Let f(x) = ax + b and g(x) = cx^2 + dx, where a, b, c, and d are constants. Compute f ◦ g and g ◦ f. Determine for which constants a, b, c, and d it is true that f ◦ g = g ◦ f. (Hint: Note that polynomials dnx n + dn−1x n−1 + · · · + d1x + d0 and enx n + en−1x n−1 + · · · + e1x + e0 are equal as functions if and only if dn = en, dn−1 = en−1, . . . , d1 = e1, d0 = e0.)

Answer:

x=-2

Step-by-step explanation:

Divide 14 by -7, then plug in the result(-2) in place of x.  Run this new equation(-7*-2=14) through your calculator to be sure it's correct.

Answer:

(1. 8/12 and 16/24)  (2.6/24 and 1/4)  (3.1/2and 6/12)   (4.12/32 and 3/8)   (5. 25) 6 is 3   7. is 21 .  8. is 2   9. is 1/5  10. is 1/3 . 11. is 1/4 .  12. is 3/5

Step-by-step explanation:

can i please get brainliest? thanks!

hope this helps

ANSWER:

The Pythagorean theorem is used to find the hypotenuse of a triangle.

=> AC² = AB² + BC²

In squares, the length of the side can be calculated by finding the square root of the area.

Answer:

A?

Step-by-step explanation:

It's super tiny

Answer

definitely C.  If two sentences have the same truth value, then they are equivalent.

Step-by-step explanation:

Answer:

7

Step-by-step explanation:

This is so because (+) × (-) is (-).

Just like (+) × (+) is (+) and (-) × (-) is (-).

So 10 + (-3) = 10 - 3 = 7.

Answer:

\(7\)

Step-by-step explanation:

\(10 + ( - 3) \\ \\ (+) \: \times \: (- )= (- ) \\ so \: 10 - 3 \\ = 7\)

we need to solve the inequality 7x - 8 ≤ -15 and graph the solution on the number line.

To solve the inequality, we start by isolating the variable x. We add 8 to both sides of the inequality, which gives us 7x ≤ -7. Then, we divide both sides by 7 to obtain x ≤ -1.

The solution to the inequality is x ≤ -1. On the number line, we represent this solution by shading all values to the left of and including -1. This indicates that any value of x that is less than or equal to -1 satisfies the inequality.

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\(x^2+10x+22=13\\x^2+10x+22-13=0\\x^2+10x+9=0\\\)

Now completing the square.

\((x^2+10x)+9=0\\(x^2+10x+25)+9-25=0\\(x+5)^2-16=0\\(x+5)^2=16\)

(x+5)² = 16 is the rewritten equation... For first question btw.

For the solutions to the equation -

\(x+5=\sqrt{16},-\sqrt{16} \\x+5=4,-4\\x=4-5,-4-5\\x=-1,-9\)

x = -1,-9 is the solution, for the second question.

The probability that the sum of the two is even and at least one of the dice shows 5 is 1/18.

There are a total of 36 possible outcomes when you roll two dice, since each die has 6 possible outcomes.

To find the number of outcomes where the sum of the dice is even, we can divide the 36 outcomes into two groups: outcomes where the sum is even, and outcomes where the sum is odd. There are 18 even sums (2, 4, 6, 8, 10, 12) and 18 odd sums (1, 3, 5, 7, 9, 11). So the probability that the sum of the dice is even is 18/36, or 1/2.

Now, we need to find the number of outcomes where at least one of the dice shows 5. There are 4 outcomes where the first die shows 5 (5,5; 5,1; 5,2; 5,3) and 4 outcomes where the second die shows 5 (1,5; 2,5; 3,5; 4,5). There are also 4 outcomes where both dice show 5. However, we have counted each of these outcomes twice (once as an outcome where the first die shows 5, and once as an outcome where the second die shows 5). So we need to subtract these 4 outcomes from our total. This leaves us with a total of 8 outcomes where at least one of the dice shows 5.

