Write the standard form of the equation and the general form of the equation of the circlewith radius r and center (h.k). Then graph the circle.r= 10; (h,k) = (8,6)The standard form of the equation of this circle isThe general form of the equation of this circle is(Simplify your answer.)Graph the circle.-20 -18Click toenlargegraph
Answers
To solve this problem, we will first find the standard form of the circle equation. Given a circle of radius r and center (h,k), the standard form of the circle equation would be
[tex](x-h)^2+(y-k)^2=r^2[/tex]In our case, we have h=8 , k=6 and r=10. So the equation for the given circle would be
[tex](x-8)^2+(y-6)^2=10^2=100[/tex]The general form of the circle equation can be obtained from expanding the squares on the left side of the equality sign. Recall that
[tex](a-b)^2=a^2-2a\cdot b+b^2[/tex]So, applying this to the standard equation we get
[tex](x-8)^2=x^2-16x+64[/tex][tex](y-6)^2=y^2-12y+36[/tex]So our equation becomes
[tex]x^2-16x+64+y^2-12y+36=100[/tex]Operating on the left side, we have
[tex]x^2-16x+y^2-12y+100=100[/tex]By subtracting 100 on both sides, we get
[tex]x^2-16x+y^2-12y=0[/tex]which the general form of the equation of the given circle.
Using a graphing tool, we have that the circle's graph would be
Related Questions
Eddie brought his DVD set to the secondhand store to sell he was paid for all three DVDs in the set before he left Eddie used $15.70 of his earnings to purchase a pair of headphones had $2.30 remaining which equation can you use to find the amount of money Eddie received for each DVD
15.70(d-3)=2.30
3d-15.70=2.30
15.70d-3=2.30
3(d-15.70)=2.30
Answers
Answer:
3d - 15.70 = 2.30
Step-by-step explanation:
We don't know the cost of one DVD, so let's use d to represent this unknown variable. Eddie sold 3 DVDs, so 3 multiplied by d equals his total earnings.
Eddie then used $15.70 of his earnings to buy a pair of headphones. We can represent this by subtracting 15.70 from the total earnings (3d).
After buying/subtracting the price of the headphones from his total earnings, Eddie had $2.30 left over, which can be represented by making 3d - 15.70 equal 2.30.
So, the final equation turns out to be: 3d - 25.70 = 2.30
:)
In basketball, " one on one" free throw shooting ( commonly called foul shooting) is done as follows: if the player makes the first shot(1point), she is given a second shot. If she misses the first shot, she is not given a second shot. Christine, a basketball player, has a 70% free throw record. (she makes 70% of her free throws). Find the probability that, given one-on-one free throw shooting opportunity, Christene will score one point.
Answers
If she will be able to shoot the first shot and miss the second shot, then she will obtain 1 point.
Thus, the probability that Christine will get the first shot is as follows:
[tex]P(1pt)=(0.7)(0.3)=0.21[/tex]where the first factor is the probability that she will shoot the first shot and the second factor is the probability that she missed the second shot. Thus, the probability of obtaining 1 point is 21% or 0.21.
Is x5 + x2 + x a polynomial? Explain why or why not.
Answers
A polynomial is a mathematical expression formed by variables and coefficients, that involves only the operations of addition, subtraction, multiplication and non-negative integer exponentiation of variables.
The expression:
[tex]x^5+x^2+x[/tex]Is formed by the addition of three terms, each consisting of the variable x raised to a positive integer quantity. Therefore, the given expression is a polynomial.
at Kelly's school, 2/3 of the play ground is covered by grass, and 3/5 of the grassy area is a baseball field. how much of the school playground is baseball feild?
Answers
At Kelly's school, 2/3 of the playground is covered by grass, and 3/5 of the grassy area is a baseball field.
How much of the school playground is the baseball field?
SOLUTION
2/3 of the playground is covered by grass and 3/5 of the grassy area is a baseball field.
