertex form of the equation?
y = -x^2 + 12x - 4
Hey!
-------------------------------------
Formula's:
x_v = -b/2a
y_v = ax² + bx + c
-------------------------------------
Parabola Params:
a = -1
b = 12
c = 4
-------------------------------------
Solve for [tex]x_{v}[/tex]
[tex]x_{v} = \frac{-b}{2a}[/tex]
[tex]x_{v} = \frac{-12}{2(-1)}[/tex]
[tex]x_{v} = -6[/tex]
-------------------------------------
Solve for [tex]y_{v}[/tex]
Use -6 for [tex]x_{v}[/tex]
[tex]y_{v} = -6^2 + 12(6) - 4\\[/tex]
[tex]y_{v} = -36 + 72 - 4[/tex]
[tex]y_{v} = 32[/tex]
-------------------------------------
Answer:
(6, 32)
-------------------------------------
Hope This Helped! Good Luck!
-7y-4x=1
7y-2x=53
solve for system of equations
Answer:
x = -9
y = 35/7
Step-by-step explanation:
Given equations :-
-7y - 4x = 1 ....... ( i )
7y - 2x = 53 ........ ( ii )
From ( i )
-7y = 1 + 4x
[tex]y = \frac{ - (1 + 4x)}{7} [/tex]
..........( iii )
From ( ii )
7y - 2x = 53
7y = 53 + 2x
[tex]y = \frac{53 + 2x}{7} [/tex]
.......( iv )
Equating both ( iii ) & ( iv )
y = y
[tex] \frac{ - (1 + 4x)}{7} = \frac{53 + 2x}{7} [/tex]
-(1 + 4x ) = 53 + 2x
-1 -4x = 53 + 2x
-1 - 53 = 2x + 4x
-54 = 6x
-54/6 = x
-9 = x
Also,
[tex]y = \frac{ - (1 + 4x)}{7 } \\ \\ y = - \frac{(1 + 4( - 9))}{7} [/tex]
[tex]y = - \frac{ (1 - 36)}{7} \\ \\ y = - \frac{( - 35)}{7} \\ \\ y = \frac{35}{7} [/tex]
| A retiree invests $8,000 in a savings plan that pays
4% per year. What will the account balance be at
the end of the first year?
Answer:
8320
Step-by-step explanation:
(100%+4%)8000=8320
The balance at the end of the first year of the investment will be $8320, which includes the initial investment of $8000 and the interest earned, calculated as 4% of the initial investment.
Explanation:This is a basic interest problem in mathematics. To calculate the balance at the end of the first year, we need to add the initial investment to the amount earned through interest. The interest earned is the product of the initial investment and the interest rate.
In this case, the initial investment is $8000 and the interest rate is 4% per year or 0.04 in decimal form. So, the interest earned would be the product of $8000 and 0.04, which equals $320.
Therefore, the balance at the end of the first year would be the initial investment plus the interest earned, which is $8000 + $320 = $8320.
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What are the coordinates of the hole in the graph of the function?
Answer:
The answer to your question is: (5/2, -6)
Step-by-step explanation:
Given f(x) = (6x - 36) / (2x² - 17x + 30)
Factorize both, numerator and denominator
Numerator = 6(x - 6)
Denominator = 2x² - 12x - 5x + 30
= 2x(x - 6) - 5(x - 6)
= (x - 6) (2x - 5)
Now f(x) = 6(x - 6) / (x - 6) (2x - 5) Cancel (x - 6)
f(x) = 6 / 2x - 5
Find the hole 2x - 5 = 0
2x = 5
x = 5/2
In 5/2 there is a hole, in y is approximately -6
A major purpose of preparing closing entries is to a) adjust the asset accounts to their correct current balances. b) update the Retained Earnings account. c) close out the Supplies account. d) zero out the liability accounts.
Answer:
b) update the Retained Earnings account.
Step-by-step explanation:
A major purpose of preparing closing entries is to - update the Retained Earnings account.
Retained earnings are defined as those profits, that a company has earned to date minus any dividends or other money paid to investors.
Whenever we make an entry to the accounting records, that affects a revenue or expense account, this retained earning amount is adjusted.
The major purpose of preparing closing entries is to update the Retained Earnings account by transferring the balances of temporary accounts to it, effectively resetting these accounts for the next period. This ensures that the company's financial statements only reflect the transactions of the current period.
Explanation:The student's question relates to the major purpose of preparing closing entries in accounting. The correct answer to the question is b) update the Retained Earnings account. Closing entries are an essential part of the accounting cycle that serves to transfer the balances of temporary accounts (like revenues, expenses, dividends/distributions, and income summary) to the Retained Earnings account to reflect the changes that occurred over the period. This process resets the balance of the temporary accounts to zero, ready for the next accounting period, while updating the balance of the Retained Earnings to reflect the net income or loss that was earned or incurred during the period.
