Solve each inequality given that the function f is increasing over its domain.
Answers
Therefore, the solution to the inequality is: -8 < x ≤ 1 or x = 2. In interval notation, we can write the solution as: (-8, 1] ∪ {2}.
What is inequality?Inequality is a mathematical statement that compares two quantities or expressions and indicates that one is greater than, less than, or not equal to the other. An inequality is typically expressed using symbols such as < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to).
Here,
Since the function f is increasing, we know that if x₁ < x₂, then ƒ(x₁) < ƒ(x₂). To solve the inequality, we need to isolate x on one side of the inequality. We'll start by applying the function ƒ to both sides of the inequality:
ƒ(4x − 3) ≥ ƒ(2 − x²)
Since ƒ is increasing, we can apply it to each side of the inequality without changing the direction of the inequality:
4x − 3 ≥ 2 − x²
Next, we'll simplify the right side of the inequality by expanding the square:
4x − 3 ≥ 2 + x²
Now we'll move all the terms to one side of the inequality:
x² - 4x + 5 ≤ 0
We can factor the quadratic expression to get:
(x - 2)(x - 2 + 1) ≤ 0
Simplifying further:
(x - 2)(x - 1) ≤ 0
Now we need to determine the sign of the expression (x - 2)(x - 1) over the domain D = (-8, 4). We can do this by using a sign chart:
x x - 2 x - 1 (x - 2)(x - 1)
-8 -10 -9 +90
-1 -3 -2 +2
1 -1 0 0
4 2 3 -6
We see that (x - 2)(x - 1) is negative (less than zero) over the interval (-8,1) and positive (greater than zero) over the interval (1,4).
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Related Questions
help???? thanks! i do not get this at all!!
Answers
Answer:
S = [tex]\sqrt{3}[/tex]
H = 2
It's D
Step-by-step explanation:
Considering the attached picture
h = 1 x 2 = 2
s = 1 x [tex]\sqrt{3}[/tex] = [tex]\sqrt{3}[/tex]
Mrs. Myrtle wants to buy this plastic toy for Willow. How much plastic would it take to construct the following toy? [Assume toy is solid plastic]
9m
13 m
Radius: type your answer...
type your answer...
Height of the Cone: type your answer....
; Total: type your answer...
Cone: type your answer...
; Hemisphere:
Answers
Answer:
Step-by-step explanation:
6
Three security guards: A, B and C, who guard a jewellery shop, work under the following conditions:
None of the guards is called to work for three consecutive days.
Guard A does not work on Tuesday, Thursday and Sunday.
Guard B does not work on Monday, Wednesday and Saturday.
Guard C does not work on Tuesday, Thursday and Sunday.
On which day of the week are all of them present for work?
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7. Groceries Galore sells bags of chips from two different companies., The mean absolute deviation of the number of chips in each bag from Company A is higher than the mean absolute deviation of the number of chips in each bag from Company B. Heather believes this means that Company A tends to sell more chips in each bag. Do you agree? Explain why or why not. CHIPS
Answers
Answer:
Always need to do research to improve your studies
The mean absolute deviation (MAD) is a measure of the variability of a dataset. It measures how spread out the data is by calculating the average distance between each data point and the mean of the dataset. So, if the MAD of the number of chips in each bag from Company A is higher than the MAD of the number of chips in each bag from Company B, it means that the data points in the dataset of Company A are more spread out than those in the dataset of Company B.
However, the MAD alone cannot tell us whether Company A tends to sell more chips in each bag than Company B. To determine this, we would need to look at other measures such as the mean, median, or mode of the datasets.
For example, if the mean number of chips in each bag from Company A is higher than the mean number of chips in each bag from Company B, then Heather's belief would be supported. On the other hand, if the mean number of chips in each bag from Company A is lower than the mean number of chips in each bag from Company B, then her belief would be contradicted.