The area of the school playground which is baseball field =
[tex]\frac{2}{3}\text{ x }\frac{3}{5}\text{ = }\frac{6}{15\text{ }}\text{ = }\frac{2}{5}[/tex]CONCLUSION :
[tex]\frac{2}{5}\text{ of the school field = Area of the Basket Ball Field.}[/tex]
A retail clothing store offers customers an opportunity to open up a credit card during checkout. One location of the retail clothing store states that the number of credit cards, A, that are opened t months since January can be modeled by the function A(t) = 15 + 3t. The number of credit cards opened at another location, B, is defined by the function B(t) = 25 − t. What is an expression that can be used to determine the total amount of credit cards opened at the two locations?
(A + B)(t) = 40 + 4t
(A + B)(t) = 40 + 2t
(A − B)(t) = −10 + 2t
(A − B)(t) = −10 + 4t
Answers
The expression that is useful to determine the total amount of credit cards opened at the two locations is (A + B)(t) = 40 + 2t so option (B) is correct.
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
A statement expressing the equality of two mathematical expressions is known as an equation.
As per the given,
The amount in location A is given as
A(t) = 15 + 3t
The amount in location B is given as
B(t) = 25 − t
The total amount combined between A and B is given as,
(A + B)(t) = 15 + 3t + 25 - t
(A + B)(t) = 15 + 25 + 3t - t
(A + B)(t) = 40 + 2t
Hence "The expression that is useful to determine the total amount of credit cards opened at the two locations is (A + B)(t) = 40 + 2t".
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A data set is summarized in the frequency table below. Using the table, determine the number of values less than or equal to 6.ValueFrequency152332435463788397108113Give your answer as a single number. For example if you found the number of values was 14, you would enter 14.
Answers
The number of values less than or equal to 6 is 5 + 3 +2 +3 +4 +3 = 20
Line AB and line DA are?
Answers
Answer:
perpendicular
Step-by-step explanation:
Square
Rectangle
Right triangle
Cube
Rectangular prism
are all examples of perpendicular shapes
i hope this helped
have a good day ^^
Describe a series of transformations that takes triangle ABC to triangle A’B’C’
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Notice that if the triangle ABC is reflected over the X axis (red), and then reflected over the Y axis (green), we just would have to translate the triangle two units to the right (blue) to get A'B'C':
The diameter of circle is 20 inches. find the circumference in terms of pi
Answers
The below formula is used to find the circumference of a circle;
[tex]C=2\pi r[/tex]But we know that the diameter of a circle is expressed as;
[tex]d=2r[/tex]Let's replace 2r with d in the 1st equation, we'll then have;
[tex]C=\pi d[/tex]We've been told that the diameter of the circle is 20inches, if we substitute this value into our equation, we'll have;
[tex]C=20\pi[/tex]Rewrite y + 1 = -2x – 3 in standard form
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The algebraic expressions can be written as
[tex]a+b+c=0[/tex]The given expression is,
[tex]\begin{gathered} y+1=-2x-3 \\ -2x-y=1+3 \\ -2x-y=4 \\ -2x-y-4=0 \\ 2x+y+4=0 \end{gathered}[/tex]can you please give me any examples on how to do this
Answers
we can take two numbers of the sequence and subtract them to see the difference
so
[tex]1.9-1.2=0.7[/tex]the sequence adds 0.7 each step
the next 3 terms are
[tex]3.3+0.7=4[/tex][tex]4+0.7=4.7[/tex][tex]4.7+0.7=5.4[/tex]Fowler has a collection of marbles of different sizes and colors. Big Small Red 9 9 Green 14 9 Purple 9 6 Blue 0 10 What is the probability that a randomly selected marble is not red or is not small? Simplify any fractions.
Answers
From the given table, the following are observed:
No. of marbles that are not red and not small = No. of Big Green and Big Purple
= 14 + 9
= 23 Marbles
Total number of marbles = 9 + 14 + 9 + 9 + 9 + 6 + 10 = 66 Marbles
We get,
[tex]\text{ Probability of getting a marble that is not red or not small = }\frac{\text{ 23 Marbles}}{66\text{ Marbles}}[/tex][tex]\text{ = }\frac{23}{66}[/tex]We can no longer simplify 23/66. Therefore, 23/66 is the answer.