So the probability that at least one of the dice shows 5 is 8/36, or 2/9.

Finally, we want to find the probability that the sum of the dice is even and at least one of the dice shows 5. Since these two events are independent (the fact that the sum is even does not affect the probability that one of the dice shows 5, and vice versa), we can find the probability of both events occurring by multiplying the probabilities of the individual events.

So the probability that the sum of the dice is even and at least one of the dice shows 5 is (1/2) * (2/9) = 1/18.

Therefore the answer is: 1/18.

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There are approximately 0.976 grams of carbon (C) in the given sample of the carbohydrate (C9H11O6).

To find the number of grams of carbon (C) in the sample, we need to first determine the molar mass of the carbohydrate.

The molar mass of C is approximately 12.01 g/mol. The molar mass of H is approximately 1.01 g/mol, and the molar mass of O is approximately 16.00 g/mol.

The molar mass of the carbohydrate (C9H11O6) can be calculated as follows:

(9 * 12.01 g/mol) + (11 * 1.01 g/mol) + (6 * 16.00 g/mol) = 162.18 g/mol

Next, we can use the Avogadro's number (6.022 x 10^23) to determine the number of moles in the sample:

4.9 x 10^22 molecules / 6.022 x 10^23 molecules/mol = 0.0813 mol

Finally, we can calculate the mass of carbon in the sample by multiplying the number of moles by the molar mass of carbon:

0.0813 mol * 12.01 g/mol ≈ 0.976 g

So, there are approximately 0.976 grams of carbon in this sample.
There are approximately 0.976 grams of carbon (C) in the given sample of the carbohydrate (C9H11O6).

To find the mass of carbon in the sample, we first need to determine the molar mass of the carbohydrate. The molar mass of an element or compound is the mass of one mole of that substance. In this case, we have the formula C9H11O6, which indicates that the carbohydrate consists of 9 carbon atoms (C), 11 hydrogen atoms (H), and 6 oxygen atoms (O).

The molar mass of carbon is approximately 12.01 g/mol, hydrogen is approximately 1.01 g/mol, and oxygen is approximately 16.00 g/mol. To calculate the molar mass of the carbohydrate, we multiply the number of atoms of each element by their respective molar masses and sum them up.

Therefore, the molar mass of C9H11O6 is

(9 * 12.01 g/mol) + (11 * 1.01 g/mol) + (6 * 16.00 g/mol) = 162.18 g/mol.

Next, we can use the Avogadro's number (6.022 x 10²³) to determine the number of moles in the given sample. The given sample contains 4.9 x 10²²molecules of the carbohydrate. Dividing this by Avogadro's number gives us the number of moles:

4.9 x 10 molecules / 6.022 x 10^23 molecules/mol = 0.0813 mol.

Finally, we can calculate the mass of carbon in the sample by multiplying the number of moles by the molar mass of carbon: 0.0813 mol * 12.01 g/mol ≈ 0.976 g. Therefore, there are approximately 0.976 grams of carbon in the given sample.

In conclusion, the sample of the carbohydrate (C9H11O6) contains approximately 0.976 grams of carbon (C).

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The number of doctors that would be needed for the 35-bed wing is 38.

The first step is to determine the number of doctors needed for one bed : 463 / 423

The second step is to multiply the ratio gotten in the first step by the number of beds in the new wing: (463 / 423) x 35 = 38.21 = 38

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well, from 1990 to 1998 is 8 years, and we know the amount went from $4800 to $6300, let's check for the rate of growth.

\(\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$6300\\ P=\textit{initial amount}\dotfill &\$4800\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{years}\dotfill &8\\ \end{cases} \\\\\\ 6300=4800(1 + \frac{r}{100})^{8} \implies \cfrac{6300}{4800}=(1 + \frac{r}{100})^8\implies \cfrac{21}{16}=(1 + \frac{r}{100})^8\)

\(\sqrt[8]{\cfrac{21}{16}}=1 + \cfrac{r}{100}\implies \sqrt[8]{\cfrac{21}{16}}=\cfrac{100+r}{100} \\\\\\ 100\sqrt[8]{\cfrac{21}{16}}=100+r\implies 100\sqrt[8]{\cfrac{21}{16}}-100=r\implies \stackrel{\%}{3.46}\approx r\)

now, with an initial amount of $4800, up to 2007, namely 17 years later, how much will that be with a 3.46% rate?