'T-accounts' help visualize the transactions and balances of accounts including assets, liabilities, and equity. When closing entries are made, we are not adjusting asset or liability accounts (which are permanent accounts) directly. Instead, we close temporary accounts to a permanent equity account, typically Retained Earnings, based on the principle that assets will always equal liabilities plus net worth.
The steps in preparing closing entries generally involve: first, closing all revenue accounts to Income Summary; second, closing all expense accounts to Income Summary; third, closing the Income Summary account to Retained Earnings; and lastly, closing any dividends or distributions to Retained Earnings.
PLEASE HELP What is the equation in standard form of a parabola with a focus of (-3,2) and a directrix of y=4.
Graphing y = 4 and the focus, we clearly see that the equation we need has the form (x - h)^2 = 4a(y - k).
We need to find a, h and k.
The focus is half way between the vertex and directrix.
You know that y = 4 is 2 units away from the focus and the focus is 2 units down from the focus. So, our vertex is (-3, 0).
From the vertex to the directrix, there are 4 units. Half that distance is the value of a. So, a = 2.
We have all that is needed to form our equation.
(x - (-3))^2 = 4(2)(y - 0)
(x + 3)^2 = 8y
Done.
Answer:
i think this will help (x - (-3))^2 = 4(2)(y - 0)
(x + 3)^2 = 8y
Step-by-step explanation:
An irrational number is a real number and an integer.
True
False
Answer:
A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational. The square root of 2 is an irrational number because it can't be written as a ratio of two integers
hope it helps!!
Step-by-step explanation:
Rationalize the denominator and simplify:
Answer:
∛(2b)
Step-by-step explanation:
[tex]\displaystyle\frac{\sqrt[3]{4b^2}}{\sqrt[3]{2b}}=\sqrt[3]{\frac{4b^2}{2b}}=\sqrt[3]{2b}[/tex]
Billy estimates that they will sell approximately 250 burgers thisweekend how much burger meat and fries (in pounds) should he order to be prepared for this weekend
Answer:
300 Burgers
Step-by-step explanation:
Since the question gives us an approximation of 250 burgers, we can round the number to 300. This is only because they had a estimation of 250 and would like to be prepared for the weekend. Being prepared would consist of ordering more than expected to make sure the burgers do not run out.
Keira and Javier are grouping equations based on the number of operations needed to solve them in order to solve them in order to solve the equation -5(2+x)=53
To solve the equation -5(2+x)=53, we need to use the order of operations and follow the steps mentioned in the detailed answer. The solution to the equation is x=-12.6.
Explanation:In order to solve the equation -5(2+x)=53, we need to use the order of operations. Let's break it down step by step:
First, we need to distribute the -5 to both 2 and x. This gives us -10-5x=53.Next, we combine like terms by adding 10 to both sides of the equation, which gives us -5x=63.Finally, we divide both sides of the equation by -5 to isolate the variable x. This gives us x=-12.6.So, the solution to the equation -5(2+x)=53 is x=-12.6.
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Britney is going to the candy store to buy 20 pieces of candy. She is going to buy chocolate candy and caramel candy. Each piece of chocolate candy costs 50 cents, and each piece of caramel candy costs 10 cents. You know that Britney spent $6.80 and bought 20 pieces of candy. She bought ______ pieces of chocolate.
Given the coordinates of the vertices of a quadrilateral, determine whether it is a square, a rectangle, or a parallelogram. Then find the perimeter of the quadrilateral. A(6, –4), B(11, –4), C(11, 6), D(6, 6)
Answer:
The answer to your question is: it's a rectangle
perimeter = 30 u
Step-by-step explanation:
d = √((x2.x1)² + (y2-y1)²)
Now, calculate the distances AB, BC, CD, AD
dAB = √((11-6)² + (-4+4)² = √5² = 5
dBC = √((11-11)² + (-4-6)² =√10 = 10
dCD = √((6-11)² + (6-6)² = √5² = 5
dAD = √((6-6)² + (6+4)² = √10² = 10
From the results we conclude that is a rectangle because two sides have the same length and the other two also measure the same. We can draw it to confirm this.
Perimeter = dAB + dBC + dCD + dAD
= 5 + 10+ 5 + 10 = 30 units
To determine the shape of the quadrilateral, calculate the distances between the vertices and check for equal side lengths and 90-degree angles. The given quadrilateral is a parallelogram since the opposite sides are parallel and equal. The perimeter of the quadrilateral is 30 units.