Sara has saved $500 and wants to buy a new computer. the computer she wants costs $1400. her current job pays $17.50 per hour after taxes . use an inequality to describe the situation , solve the inequality and write a sentence describing what the solution means to Sara
Answers
She has already 500
x hours woked
pund we got:
7.50 x >= 900
And inequality could be
fx:o
[tex]500+17.5x\ge1400[/tex]because we need to gain 1400 or more
hours in order to have enough money to buy the computer
Sara needs to work at least 52 00/17.5
x >= 5r t o
The equatu
1
And then we can solve for x a
500+17.5x>>= 1400
utioen x
For this50 case we can do this:
500+ 17.
A straight line l1 with equation 5x - 7 = 0 cuts the x axis at point A. Straight line l2 is perpendicular to straight line l1 and passes through point A. What is the coordinates of point A and the equation of the straight line l2?
Answers
Point A has a coordinate of (7/5, 0) while the straight line l2 is represented by the equation y = 0
The coordinates of point AThe equation of line l1 is given as
5x - 7 = 0
It cuts the x-axis at point A
This means that
5A - 7 = 0
Solve for A
5A = 7
So, we have
A = 7/5
Rewrite as
A = (7/5, 0)
The equation of the straight line l2From the question, we have
Lines l1 and l2 are perpendicular lines
The equation 5x - 7 = 0 has no y variable
So, the slope is undefined
The slopes of perpendicular lines are represented as follows
Slope 1 * Slope 2 = -1
So, we have
Slope 2 = -1/Slope 1
This gives
Slope 2 = -1/undefined
Evaluate
Slope 2 = 0
This means that l2 has a slope of 0
The equation of l2 is calculated as
y = m(x - x₁) + y₁
In this case,
A = x₁ and y₁ = 0
So, we have
y = m(x - A)
This gives
y = 0 * (x - 7/5)
Evaluate
y = 0
Hence, the equation of the straight line l2 is y = 0
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xyx2xy1645256720 2258 484 1,2762873 7842,044 3294 1,0243,008 45141 2,025 6,345 ∑x=143 ∑y=411 ∑x2=4,573 ∑xy=13,393 Which regression equation correctly models the data?y = 2.87x + 0.12y = 2.87x + 11.85y = 3.39x – 14.75y = 3.39x – 9.24
Answers
We are asked to identify the correct regression equation.
The regression equation is given by
[tex]y=bx+a[/tex]Where the coefficients a and b are
[tex]a=\frac{\sum y\cdot\sum x^2-\sum x\cdot\sum xy}{n\cdot\sum x^2-(\sum x)^2}[/tex][tex]b=\frac{n\cdot\sum xy-\sum x\cdot\sum y}{n\cdot\sum x^2-(\sum x)^2}[/tex]Where n is the number of observations that is 5.
Let us substitute the following into the above formula.
∑x=143
∑y=411
∑x^2=4,573
∑xy=13,393
[tex]a=\frac{411\cdot4573-143\cdot13393}{5\cdot4573-(143)^2}=-14.75[/tex][tex]b=\frac{5\cdot13393-143\cdot411}{5\cdot4573-(143)^2}=3.39[/tex]So, the coefficients are
a = -14.75
b = 3.39
Therefore, the correct regression equation is
[tex]y=3.39x-14.75[/tex]The sum of 5 times a number and 7 equals 8. Find the number
Answers
Explanation
Let the number be x. Therefore, we will have
[tex]\begin{gathered} 5x+7=8 \\ 5x=8-7 \\ 5x=1 \\ x=\frac{1}{5} \end{gathered}[/tex]the discriminant equation How many real solution 4x^2-8x+10=-x^2-5 have?
Answers
Answer:
0 real solutions
Explanation:
First, we need to transform the equation into the form:
[tex]ax^2+bx+c=0[/tex]So, the initial equation is equivalent to:
[tex]\begin{gathered} 4x^2-8x+10=-x^2-5 \\ 4x^2-8x+10+x^2+5=-x^2-5+x^2+5 \\ 5x^2-8x+15=0 \end{gathered}[/tex]Now, the discriminant can be calculated as:
[tex]b^2-4ac[/tex]If the discriminant is greater than 0, the equation has 2 real solutions.
If the discriminant is equal to 0, the equation has 1 real solution
If the discriminant is less than 0, the equation has 0 real solutions
So, in this case, a is 5, b is -8 and c is 15. Then, the discriminant is equal to:
[tex](-8)^2-4\cdot5\cdot15=84-300=-236[/tex]Since the discriminant is less than zero, the equation has 0 real solutions
Which of the following numbers are greater than 6 and less than 8? Explainhow you know.