\(\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &4800\\ r=rate\to 3.46\%\to \frac{3.46}{100}\dotfill &0.0346\\ t=years\dotfill &17\\ \end{cases} \\\\\\ A=4800(1 + 0.0346)^{17} \implies A=4800(1.0346)^{17}\implies A \approx 8558.02\)

Answer:

14+ 8n<40

Hope this helps

Answer:

volume = 31.5w  

Step-by-step explanation:

A rectangular prism is a 3 dimensional figure that have 6 faces that are rectangular in shape. To ascertain the volume of a rectangular prism you need to know the length, width and height. The formula for the volume of a rectangular prism is represented below .

volume of a rectangular prism = LHW

where

L= length

H = height

W = width

volume = LHW

L = 7.5 ft

W = w ft

H = 4.2 ft

The expression for the volume of a rectangular prism with the following value above can be represented below.

volume = 4.2 × 7.5 × w

volume = 31.5w  

f(x) = \(4\sqrt[3]{x} - 8\)

Let's replace f(x) with y

y = \(4\sqrt[3]{x} - 8\)

Switch the x and y

x = \(4\sqrt[3]{y} - 8\)

Now, let's solve for y. Start off by adding 8 to both sides.

x + 8 = \(4\sqrt[3]{y}\)

Divide both sides by 4

\(\frac{x}{4} + 2\) = \(\sqrt[3]{y}\)

Take each side by the power of 3

\((\frac{x}{4} + 2)^{3}\) = y

Now, let's replace y with \(f^{-1}(x)\)

\(f^{-1}(x)\) = \((\frac{x}{4} + 2)^{3}\) is the inverse

A 40 gram sample of a substance that’s used for drug research has a k-value of 0.1472. The substance's half-life, in days, is approximately 4.7 days.

The half-life of a substance is the time it takes for half of the substance to decay or undergo a transformation. The half-life can be determined using the formula:

t = (0.693 / k)

where t is the half-life and k is the decay constant.

In this case, we are given that the sample has a k-value of 0.1472. We can use this value to calculate the half-life.

t = (0.693 / 0.1472) ≈ 4.7 days

Therefore, the substance's half-life, rounded to the nearest tenth, is approximately 4.7 days.

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Answer:

----------------

The longest straight line is the line connecting the two opposite corners.

This is a diagonal of the rectangle.

Use Pythagorean theorem to find the length of the diagonal, the hypotenuse of the triangle with legs 20 in and 15 in:

Hence the longest straight line is 25 in long.

First scenario: The polynomial function that satisfies the given conditions is f(x) = (x - 3)(x^2 + 4). The real zeros are x = 3, and the complex zeros are x = 2i and x = -2i. The function value f(1) = -10 is also satisfied.

Second scenario: The specific polynomial function is not provided, but it will have real coefficients and the zeros x = -3, x = 8 + 4i, and x = 8 - 4i. The function value f(1) = 260 can be confirmed using a graphing utility.

To find an nth-degree polynomial function with real coefficients that satisfies the given conditions, we can use the fact that complex zeros occur in conjugate pairs.

In the first scenario, we are given that n = 3, and the zeros are 3 and 2i. Since complex zeros occur in conjugate pairs, we know that the third zero must be -2i. We are also given that f(1) = -10.

Using this information, we can construct the polynomial function. Since the zeros are 3, 2i, and -2i, the polynomial must have factors of (x - 3), (x - 2i), and (x + 2i). Multiplying these factors, we get:

f(x) = (x - 3)(x - 2i)(x + 2i)

Expanding and simplifying this expression, we find:

f(x) = (x - 3)(x^2 + 4)

To verify the real zeros and the given function value, we can graph this function using a graphing utility. The graph will show the x-intercepts at x = 3, x = 2i, and x = -2i. Additionally, substituting x = 1 into the function will yield f(1) = -10, as required.