In conclusion, the MAD alone cannot tell us whether Company A tends to sell more chips in each bag than Company B. We would need to look at other measures of central tendency to determine this.
Simplify the expression using basic identities Cosu - cosu sin^2u
Answers
Cos u - cos u sin²u simplifies to cos³(u). We can solve it in the following manner.
Starting with the expression Cos u - cos u sin²u, we can factor out cos(u) to get:
Cos u - cos u sin²u = cos(u) (1 - sin²(u))
Using the identity sin²(u) + cos²(u) = 1, we can substitute cos²(u) for 1 - sin²(u):
cos(u) (1 - sin²(u)) = cos(u) cos²(u) = cos³(u)
Therefore, Cos u - cos u sin²u simplifies to cos³(u).
In mathematics, a factor refers to a number, algebraic expression, or function that divides or multiplies with another number, algebraic expression, or function without leaving a remainder. For example, in the expression 12x, 12 is a factor of the expression because it can be divided evenly by 12.
In algebra, factoring is the process of finding the factors of an algebraic expression, which means breaking it down into simpler parts that can be multiplied together to get the original expression. Factoring is an important skill in algebra, as it allows us to simplify and solve equations, identify common factors, and manipulate expressions more easily.
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f(x) = 1 + 4x 1
Find the value of x when f(x) = 15.
Answers
Answer:
the value of x when f(x) = 15 is 3.5.
Step-by-step explanation:
We are given that:
f(x) = 1 + 4x
We need to find the value of x when f(x) = 15.
So we can set up the equation:
1 + 4x = 15
Subtracting 1 from both sides, we get:
4x = 14
Dividing both sides by 4, we get:
x = 14/4
Simplifying, we get:
x = 3.5
Therefore, the value of x when f(x) = 15 is 3.5.
i need help with an ixl. i dont understand it at all. please help !!
Answers
The value of m∠U in the circle is 168 degrees
How to find the tangent of a circle?The tangent of a circle is the line that touches the circle at only one point.
The tangent to a circle is always perpendicular to the radius. Therefore, it forms a right angle.
Therefore, let's use the theorem to find the angle m∠U
Hence,
m∠STU = 90 degrees
m∠UTV = 90 - 84 = 6 degrees
Therefore, the triangle STV is an isosceles triangle. This means the base angle is congruent .
Hence,
m∠SVT = 84 degrees
Therefore,
m∠U = 180 - 6 - 6
m∠U = 180 - 12
m∠U = 168 degrees
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Keeshawn measured the box show below. What is the surface area of the box
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when the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
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The ratio of the first term to the common difference is 1 is to 1 (1:1).
Let's say the first term of the arithmetic progression is 'a' and the common difference is 'd'.
Then the sum of the first five terms will be,
S5 = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 5a + 10d
And the sum of the first ten terms will be,
S10 = a + (a + d) + (a + 2d)+ ... + (a + 9d)
=10a + 45d
We have been given that sum of the first 10 terms is equal to four times the sum of the first 5 terms, i.e
S10 = 4S5
Substituting the expressions for S5 and S10 in terms of a and d, we get,
10a + 45d = 4(5a + 10d)
Simplifying this equation, we get,
10a + 45d = 20a + 40d
10d = 10a
d = a
So the ratio of the first term to the common difference is 1:1 or 1/1.
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Can u do step by step on how u got it I have to show my work
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PLEASE HELP ASAP!!!!!!!
REASONING Meghana is making a kite out of nylon. The material is sold for $5.99 per square yard. She needs enough material to cover her kite on both sides.
a. What is the area of Meghana’s kite?
b. How many yards of material will Meghana need to cover both sides of her kite?
c. How much will the material cost before taxes?
Answers
Answer: The material will cost Meghana $95.84 before taxes.