Answers
Let's analyze each case and see if they are less than 8 first:
[tex]7<8,therefore(7)^{\frac{1}{2}}<8[/tex][tex]\sqrt[]{64}=8[/tex]thus
[tex]\sqrt[]{60}<\sqrt[]{64}=8[/tex][tex]\sqrt[]{60}<8[/tex]Finally,
[tex]64<80[/tex][tex]\sqrt[]{64}<\sqrt[]{80}[/tex]thus
[tex]8<\sqrt[]{80}[/tex]In sumary, the first two options are less than 8, but not the third.
Now let's see if they are greater than 6
[tex]9>7[/tex][tex]\sqrt[]{9}>\sqrt[]{7}[/tex][tex]3>\sqrt[]{7}[/tex]and
[tex]6>3>\sqrt[]{7}[/tex]thus
[tex]6>\sqrt[]{7}[/tex]Now
[tex]36<60[/tex][tex]\sqrt[]{36}<\sqrt[]{60}[/tex]and so
[tex]6<\sqrt[]{60}[/tex]Finally
[tex]\sqrt[]{80}>\sqrt[]{60}>6[/tex]thus
[tex]\sqrt[]{80}>6[/tex]In conclussion, the second and third options are greater than 6, but not the first.
x^2 = 16, therefore x = 4.
Is this a valid conclusion? If not, give a counterexample.
Answers
Describe the association in the scatter plot below.----------------The scatter plot shows (positive linear, positive linear with one outlier, negative linear, negative linear with one outlier, nonlinear, or no) association because as the plotted values of x increase, the values of y generally (decrease, increase, show no pattern or follow a nonlinear pattern).
Answers
From the given figure
The given point can form a line with a negative slope, because when
the values of x increase the values of y decrease
Then the scatter plot shows a negative linear association because
as the values of x increase, the values of y generally decrease
Mike is shopping for new clothes. He has a coupon for 20% off of his total purchase. His purchase price before the discount is $68. Let T represent the total cost after the discount. Which equation can be written to model this scenario? Select all that apply.68-0.2(68) = T68 - .20 =T68-20 = T0.8(68) = T0.2(68) =T
Answers
ANSWER
68 - 0.2(68) = T
0.8(68) = T
EXPLANATION
The coupon allows for 20% off of his total purchase.
His purchase price before the discount is $68.
To find the price after the discount, we can use two methods:
=> Find 20% of $68 and then subtract from $68 to get T.
That is:
[tex]\begin{gathered} 68\text{ - (}\frac{20}{100}\cdot\text{ 68) = T} \\ \Rightarrow\text{ 68 - 0.2(68) = T} \end{gathered}[/tex]=> Subract 20% from a total of 100% and then multiply by $68 to get T.
That is:
[tex]\begin{gathered} (100\text{ - 20)\% }\cdot\text{ 68 = T} \\ \Rightarrow\text{ 80\% }\cdot\text{ 68 = T} \\ \Rightarrow\text{ 0.8(68) = T} \end{gathered}[/tex]Those are the two answers.
Determine whether the ratios are equivalent.
2:3 and 24:36
O Not equivalent O Not equivalent
Answers
We can conclude that the given ratios 2:3 and 24:36 are equivalent.
What are ratios?A ratio in math displays how many times one number is contained in another. The ratio of oranges to lemons, for instance, is eight to six if there are eight oranges and six lemons in a bowl of fruit. The proportions of oranges to the total amount of fruit are 8:14 for oranges and 6:8 for lemons, respectively.So, ratios are equivalent or not:
2:3 and 24:362/3 = 24/362/3 = 2/3 (Divide by 12)Then, 2:3 :: 2:3
Therefore, we can conclude that the given ratios 2:3 and 24:36 are equivalent.
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Absolute risk is defined as the proportion or percentage of people in a group for whom an undesirable event occurs. In college classrooms, students typically can choose their own seats. Professors have noticed a difference in grades between students who choose to sit in the front and those who choose to sit in the back. For example, in one math class, 9 of the 20 students who sat in the back failed the class, but only 3 of the 20 students who sat in the front failed the class. What was the absolute risk of failing the class for students who sat in the back? For students who sat in the front? Give your answers as fractions, proportions, and percents.
