Answer:
[tex]\frac{x+6}{2x^2-9x+5}=-\sum_{n=0}^{\infty} [(-2)^{n}x^{n} + \frac{x^{n}}{5^{n+1}}][/tex]
when:
[tex]|x|<\frac{1}{2}[/tex]
Step-by-step explanation:
In order to solve this problem, we must begin by splitting the function into its partial fractions, so we must first factor the denominator.
[tex]\frac{x+6}{2x^2-9x+5}=\frac{x+6}{(2x+1)(x-5)}[/tex]
Next, we can build our partial fractions, like this:
[tex]\frac{x+6}{(2x+1)(x-5)}=\frac{A}{2x+1}+\frac{B}{x-5}[/tex]
we can then add the two fraction on the right to get:
[tex]\frac{x+6}{(2x+1)(x-5)}=\frac{A(x-5)+B(2x+1)}{(2x+1)(x-5)}[/tex]
Since we need this equation to be equivalent, we can get rid of the denominators and set the numerators equal to each other, so we get:
[tex]x+6=A(x-5)+B(2x+1)[/tex]
and expand:
[tex]x+6=Ax-5A+2Bx+B[/tex]
we can now group the terms so we get:
[tex]x+6=Ax+2Bx-5A+B[/tex]
[tex]x+6=(Ax+2Bx)+(-5A+B)[/tex]
and factor:
[tex]x+6=(A+2B)x+(-5A+B)[/tex]
so we can now build a system of equations:
A+2B=1
-5A+B=6
and solve simultaneously, this one can be solved by substitution, so we get>
A=1-2B
-5(1-2B)+B=6
-5+10B+B=6
11B=11
B=1
A=1-2(1)
A=-1
So we can use these values to build our partial fractions:
[tex]\frac{x+6}{(2x+1)(x-5)}=\frac{A}{2x+1}+\frac{B}{x-5}[/tex]
[tex]\frac{x+6}{(2x+1)(x-5)}=-\frac{1}{2x+1}+\frac{1}{x-5}[/tex]
and we can now use the partial fractions to build our series. Let's start with the first fraction:
[tex]-\frac{1}{2x+1}[/tex]
We can rewrite this fraction as:
[tex]-\frac{1}{1-(-2x)}[/tex]
We can now use the following rule to build our power fraction:
[tex]\sum_{n=0}^{\infty} ar^{n} = \frac{a}{1-r}[/tex]
when |r|<1
in this case a=1 and r=-2x so:
[tex]-\frac{1}{1-(-2x)}=-\sum_{n=0}^{\infty} (-2x)^n[/tex]
or
[tex]-\frac{1}{1-(-2x)}=-\sum_{n=0}^{\infty} (-2)^{n} x^{n}[/tex]
for: |-2x|<1
or: [tex] |x|<\frac{1}{2} [/tex]
Next, we can work with the second fraction:
[tex]\frac{1}{x-5}[/tex]
On which we can factor a -5 out so we get:
[tex]-\frac{1}{5(1-\frac{x}{5})}[/tex]
In this case: a=-1/5 and r=x/5
so our series will look like this:
[tex]-\frac{1}{5(1-\frac{x}{5})}=-\frac{1}{5}\sum_{n=0}^{\infty} (\frac{x}{5})^n[/tex]
Which can be simplified to:
[tex]-\frac{1}{5(1-\frac{x}{5})}=-\sum_{n=0}^{\infty} \frac{x^n}{5^(n+1)}[/tex]
when:
[tex]|\frac{x}{5}|<1[/tex]
or
|x|<5
So we can now put all the series together to get:
[tex]\frac{x+6}{2x^2-9x+5}=-\sum_{n=0}^{\infty} [(-2)^{n}x^{n} + \frac{x^{n}}{5^{n+1}}}[/tex]
when:
[tex]|x|<\frac{1}{2}[/tex]
We use the smallest interval of convergence for x since that's the one the whole series will be defined for.
What is the solution set of the equation x2+3*-4=6
Answer:
x=9
Step-by-step explanation:
Plz help I’ll mark you
Answer:
The option B, c2=a2+b2−2ab• cos(B) is the right answer.