In the second scenario, we are given that n = 3 and the zeros are -3 and 8 + 4i. Again, since complex zeros occur in conjugate pairs, we know that the third zero must be 8 - 4i. We are also given that f(1) = 260.

Using this information, we can construct the polynomial function. The factors will be (x + 3), (x - (8 + 4i)), and (x - (8 - 4i)). Multiplying these factors, we get:

f(x) = (x + 3)(x - (8 + 4i))(x - (8 - 4i))

Expanding and simplifying this expression may be more cumbersome due to the complex numbers involved, but the resulting polynomial will have real coefficients.

To verify the real zeros and the given function value, we can graph this function using a graphing utility. The graph will show the x-intercepts at x = -3, x = 8 + 4i, and x = 8 - 4i. Substituting x = 1 into the function should yield f(1) = 260, as required.

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Answer:

Step-by-step explanation:

Given:

n= Total number of random sample of people taken=400

x= Number of people who provided yes responses= 124

P= The point of estimate of the sample of the population portion taken=?

Now,

P=x/n

P=124/400

P= 124/4×100

P= 32/100

P= 0.32 or 32%

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0.31 or 31% is the point estimate of the proportion of the population that would provide yes responses

Estimate of Proportion: If we divide the random variable, the mean, and the standard deviation by n, we get a normal distribution of proportions with P′, called the estimated proportion, as the random variable. (Recall that a proportion as the number of successes divided by n.)

a simple random sample of 400 individuals provides 124 yes responses

The point estimate of individuals that would provide yes responses is the sample proportion.

n= Total number of random sample of people taken=400

x= Number of people who provided yes responses= 124

P= The point of estimate of the sample of the population portion taken

The sample proportion is calculated by dividing number of yes responses by sample size

p = x/n = 124/400= 0.31 or 31%

0.31 or 31% is the point estimate of the proportion of the population that would provide yes responses

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There are two main subtypes of quantitative data, which are discrete data and continuous data. Each subtype has different techniques for analysis.

1. Discrete data: This type of data represents whole numbers or counts that can only take specific values. Examples include the number of students in a class or the number of cars in a parking lot. Techniques used to analyze discrete data include:
  a. Frequency distribution: A summary of the number of times each value appears in the dataset.

 b. Measures of central tendency: Calculating the mean, median, and mode of the dataset to understand its central point.

 c. Measures of dispersion: Assessing the range, variance, and standard deviation to understand the spread of the data.

2. Continuous data: This type of data represents measurements that can take any value within a range, such as height, weight, or temperature. Techniques used to analyze continuous data include:

 a. Histogram: A graphical representation of the distribution of the data, using bars to represent the frequency of values within specific intervals.

  b. Probability density function: A mathematical function that describes the probability of a continuous random variable falling within a specific range.

  c. Regression analysis: A statistical method for analyzing the relationship between a dependent variable and one or more independent variables.

Both discrete and continuous data can also be analyzed using inferential statistics, such as hypothesis testing and confidence intervals, to make inferences about a population based on a sample of the data.

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In a Receiver Operating Characteristic (ROC) curve, the x-axis typically represents the False Positive Rate (FPR), and the y-axis represents the True Positive Rate (TPR).

The False Positive Rate (FPR) is the proportion of negative instances that are incorrectly classified as positive. It is calculated as:

FPR = FP / (FP + TN)

where FP represents the number of false positives (negative instances incorrectly classified as positive) and TN represents the number of true negatives (correctly classified negative instances).

The True Positive Rate (TPR), also known as Sensitivity or Recall, is the proportion of positive instances that are correctly classified as positive. It is calculated as

TPR = TP / (TP + FN)

where TP represents the number of true positives (correctly classified positive instances) and FN represents the number of false negatives (positive instances incorrectly classified as negative).