Answers
Given in the scenario:
a.) 9 of the 20 students who sat in the back failed the class.
b.) 3 of the 20 students who sat in the front failed the class.
A.) The absolute risk of failing the class for students who sat in the back.
In the back, 9 of the 20 students who sat in the back failed the class.
The absolute risk in proportion = 9:20
The absolute risk in fraction = 9/20
The absolute risk in percentage = (9 ÷ 20) x 100 = 0.45 x 100 = 45%
B.) The absolute risk of failing the class for students who sat in the front.
In the front, 3 of the 20 students who sat in the front failed the class.
The absolute risk in proportion = 3:20
The absolute risk in fraction = 3/20
The absolute risk in percentage = (3 ÷ 20) x 100 = 0.15 x 100 = 15%
Find the slope, if it exists, of the line containing the pair of points. (−2,−6) and (−15,−7)
Answers
The linear regression for a given data set has the form
[tex]y=a+bx[/tex]where the values a and b can be solved using the equation
[tex]\begin{gathered} a=\frac{(\sum y)(\sum x^2)-(\sum x)(\sum xy)}{n(\sum x^2)-(\sum x)^2} \\ b=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^2)-(\sum x)^2} \end{gathered}[/tex]Based on the given data set, we have n equals 5. We will solve for the values of the summation first. We have the following
[tex]\begin{gathered} \sum y=4+4+6+6+8=28 \\ \sum x=1+3+5+7+9=25 \\ \sum xy=(1\cdot4)+(3\cdot4)+(5\cdot6)+(7\cdot6)+(8\cdot9)=160 \\ \sum x^2=1^2+3^2+5^2+7^2+9^2=165^{} \\ (\sum x)^2=25^2=625 \end{gathered}[/tex]Using these values to compute for the values of a and b, we get
[tex]\begin{gathered} a=\frac{(28\cdot165)-(25\cdot160)}{5(165)-625}=\frac{31}{10}=3.1 \\ b=\frac{5(160)-(28\cdot25)}{5(165)-625}=\frac{1}{2}=0.5 \end{gathered}[/tex]Take note that the problem wants us to reduce the numbers to the nearest tenth. Hence, the linear regression for the given data set is written as
[tex]y=3.1+0.5x[/tex]consider the parent function f(x)=x^2. a. graph y=f(x). b. write an equation for f(1/2x). Then sketch a graph of y=f(1/2x) and describe the transformation. c.write an equation for f(3x). Then sketch a graph of y=f(3x) and describe the transformation.
Answers
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
f(x)=x²
f(1/2x) = ?
graph of y=f(1/2x) = ?
Step 02:
b. f(1/2x)
x ===> 1/2x
[tex]f\text{ (1/2 x) = (}\frac{1}{2}x)^2=\frac{1}{4}x^2[/tex]Step 03:
c. Graph:
We give values to x, and we obtain the values of y.
f(x) = 1/4 x²
e.g.
if x = 4
y = 1/4 (4)² = 1/4 * 16 = 4
That is the solution for b. and c.
Solve the following system of equations by graphing3x+5y=10y=-x+4
Answers
ANSWER
The point of intersection of the two equations is (5, - 1)
The graph is
STEP BY STEP EXPLANATION
Step 1: The given equations are:
3x + 5y = 10
y= -x + 4
Step 2: Assume values for x in a table (example -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5) to determine the corresponding values for y for both equations
Step 3: Graph the equations and locate the intersection of the two equations
Which recipe makes more cookies per cup of chocolate chips?
Answers
We are given the recipes of Grandma and Betty Potter box.
We are asked to find out which of them makes more cookies per cup of chocolate chips.
Let us find the unit rate for both of them and compare which is greater.
Grandma:
She uses 1 1/2 cups of chocolate chips to make 24 cookies.
The unit rate is
[tex]\frac{24}{1\frac{1}{2}}=\frac{24}{\frac{3}{2}}=24\times\frac{2}{3}=16[/tex]So the unit rate is 16 cookies per chocolate chip.
Betty Potter box:
60 cookies are made using 3 chocolate chips.