When P(x) is divided by (x - 1) and (x + 3), the remainders are 4 and 104 respectively. When P(x) is divided by x² - x + 1 the quotient is x² + x + 3 and the remainder is of the form ax + b. Find the remainder.
Answer:
The remainder is 3x - 4
Step-by-step explanation:
[Remember] [tex]\frac{Dividend}{Divisor} = Quotient + \frac{Remainder}{Divisor}[/tex]
So, [tex]Dividend = (Quotient)(Divisor) + Remainder[/tex]
In this case our dividend is always P(x).
Part 1
When the divisor is [tex](x - 1)[/tex], the remainder is [tex]4[/tex], so we can say [tex]P(x) = (Quotient)(x - 1) + 4[/tex]
In order to get rid of "Quotient" from our equation, we must multiply it by 0, so [tex](x - 1) = 0[/tex]
When solving for [tex]x[/tex], we get
[tex]x - 1 = 0\\x - 1 + 1 = 0 + 1\\x = 1[/tex]
When [tex]x = 1[/tex],
[tex]P(x) = (Quotient)(x - 1) + 4\\P(1) = (Quotient)(1 - 1) + 4\\P(1) = (Quotient)(0) + 4\\P(1) = 0 + 4\\P(1) = 4[/tex]
--------------------------------------------------------------------------------------------------------------
Part 2
When the divisor is [tex](x + 3)[/tex], the remainder is [tex]104[/tex], so we can say [tex]P(x) = (Quotient)(x + 3) + 104[/tex]
In order to get rid of "Quotient" from our equation, we must multiply it by 0, so [tex](x + 3) = 0[/tex]
When solving for [tex]x[/tex], we get
[tex]x + 3 = 0\\x + 3 - 3 = 0 - 3\\x = -3[/tex]
When [tex]x = -3[/tex],
[tex]P(x) = (Quotient)(x + 3) + 104\\P(-3) = (Quotient)(-3 + 3) + 104\\P(-3) = (Quotient)(0) + 104\\P(-3) = 0 + 104\\P(-3) = 104[/tex]
--------------------------------------------------------------------------------------------------------------
Part 3
When the divisor is [tex](x^2 - x + 1)[/tex], the quotient is [tex](x^2 + x + 3)[/tex], and the remainder is [tex](ax + b)[/tex], so we can say [tex]P(x) = (x^2 + x + 3)(x^2 - x + 1) + (ax + b)[/tex]
From Part 1, we know that [tex]P(1) = 4[/tex] , so we can substitute [tex]x = 1[/tex] and [tex]P(x) = 4[/tex] into [tex]P(x) = (x^2 + x + 3)(x^2 - x + 1) + (ax + b)[/tex]
When we do, we get:
[tex]4 = (1^2 + 1 + 3)(1^2 - 1 + 1) + a(1) + b\\4 = (1 + 1 + 3)(1 - 1 + 1) + a + b\\4 = (5)(1) + a + b\\4 = 5 + a + b\\4 - 5 = 5 - 5 + a + b\\-1 = a + b\\a + b = -1[/tex]
We will call [tex]a + b = -1[/tex] equation 1
From Part 2, we know that [tex]P(-3) = 104[/tex], so we can substitute [tex]x = -3[/tex] and [tex]P(x) = 104[/tex] into [tex]P(x) = (x^2 + x + 3)(x^2 - x + 1) + (ax + b)[/tex]
When we do, we get:
[tex]104 = ((-3)^2 + (-3) + 3)((-3)^2 - (-3) + 1) + a(-3) + b\\104 = (9 - 3 + 3)(9 + 3 + 1) - 3a + b\\104 = (9)(13) - 3a + b\\104 = 117 - 3a + b\\104 - 117 = 117 - 117 - 3a + b\\-13 = -3a + b\\(-13)(-1) = (-3a + b)(-1)\\13 = 3a - b\\3a - b = 13[/tex]
We will call [tex]3a - b = 13[/tex] equation 2
Now we can create a system of equations using equation 1 and equation 2
[tex]\left \{ {{a + b = -1} \atop {3a - b = 13}} \right.[/tex]
By adding both equations' right-hand sides together and both equations' left-hand sides together, we can eliminate [tex]b[/tex] and solve for [tex]a[/tex]
So equation 1 + equation 2:
[tex](a + b) + (3a - b) = -1 + 13\\a + b + 3a - b = -1 + 13\\a + 3a + b - b = -1 + 13\\4a = 12\\a = 3[/tex]
Now we can substitute [tex]a = 3[/tex] into either one of the equations, however, since equation 1 has less operations to deal with, we will use equation 1.