Explanation:To determine whether the given quadrilateral is a square, a rectangle, or a parallelogram, we can use the properties of each shape. A square is a quadrilateral with all sides equal in length and all angles equal to 90 degrees. A rectangle is a quadrilateral with opposite sides equal in length and all angles equal to 90 degrees. A parallelogram is a quadrilateral with opposite sides parallel and equal in length. To find the perimeter of the quadrilateral, we can calculate the sum of the lengths of all four sides.
Calculating the distances between each pair of given points:
Distance between A and B: √[(11-6)² + (-4-(-4))²] = √(5² + 0) = √25 = 5Distance between B and C: √[(11-11)² + (6-(-4))²] = √(0 + 100) = 10Distance between C and D: √[(6-11)² + (6-6)²] = √((-5)² + 0) = √25 = 5Distance between D and A: √[(6-6)² + (6-(-4))²] = √(0 + 100) = 10The distances between the vertices are: AB = 5, BC = 10, CD = 5, and DA = 10. Therefore, the perimeter of the quadrilateral is 5 + 10 + 5 + 10 = 30 units.
Write the equation of the line perpindicular to the graph of 2x-5y=0 that passes through the point (-2,3)
Answer:
5x-2y=-4
Step-by-step explanation:
2x-5y=0
eq. of line perpendicular to ax+by=c is bx-ay=d,d is calculated by the given condition.
line perpendicular to 2x-5y=0 is 5x+2y=c
if it passes through (-2,3) ,then
5*-2+2*3=c
c=-10+6=-4
reqd. eq. is 5x+2y=-4
A sample of n = 9 college students is used to evaluate the effectiveness of a new Study Skills Workshop. Each student’s grade point average (GPA) is recorded for the semester before the workshop and for the semester after the workshop. The average GPA improved by MD = 0.60 points with s2 = 0.09. The researcher would like to use the sample to estimate how much effect the workshop would have for the entire college population. Which of the following is the 80% confidence interval for these data?
A) μD = 0.60 ± 0.09( 1.860)
B) μD = 0.60 ± 0.10(1.397)
C) μD = 0.60 ± 0.01(1.397)
D) μD = 0.60 ± 0.10( 1.860)
Answer:
B) μD = 0.60 ± 0.10(1.397)
Step-by-step explanation:
The confidence interval is given by:
[tex]MD±t_{\alpha/2, n-1} \frac{s}{\sqrt{n} }[/tex]
Where
MD=60
n=9
df=n-1=8
[tex]t_{\alpha/2, n-1}=1.397[/tex]
[tex]s=\sqrt{0.09} =0.3[/tex]
Then the confidence interval is
μD=0.60±1.397*(0.3/√9)
μD=0.60±0.10*(1.397)
Three parking attendants are required to park cars for each blue lot , while two can handle each red lot. Of the 15 lots that will be used for tonight, 60% are blue and 40% are red. How many parking attendants are required for tonight's event
Answer:
[tex]39\ parking\ attendants[/tex]
Step-by-step explanation:
Let
x -----> the number of blue lots
y-----> the number of red lots
we know that
[tex]x+y=15[/tex] ------> equation A
we have that
[tex]x=0.60(15)=9\ blue\ lots[/tex]
[tex]y=0.40(15)=6\ red\ lots[/tex]
The total parking attendants required for tonight's event is equal to the product of the number of blue lots by three plus the product of the number of red lots by two
so
[tex]9(3)+6(2)=39\ parking\ attendants[/tex]
Answer:
39
Step-by-step explanation:
You estimate that you have completed 1/4 of the work your boss expects this week. What precent if your work is complete?
Answer:
You have completed 25% of your work.
Step-by-step explanation:
Consider the provided information.
You estimate that you have completed 1/4 of the work.
To find the percentage of work complete simplify the fraction and multiply it with 100.
[tex]\frac{1}{4}=0.25[/tex]
Now multiply it by 100.
[tex]0.25\times 100=25\%[/tex]
Hence, you have completed 25% of your work.
What value of x makes the equation true?
1/2 (4x−5)+5/2=10 (Btw, the numbers 1/2 and 5/2 are fractions)
A. 5
B. 8
C. 12
D. 20
Answer:
x= 5 fam trust
Step-by-step explanation:
plug it in brainliest plzzzzzzzzzzzzzzzzz
Choose all of the angles that are coterminal with -150°.
-510°
-210°
150°
210°
570°
Answer:
210, 150
Step-by-step explanation:
Answer:
-510, 210 and 570.
Step-by-step explanation:
I'm on the same assignment right now! If you draw them you will see that those are the answers.
At the end of year X, automobile installment credit accounted for 36 percent of all outstanding consumer installment credit. At that time automobile finance companies extended $57 billion of credit, or \small \frac{1}{3} of the automobile installment credit. How many billion dollars of consumer installment credit was outstanding at that time?A. 62B. 171C. 475D. 513E. 684
Answer:
option (c) 475
Step-by-step explanation:
let the automobile installment credit be 'C'
Given:.