Therefore, the labels on the x-axis and y-axis in an ROC curve indicate the False Positive Rate (FPR) and True Positive Rate (TPR), respectively.

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At least 75% of the flights (or 15 out of the 20 flights) will last between 3 days and 11 days, according to Chebyshev's Theorem.

Chebyshev's Theorem states that for any given number k greater than 1, at least (\(1-\frac{1}{k^2}\)) of the data values in any data set will fall within k standard deviations of the mean.

In this case, we can use Chebyshev's Theorem to determine the minimum number of flights that lasted between 3 and 11 days.

Given:

Mean (μ) = 7 days

Standard Deviation (σ) = 2 days

To find the number of flights within the range of 3 to 11 days, we need to calculate how many standard deviations away from the mean these values are.

Lower Bound:

Value = 3 days

Number of standard deviations away from the

\(mean = \frac{(Value - Mean)}{ Standard Deviation}\)

\(mean =\frac{(3 - 7) }{2}\)

\(mean =\frac{-4}{2}\)

\(mean = -2\)

Upper Bound:

Value = 11 days

Number of standard deviations away from the

\(mean = \frac{(Value - Mean)}{Standard Deviation}\)

\(mean = \frac{(11 - 7)}{2}\)

\(mean = \frac{4}{2}\)

\(mean = 2\)

According to Chebyshev's Theorem, the minimum proportion of data values within k standard deviations of the mean is given by \((1- \frac{1}{k^2} )\).

So, we need to determine the proportion of data within 2 standard deviations, which is k = 2.

Proportion within 2 standard deviations = \(1-\frac{1}{2^2}\)

\(=1-\frac{1}{4}\)

\(= 1 - 0.25\)

\(= 0.75\)

Now, we can find the percentage of \(0.75\):

\(= 0.75\times 100\)

\(= 75\%\)

Therefore, at least 75% of the flights (or 15 out of the 20 flights) will last between 3 days and 11 days, according to Chebyshev's Theorem.

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The statement is false because in a Poisson distribution, the probability of success does not vary from trial to trial.

The Poisson distribution is a discrete probability distribution that describes the number of times an event occurs in a fixed time interval or spatial region, given the average rate of occurrence and assuming that the events are independent and random.

The Poisson distribution has only one parameter, λ, which represents the average rate of occurrence of the event.

The probability of observing k events in the interval is given by the Poisson probability mass function:

P(k) = (e^(-λ) * λ^k) / k!

where e is the base of the natural logarithm, and k is a non-negative integer.

The Poisson distribution assumes that the probability of observing an event at any given point in time or space is constant, and that the events are independent of each other.

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Answer:

40/3

\(15 \times x = 200 \\ 15x = 200 \\ x = 200 \div 15 \\ x = \frac{40}{3} \)

The painter should work at least 13.33 hours to earn $200.

An inequality is a mathematical statement showing the non-equal comparison between two expressions.

Given that, a painter earns $15 per hour. We need to find the minimum number of hours he must work to earn at least $200,

So, let the minimum number of hours be x,

15x ≥ 200

This will be the inequality,

To find the minimum number of hours we will find the value of x,

15x ≥ 200

Divide by 15

x ≥ 200/15

x ≥ 13.33

Hence, the painter should work at least 13.33 hours to earn $200.

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Step-by-step explanation:

the area of a parallelogram is

baseline × height

to multiply (like the other answer) just length and width, that only works for real rectangles, as then due to the 90° angles the length is also the height to the width and vice versa.

so the area here is

12 × 12 = 144 in²

Answer:

Answer = tan 3pi/10

Step-by-step explanation:

Hope that helps! :)

Our final rational function becomes: g(x) =\([(x + 4)(ax + b)(x + 3)^2(x + 1)] / [(x + 4)(cx + d)(x - 1)^2]\)

To create a rational function g(x) that satisfies the given properties, we can start by considering the horizontal asymptote and the hole.