So substituting [tex]a = 3[/tex] into equation 1:
[tex]3 + b = -1\\3 - 3 + b = -1 - 3\\b = -4[/tex]
Now that we have both of the values for [tex]a[/tex] and [tex]b[/tex], we can substitute them into the expression for the remainder.
So substituting [tex]a = 3[/tex] and [tex]b = -4[/tex] into [tex]ax + b[/tex]:
[tex]ax + b\\= (3)x + (-4)\\= 3x - 4[/tex]
Therefore, the remainder is [tex]3x - 4[/tex].
PLEASE HELP!!! WILL GIVE BRAINLIEST!!!!!
Step-by-step explanation:
[tex]g(x) = 3^{\frac{x}{2}}[/tex]
For [tex]x = -2[/tex], we get
[tex]g(-2) = 3^{\frac{-2}{2}} = 3^{-1} = \frac{1}{3}[/tex]
work out the area of this shape
Answer:
1000
Step-by-step explanation:
Please Help NO LINKS
[tex]V = 864\pi[/tex]
Step-by-step explanation:
Since one of the boundaries is y = 0, we need to find the roots of the function [tex]f(x)=-2x^2+6x+36[/tex]. Using the quadratic equation, we get
[tex]x = \dfrac{-6 \pm \sqrt{36 - (4)(-2)(36)}}{-4}= -3,\:6[/tex]
But since the region is also bounded by [tex]x = 0[/tex], that means that our limits of integration are from [tex]x=0[/tex] (instead of -3) to [tex]x=6[/tex].
Now let's find the volume using the cylindrical shells method. The volume of rotation of the region is given by
[tex]\displaystyle V = \int f(x)2\pi xdx[/tex]
[tex]\:\:\:\:\:\:\:= \displaystyle \int_0^6 (-2x^2+6x+36)(2 \pi x)dx[/tex]
[tex]\:\:\:\:\:\:\:= \displaystyle 2\pi \int_0^6 (-2x^3+6x^2+36x)dx[/tex]
[tex]\:\:\:\:\:\:\:= \displaystyle 2\pi \left(-\frac{1}{2}x^4+2x^3+18x^2 \right)_0^6[/tex]
[tex]\:\:\:\:\:\:\:= 864\pi [/tex]
write the following sets in the set builder form C={1,4,9,16,25}
C={ check example in book}
A cyclist completes a journey of 500 m in 22 seconds, part of the way at 10 m/s and the remainder at 50 m/s. How far does she travel at each speed. solve by forming simultaneous equation
Answer:
150 m at 10 m/s
350 m at 50 m/s
Step-by-step explanation:
x + y = 500
x/10 + y/50 = 22
~~~~~~~~~~~~~~~~~
x + y = 500
5x + y = 1100
~~~~~~~~~~~~~~~~
x + y = 500
-5x - y = -1100
-4x = -600
x = 150
y = 350
Mrs Lee used 6 Meters of material to make 3 dresses. She used 4 ties as much material for a curtain as for a dress. How much material did she use for the curtain? (Dress)
Answer:
for each dress she used 6/3 of material
=2
then for a curtain =2x4=8 materials
Will give brainliest answer
Answer:
A
Step-by-step explanation:
the proof of the answer is shown above
a/b=2/5 and b/c=3/8 find a/c
Answer:
[tex]\frac{a}{c}[/tex] = [tex]\frac{3}{20}[/tex]
Step-by-step explanation:
[tex]\frac{a}{c}[/tex] = [tex]\frac{a}{b}[/tex] × [tex]\frac{b}{c}[/tex] = [tex]\frac{2}{5}[/tex] × [tex]\frac{3}{8}[/tex] = [tex]\frac{6}{40}[/tex] = [tex]\frac{3}{20}[/tex]
A grinding stone completes 175 revolutions before coming to a stop. How many radians did the stone complete
Answer:
175 * 2 * [tex]\pi[/tex]
350[tex]\pi[/tex] radians
Step-by-step explanation:
The number of radians completed by the stone will be 350 radians.