Step-by-step explanation:
a. To find the area of Meghana's kite, we first need to convert the dimensions from feet to yards since the material is sold per square yard:
Length = 18 ft = 6 yards (since 1 yard = 3 feet)
Width = 4 ft = 4/3 yards (since 1 yard = 3 feet)
The area of the kite is the product of its length and width:
Area = Length x Width = 6 x 4/3 = 8 square yards
b. Meghana needs enough material to cover both sides of her kite, so she needs twice the area of the kite:
Area of both sides = 2 x Area of one side = 2 x 8 = 16 square yards
Therefore, Meghana needs 16 square yards of nylon material.
c. The material is sold for $5.99 per square yard, so the total cost of the material before taxes is:
Total cost = Cost per square yard x Number of square yards
Total cost = $5.99 x 16 = $95.84
Therefore, the material will cost Meghana $95.84 before taxes.
Help me with this problem please and thank you.
Answers
In statistics, the mean is a measure of central tendency that represents the average value of a set of data. It is calculated by adding up all the values in the data set and dividing by the total number of values. The mean is also known as the arithmetic mean or average.
The formula for the mean is:
mean = (sum of all values) / (total number of values)
1.To find the mean of these scores, we add them up and divide by the total number of scores:
Mean = (83 + 97 + 85 + 84 + 96 + 80 + 80 + 87 + 91) / 9 = 86
2.To find the median, we first need to put the scores in order from lowest to highest:
80, 80, 83, 84, 85, 87, 91, 96, 97
The median is the middle score, which in this case is 85.
3.To find the mode, we look for the score that occur most frequently. In this case, there are two modes: 80 and 85, each occurring twice.
4.To find the range, we subtract the lowest score from the highest score:
Range = 97 - 80 = 17
Therefore, the mean is 86, the median is 85, the mode(s) are 80 and 85, and the range is 17.
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according to a study published by a group of university of massachusetts sociologists, approximately 60% of the valium users in the state of massachusetts first took valium for psychological problems. find the probability that among the next 8 users from this state who are interviewed
Answers
the probability that among the next 8 users from this state who are interviewed, the probability of the first 6 users first took it for psychological problems is 60% is 0.03456.
As per the given information, 60% of the valium users in Massachusetts first took it for psychological problems. Now, we are supposed to find the probability that among the next 8 users from this state who are interviewed, the probability of the first 6 users first took it for psychological problems is: 60%.
What is Valium?
Valium is a prescription medicine that is used to treat the symptoms of anxiety, alcohol withdrawal, and muscle spasm. The use of Valium can lead to addiction, especially if it is not taken as prescribed by the doctor.The probability of the first 6 users first took it for psychological problems is 60%. Hence, the probability that the first user did not take it for psychological problems is: 40%.
Probability of first user did not take Valium for psychological problems = P(A) = 0.40
Probability of first user took Valium for psychological problems = P(B) = 0.60
Now, we are supposed to find the probability that among the next 8 users from this state who are interviewed, the first 6 users first took it for psychological problems is: 60%.Probability of the first 6 users first took it for psychological problems is:
Probability of the next 5 users took it for psychological problems = P(B) × P(B) × P(B) × P(B) × P(B) = (0.60)⁵
Probability of the next 2 users did not take it for psychological problems = P(A) × P(A) = (0.40)²
Thus, the probability that among the next 8 users from this state who are interviewed, the probability of the first 6 users first took it for psychological problems is 60%, is:(0.60)⁵ × (0.40)²= 0.03456 (rounded to five decimal places)
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A tiger in a zoo is 8 feet long. A scale model of the tiger is 4 inches long. What is the scale factor?
Answers
Answer:
1/24
Step-by-step explanation:
1 ft = 12 in
8 X 12 = 96
96/4 = 24
1/24 is the scale factor
Answer: 24:1 (The tiger : The tiger model)
Step-by-step explanation:
1 feet =12 inch
so 8 feet=96 inch
The tiger : The tiger model= 96:4= 24:1
What number solves this equation? × (3 + 8) = 55
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To solve the equation × (3 + 8) = 55, we need to simplify the expression inside the parentheses first, which gives us:
× (3 + 8) = × 11
Now we substitute × 11 back into the original equation:
× 11 = 55
To solve for ×, we divide both sides by 11:
× = 5
Therefore, the number that solves the equation × (3 + 8) = 55 is 5.