Given that the horizontal asymptote is y = 0, we know that the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator.

Considering the hole at (-4, -3/19), we can introduce a factor of (x + 4) in both the numerator and denominator to cancel out the common factor. This will create a hole at x = -4.

So far, we have:

g(x) = [(x + 4)(ax + b)] / [(x + 4)(cx + d)]

Next, let's consider the local minimum at (-3, -1/6) and the local maximum at (1, 1/2).

To ensure a local minimum at x = -3, we can make the factor (x + 3) squared in the denominator, so that it does not cancel out with the numerator. We can also choose a positive coefficient for the factor in the numerator to create a downward-facing parabola.

To ensure a local maximum at x = 1, we can make the factor (x - 1) squared in the denominator, and again choose a positive coefficient for the factor in the numerator.

Adding these factors, we have:

g(x) =\([(x + 4)(ax + b)(x + 3)^2] / [(x + 4)(cx + d)(x - 1)^2]\)

Finally, we consider the x-intercept at x = -1 and the y-intercept at y = 1/3.

To achieve an x-intercept at x = -1, we can set the factor (x + 1) in the numerator.

To achieve a y-intercept at y = 1/3, we set the numerator constant to 1/3.

Multiplying these factors, our final rational function becomes:

g(x) = \([(x + 4)(ax + b)(x + 3)^2(x + 1)] / [(x + 4)(cx + d)(x - 1)^2]\)

Where a, b, c, and d are coefficients that can be determined by solving a system of equations using the given properties.

Please note that without additional information or constraints, there are multiple possible rational functions that can satisfy these properties. The function provided above is one possible solution that meets the given conditions.

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1) 2 1/3 is also equal to 7/3

4 1/5 is also equal to 21/5

you need to make the denominators the same so you have to find the lowest common multiple, which is 15 since both 5 and 3 can be times by to make 15

you need to also times the numerators by the number you're timesing the denominator by

7/3 x 5 = 35/15

21/5 x 3 = 63/15

now add those together

35/15 + 63/15 = 98/15

now just convert 98/15 back to a mixed fraction

98/15 = 6 8/15

the answer is..

\(6 \frac{8}{15} \)

I'll do the others in a different answer!!

The number of weeks needed to get $850 is 21 weeks.

An equation is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe a scenario.

Let the number of weeks be represented by w

Amount saved = $24

Amount needed = $850

Number of weeks = w.

This will be illustrated as:

24 + 40w = 850

Collect like terms

40w = 850 - 24

40w = 826

Divide

w = 826 / 40

w = 20.65 = 21 approximately

There'll be 21 weeks.

The complete question is written below.

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You have saved $24 towards the purchase of your own laptop. you get a job at the corner store cleaning after hours on sundays that pays $40 per week. the laptop you want costs $850 and includes everything you'll need to start college. How may weeks are needed?

We can say with 95% confidence that the true proportion of checking account customers who also have savings accounts in the population lies between 0.25 and 0.35.

a. To find the sample proportion of checking account customers who also have savings accounts, we need to divide the number of customers who have both types of accounts by the total number of checking account customers in the sample. Let's say the bank selected 500 checking account customers and found that 150 of them also had savings accounts. Then, the sample proportion would be:

150/500 = 0.3

So, 30% of the checking account customers in the sample also had savings accounts.

b. To find the standard error of the sample proportion, we use the formula:

SE = sqrt(p*(1-p)/n)

where p is the sample proportion (0.3 in this case), and n is the sample size (500). Plugging in the numbers, we get:

SE = sqrt(0.3*(1-0.3)/500) = 0.025

So, the standard error is 0.025.

c. To find a 95% confidence interval for the population proportion of checking account customers who also have savings accounts, we use the formula:

CI = p ± z*(SE)

where z is the z-score corresponding to a 95% confidence level (which is 1.96), and SE is the standard error we calculated in part b. Plugging in the numbers, we get:

CI = 0.3 ± 1.96*(0.025) = (0.25, 0.35)

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