What is an angle in radians?The angle subtended from a circle's centre that intercepts an arc with a length equal to the circle's radius is known as a radian.
Given that a grinding stone completes 175 revolutions before coming to a stop.
The number of the revolutions in radians will be calculated as:-
Multiply the number by 2π to convert it into the radians.
Number of revolutions = 175 x 2 x π
Number of revolutions = 350 radians
Therefore, the number of radians completed by the stone will be 350 radians.
To know more about an angle in radians follow
https://brainly.com/question/19758686
#SPJ2
add 7/8 + 2 3/24 + 6 1/6
Answer:
9 4/24 or 9 1/6
Find the LCM(lowest common multiple) of 8, 24 and 6.The LCM of 8, 24 and 6 is 24.We now want to turn all the denominators into 24 so we are going multiply 7/8 by 3 and 1/6 by 4. We won't need to turn the denominator of 3/24 into 24 because it's already 24Whatever you do to the denominator you have to do to the numerator, so you also have to multiply the numerator of 7/8 by 3 and the numerator of 1/6 by 4That now results in 21/8 + 2 3/24 + 6 4/24Now you have to add all the fractions together which is going to equal to 28/24Because 28/24 is more than the whole, subtract 28 from 24 which gives us 4. That 4 is now our new numeratorWe are now going to all the whole numbers 6+2+1=9. Incase you're wondering, the '1' came from the 28/24The answer you should get should be 9 4/24 or if it should be simplified it would be 9 1/6
The data on the box plot describes the weight of several students in sixth grade. Which of the following statements are true about the data set? Select all that apply.
One-fourth of the students weigh between 90 and 101 pounds.
One-half of the students weigh between 75 and 90 pounds.
The median weight of the sixth graders is 85 pounds.
One-fourth of the students weigh less than 75 pounds.
One-fourth of the students weigh more than 75 pounds.
The total range of weight is 40 pounds.
Answer:
Step-by-step explanation:
B
A film distribution manager calculates that 5% of the films released are flops. If the manager is right, what is the probability that the proportion of flops in a sample of 572 released films would be greater than 6%
Answer:
0.1357 = 13.57% probability that the proportion of flops in a sample of 572 released films would be greater than 6%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
A film distribution manager calculates that 5% of the films released are flops.
This means that [tex]p = 0.05[/tex]
Sample of 572
This means that [tex]n = 572[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.05[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.05*0.95}{572}} = 0.0091[/tex]
What is the probability that the proportion of flops in a sample of 572 released films would be greater than 6%?
1 subtracted by the p-value of Z when X = 0.06. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.06 - 0.05}{0.0091}[/tex]
[tex]Z = 1.1[/tex]
[tex]Z = 1.1[/tex] has a p-value of 0.8643
1 - 0.8643 = 0.1357
0.1357 = 13.57% probability that the proportion of flops in a sample of 572 released films would be greater than 6%
What does p(B/A) represent?
Answer:
I believe you're asking about P(B|A).
Step-by-step explanation:
So,
P(B|A) represents the probability of event B occurring after it is assumed that event A has already occurred.
P(B|A) means "Event B given Event A" . In other words, event A has already happened, now what is the chance of event B? P(B|A) is also called the "Conditional Probability" of B given A.
The 100th term of 8, 8^4, 8^7, 8^10, …
Answer:
[tex]8^{298} \\8^{3(n-1)+1}[/tex]
Step-by-step explanation:
Answer:
8^298
Step-by-step explanation:
n = 1, 8^(1 + 0 * 3)
n = 2, 8^(1 + 1 * 3)
n = 3, 8^(1 + 2 * 3)
n = 4, 8^(1 + 3 * 3)
The exponent of 8 is 1 added to product of 1 less than the term number multiplied by 3.
n = n, 8^(1 + [n - 1] * 3) = 8^(1 + 3n - 3) = 8^(3n - 2)
For n = 100, the exponent is
3n - 2 = 3(100) - 2 = 300 - 2 = 298
Answer: 8^298
Identify the domain of the function shown in the graph.