Automobile installment credit accounted for 36 percent of all outstanding consumer installment credit
Now,
57 billion is [tex]\frac{1}{3}[/tex] of automobile installment credit
or
57 billion = [tex]\frac{1}{3}[/tex] × C
or
C = 57 × 3
or
C = 171 billion
now,
installment credit accounted = 36% of all outstanding consumer installment credit
or
171 billion = 0.36 × all outstanding consumer installment credit
or
All outstanding consumer installment credit = $475 billion
Hence, the correct answer is option (c) 475
Find P(2) using this probability distribution. Round to 2 decimal places as needed.
Answer:
P (2) = 0.1
Step-by-step explanation:
The table has X and P(X), you have to search the X and see its P(X) and thats the probability
During the year, FastDry Corporation has $ 340,000 in revenues, $ 155,000 in expenses, and $ 12,000 in dividend declarations and payments. Net income for the year was:
Answer:
$185,000
Step-by-step explanation:
We have been been given that during the year, Fast Dry Corporation has $ 340,000 in revenues, $ 155,000 in expenses, and $ 12,000 in dividend declarations and payments.
We will use following formula to solve our given problem.
[tex]\text{Net income}=\text{Revenues}-\text{Expenses}[/tex]
[tex]\text{Net income}=\$340,000-\$155,000[/tex]
[tex]\text{Net income}=\$185,000[/tex]
Therefore, the net income for the year was $185,000.
Assume that a procedure yields a binomial distribution with a trial repeated n = 20 times. Use either the binomial probability formula (or technology) to find the probability of k = 14 successes given the probability p = 0.72 of success on a single trial.
Answer:
the probability is 0.1879
Step-by-step explanation:
If a procedure yields a binomial distribution, the probability of having k successes is given by:
[tex]P(k)=nCk*p^{k} *(1-p)^{n-k}[/tex]
Where nCk is calculated as:
[tex]nCk=\frac{n!}{k!(n-k)!}[/tex]
Additionally, n is the number of trials and p is the probability of success in every trial.
Replacing, k by 14, n by 20 and p by 0.72 we get:
[tex]20C14=\frac{20!}{14!(20-14)!}=38,760[/tex]
[tex]P(k)=20C14*0.72^{14} *(1-0.72)^{20-14}[/tex]
[tex]P(k)=38,760*0.72^{14} *(1-0.72)^{20-14}\\P(k)=0.1879[/tex]
So, the probability is 0.1879
To find the probability of 14 successes given a binomial distribution with a trial repeated 20 times and a probability of success of 0.72 on a single trial, use the binomial probability formula.
Explanation:To find the probability of 14 successes given a binomial distribution with a trial repeated 20 times and a probability of success of 0.72 on a single trial, we can use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Plugging in the values, we get:
P(X = 14) = C(20, 14) * 0.72^14 * (1 - 0.72)^(20 - 14)
Calculating this expression will give you the probability of 14 successes out of 20 trials.
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What is the geometric relationship between u, minusv, and uminusv?
A. The vectors u, minusv, and uminusv form a right triangle.
B. The vectors u, minusv, and uminusv form a parallelogram whose other vertex is at 0.
C. The vectors u, minusv, and uminusv form an equilateral triangle.
D. The vectors u, minusv, and uminusv form a parallelogram whose other vertex is at uplusv.
Answer:
C. The vectors u, minusv, and uminusv form a parallelogram whose other vertex is at 0.
Step-by-step explanation:
We draw the vector u=A at the origin of a Cartesian plane. then at the same point, we draw -v = -B. To find the vector that represents u-v = A-B, straight lines are drawn parallel to each vector, forming a parallelogram. The resulting vector will be the diagonal of the parallelogram that begins at the origin of the plane.
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 14 defectives. Calculate a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool. Round the answers to 4 decimal places.
Answer: [tex](0.0228\ ,0.0706)[/tex]
Step-by-step explanation:
Given : Sample size : n= 300
The sample proportion of defectives : [tex]\hat{p}=\dfrac{14}{300}=0.0467[/tex]
Significance level for 95% confidence level =[tex]\alpha=1-0.95=0.05[/tex]
Critical z-value:[tex]z_{\alpha/2}=\pm1.96[/tex]
Confidence interval for population proportion :
[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
[tex]= 0.0467\pm (1.96)\sqrt{\dfrac{0.0467(1-0.0467)}{300}}[/tex]
[tex]\approx\ 0.0467\pm 0.0239\\\\=(0.0467-0.0239\ , \ 0.0467-0.0239)\\\\=(0.0228\ ,0.0706)[/tex]
Hence, a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool= [tex](0.0228\ ,0.0706)[/tex]