80 cookies are made using 4 chocolate chips.
200 cookies are made using 10 chocolate chips.
The unit rate is
[tex]\begin{gathered} \frac{60}{3}=20 \\ \frac{80}{4}=20 \\ \frac{200}{10}=20 \end{gathered}[/tex]So the unit rate is 20 cookies per chocolate chip.
As you can see, the recipe of Better Potter box makes more cookies per cup of chocolate chip.
A bag contains 8 red marbles, 7 blue marbles and 6 green marbles. If three marbles are drawn out of the bag without replacement, what is the probability, to the nearest 10th of a percent, that all three marbles drawn will be red?
Answers
SOLUTION
Given the question, the following are the solution steps to answer the question.
STEP 1: Write the formula for probability
[tex]Probability=\frac{number\text{ of required outcomes}}{number\text{ of total possible outcomes}}[/tex]STEP 2: Write the outcomes of the events
[tex]\begin{gathered} number\text{ of red marbles}\Rightarrow n(red)\Rightarrow8 \\ number\text{ of blue marbles}\Rightarrow n(blue)\Rightarrow7 \\ number\text{ of green marbles}\Rightarrow n(green)\Rightarrow6 \\ number\text{ of total marbles}\Rightarrow n(total)\Rightarrow21 \end{gathered}[/tex]STEP 3: Write the formula for getting the probability that all three marbles drawn will be red
[tex]Pr(Red\text{ and Red and Red\rparen}\Rightarrow Pr(red)\times Pr(red)\times Pr(red)[/tex]STEP 4: Calculate the probability
[tex]\begin{gathered} Pr(all\text{ three are reds\rparen}\Rightarrow\frac{8}{21}\times\frac{7}{20}\times\frac{6}{19} \\ =\frac{336}{7980}=0.042105263 \\ To\text{ percentage will be to multiply by 100} \\ 4.210526316\% \\ To\text{ the nearest tenth will be:} \\ \approx4.2\% \end{gathered}[/tex]Hence, the probability, to the nearest 10th of a percent, that all three marbles drawn will be red is 4.2%
Zales sells diamonds for $1,100 that cost $800. What is Zales’s percent markup on selling price? Check the selling price.
Answers
Zales's percent markup on the selling price as required in the task content is; 37.5%.
Percentages and markup priceIt follows from the task content that the percent markup on the selling price be determined according to the given data.
Since the cost of diamonds is; $800 while the diamonds sell for $1,100. It follows that the markup on the selling price of the diamonds is;
Markup = Selling price - Cost price.
Hence, we have;
Markup = 1,100 - 800.
Therefore, the markup is; $300.
On this note, the percent markup can be determined as follows;
= (300/800) × 100%.
= 37.5%.
Ultimately, the percent markup on the diamonds is: 37.5%.
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clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. before treatment 19 subjects had a mean wake time of 100.0 min after treatment the 19 subjects had a mean wake time of 71.6 min and a standard deviation of 20.4 min assume that the 19 sample value appears to be from a normally distributed population and construct a 99% confidence interval estimate of the mean wake time for a population with drug treatment what does the result suggest about the wake time of 100.0 min before the treatment does the drug appears to be effective
Answers
We have to calculate a 99% confidence interval for the mean.
The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.
The sample mean is M=71.6.
The sample size is N=19.
When σ is not known, s divided by the square root of N is used as an estimate of σM:
[tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{20.4}{\sqrt{19}}=\dfrac{20.4}{4.359}=4.68[/tex]The degrees of freedom for this sample size are:
[tex]df=n-1=19-1=18[/tex]The t-value for a 99% confidence interval and 18 degrees of freedom is t=2.878.
The margin of error (MOE) can be calculated as:
[tex]MOE=t\cdot s_M=2.878\cdot4.68=13.471[/tex]Then, the lower and upper bounds of the confidence interval are:
[tex]\begin{gathered} LL=M-t\cdot s_M=71.6-13.471=58.129 \\ UL=M+t\cdot s_M=71.6+13.471=85.071 \end{gathered}[/tex]The 99% confidence interval for the mean is (58.129, 85.071). This interval does not include the value 100, so we can conclude that there is statistical evidence that the treatment reduces the mean wake time.