A. -2 ≤ x ≤ 2
B. {-2,2}
C. x is all real numbers.
D. x > -2
Answer:
C. x is all real numbers
Step-by-step explanation:
Think of domain as how far the graph expands on the x-axis as asymptotes as the limits. So in this case, the graph extends infinitely on the x-axis; so it should be all real numbers.
Which of the following rational functions is graphed below?
Answer:
the answer is d
Step-by-step explanation:
because when we put-1 from x the equation hasn't any value
What is the median of the following set of values? 7, 21, 19, 15, 19, 14, 15, 19
The median of the following set of values is equals to 17.
What are median?Median represents the middle value of the given data when arranged in a particular order. The mean is the average value which can be calculated by dividing the sum of observations by the number of observations
We are given that the median of the following set of values
7, 21, 19, 15, 19, 14, 15, 19
Line them up in order first.
7, 14, 15, 15, 19, 19, 19, 21
Here the middle value are 15 and 19.
The median is 15 and 19. OR 17,
Therefore, 15 + 19 = 34/2 which equals to 17.
Learn more about mean and median;
https://brainly.com/question/17060266
#SPJ2
Acellus
First, find the surface area of the yellow prism.
3 cm 3 cm
: ?
4 cm
3 cm
front: [?]
back: [ ]
right: [ ]
left: [ ]
3cm
3cm
4cm
top:[]
5 cm
TOTAL: [ ]
Note: The bottom will not be
included because this is whern
5 cm
Answer:
57 cm²
Step-by-step explanation:
Surface area of the yellow prism = front + back + right + left + top
✔️Area of the front = L * W
L = 4 cm
W = 3 cm
Area of the front = 4*3 = 12 cm²
✔️Area of the back = L * W
L = 4 cm
W = 3 cm
Area of the back = 4*3 = 12 cm²
✔️Area of the right face = L * W
L = 4 cm
W = 3 cm
Area of the right face = 4*3 = 12 cm²
✔️Area of the left face = L * W
L = 4 cm
W = 3 cm
Area of the left face = 4*3 = 12 cm²
✔️Area of the top = L * W
L = 3 cm
W = 3 cm
Area of the top = 3*3 = 9 cm²
✅Total = 12 + 12 + 12 + 12 + 9 = 57 cm²
A Roper survey reported that 65 out of 500 women ages 18-29 said that they had the most say when purchasing a computer; a sample of 700 men (unrelated to the women) ages 18-29 found that 133 men said that they had the most say when purchasing a computer. What is the 99% confidence interval for the difference of the two proportions
Answer:
[tex]Z=-2.87[/tex]
Step-by-step explanation:
From the question we are told that:
Probability on women
[tex]P(W)=65 / 500[/tex]
[tex]P(W) = 0.13[/tex]
Probability on women
[tex]P(M)=133 / 700[/tex]
[tex]P(M) = 0.19[/tex]
Confidence Interval [tex]CI=99\%[/tex]
Generally the equation for momentum is mathematically given by
[tex]Z = \frac{( P(W) - P(M) )}{\sqrt{(\frac{ \sigma_1 * \sigma_2 }{(1/n1 + 1/n2)}}})[/tex]
Where
[tex]\sigma_1=(x_1+x_2)(n_1+n_2)[/tex]
[tex]\sigma_1=\frac{( 65 + 133 )}{ ( 500 + 700 )}[/tex]
[tex]\sigma_1=0.165[/tex]
And
[tex]\sigma_2=1 - \sigma = 0.835[/tex]
Therefore
[tex]Z = \frac{( 0.13 - 0.19)}{\sqrt{\frac{( 0.165 * 0.835}{ (500 + 700) )}}}[/tex]
[tex]Z=-2.87[/tex]
Frans paid R9600 as interest on a loan he took 5 years ago at 16% rate. What's was the amount he took as loan? Yeah
Answer:
5555 Lakh rupoes maybe hope it helps
The amount Frans took as loan = R12000
What is simple interest?"It is the interest that is only calculated on the initial amount of the loan."
Formula for simple interest:[tex]SI=\frac{P\times R\times T}{100}[/tex]
where, P: principal amount
T : period
R: rate of interest
For given question,
SI = 9600
T = 5 years
R = 16%
We need to find the principal amount.
Using simple interest formula,
[tex]\Rightarrow SI=\frac{P\times R\times T}{100}\\\\\Rightarrow P=\frac{SI\times 100}{R\times T}\\\\\Rightarrow P=\frac{9600\times 100}{5\times 16}\\\\\Rightarrow P=12000[/tex]
Therefore, the amount Frans took as loan = R12000
Learn more about the simple interest here:
https://brainly.com/question/22621039
#SPJ3
Previous Question Question 17 of 20 Next Question Based on the regression model, the expected daily production volume with 112 factory workers is 118,846 units. The human resource department noted that 123,415 units were produced on the most recent day on which there were 112 factory workers. What is the residual of this data point
Answer:
4,569 units
Step-by-step explanation:
Given :
Measured value = 123,415 units
Expected value = 118,846 units
Residual is the difference between the measured and expected value :
Residual = Measured value - Expected value
Residual = 123,415 units - 118,846 units
Residual = 4,569 units
I need help with this, please.
Answer:
it can not cleared clear but it can not cleared
√x²+2√3 +3 =0
[tex] \sqrt{x^{2} + 2 \sqrt{3} + 3} = 0[/tex]
solve x
Square both sides:
x^2+2sqrt3+3=0
x^2=-2sqrt(3)-3
x=sqrt(2sqrt(3)+3)i
or
x=-sqrt(2sqrt(3)+3)i
Which of the following statements provides the correct freezing and boiling points of water on the Celsius and Fahrenheit temperature scales?
The freezing point of water is 0°C or 32°F while the boiling point of water is 100°C or 212°F.
TemperatureTemperature is the measure of the degree of hotness or coldness of a substance or place. It is usually expressed Fahrenheit and Celsius scale. Temperature indicates the direction of heat flow.
The freezing point of water is 0°C or 32°F while the boiling point of water is 100°C or 212°F.
Find out more on Temperature at: https://brainly.com/question/24746268
Find the missing number?
Answer:
65 solve theprob
Step-by-step explanation:
sinolove ko po yan paki brainly
What does it mean if a project has a Percent Spent of 90%, Percent Scheduled of 85%, and a Percent Complete of 95%
Answer:
It means that the project is in good shape, within budget an d it would finish early
Step-by-step explanation:
The answer to this question is pretty straight forward. If a project has the percent spent fine 90 percent, the scheduled has percentage of 85 percent and the complete is at the percentage of 95, what it means is that this project is in good shape, the project being carried out is still being done within the proposed budget and at 95% complete, it means that the project is going to finish early.
Carly is the principal at a middle school and wants to know the average IQ of all the female, seventh-grade students. She does not know anything about what the population distribution looks like. She took a simple random sample of 31 seventh-grade girls in her school and found the average IQ score in her sample was 105.8 and the standard deviation was 15. Assume that all conditions are met, construct the 96% Confidence interval for the average IQ score of all seventh-grade girls in the school.
Answer:
The 96% confidence interval for the average IQ score of all seventh-grade girls in the school is (105, 111.6).
Step-by-step explanation:
We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.
The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So
df = 31 - 1 = 30
96% confidence interval
Now, we have to find a value of T, which is found looking at the t table, with 30 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.96}{2} = 0.98[/tex]. So we have T = 2.15
The margin of error is:
[tex]M = T\frac{s}{\sqrt{n}} = 2.15\frac{15}{\sqrt{31}} = 5.8[/tex]
In which s is the standard deviation of the sample and n is the size of the sample.
The lower end of the interval is the sample mean subtracted by M. So it is 105.8 - 5.8 = 105.
The upper end of the interval is the sample mean added to M. So it is 105.8 + 5.8 = 111.6
The 96% confidence interval for the average IQ score of all seventh-grade girls in the school is (105, 111.6).