A study showed that vitamin A oil reduces pain levels in patients with Lyme disease. During each session, the oil was injected into the pain site indicated by the patient. Pain reduction was measured through self-reporting after each session.
Another study is being designed to examine whether vitamin A oil also reduces pain in patients suffering from pinched nerves in the cervical region of the back. Two hundred patients aged 20–40 years are subjects in the new study.
Part A: What is an appropriate design for the new study? Include treatments used, method of treatment assignment, and variables that should be measured. (5 points)
Part B: If the study consisted of 100 young patients and 100 old patients instead of 200 patients aged 20–40, would you change the study design? If so, how would you modify your design? If not, why not? (5 points)
Part C: Suppose the data in the following table was collected at the end of the study you described in part A.
Vitamin A Oil Placebo
Reported Pain Reduction 37 21
Total Treated 100 100
Can someone please help me so I don't fail?
Construct 90% confidence intervals for the proportion of pain reduction for each treatment group. Show all work. (5 points)
Part D: What can you conclude about the use of vitamin A oil for patients with pinched nerves in the cervical region of the back? (5 points)
Answers
the 90% confidence interval for the proportion of pain reduction for the vitamin A oil group is (0.275, 0.465) , placebo group is (0.119, 0.301).
Part A: An appropriate design for the new study would be a randomized controlled trial. The treatments used in this study would be vitamin A oil and a placebo. The method of treatment assignment would be randomized, with patients being randomly assigned to receive either vitamin A oil or the placebo. The variables that should be measured include the level of pain reduction after treatment, the age of the patients, the severity of the pinched nerve, and any side effects experienced by the patients.
Part B: If the study consisted of 100 young patients and 100 old patients instead of 200 patients aged 20–40, the study design would need to be modified. Age is an important variable that could affect the results of the study. To control for this variable, the study could be designed to have two separate treatment groups: one group of young patients and one group of old patients. This would allow for the comparison of pain reduction between the two age groups and would provide more accurate results.
Part C: To construct 90% confidence intervals for the proportion of pain reduction for each treatment group, we can use the following formula:
CI = p ± z*(√(p(1-p)/n))
where p is the proportion of patients reporting pain reduction, z is the critical value from the standard normal distribution for a 90% confidence interval (which is 1.645), and n is the total number of patients in the treatment group.
For the vitamin A oil group:
p = 0.37
n = 100
z = 1.645
CI = 0.37 ± 1.645*(√(0.37(1-0.37)/100))
CI = 0.37 ± 0.095
CI = (0.275, 0.465)
Therefore, the 90% confidence interval for the proportion of pain reduction for the vitamin A oil group is (0.275, 0.465).
For the placebo group:
p = 0.21
n = 100
z = 1.645
CI = 0.21 ± 1.645*(√(0.21(1-0.21)/100))
CI = 0.21 ± 0.091
CI = (0.119, 0.301)
Therefore, the 90% confidence interval for the proportion of pain reduction for the placebo group is (0.119, 0.301).
Part D: Based on the data collected, we can conclude that vitamin A oil is effective in reducing pain levels in patients suffering from pinched nerves in the cervical region of the back. The 90% confidence interval for the proportion of pain reduction for the vitamin A oil group (0.275, 0.465) does not overlap with the 90% confidence interval for the proportion of pain reduction for the placebo group (0.119, 0.301), indicating a statistically significant difference between the two groups. However, further research may be needed to confirm these results and to investigate any potential side effects of vitamin A oil treatment.
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Last 4 I needdd help please
Answers
Answer: Number 10 is you are a rectangle
Step-by-step explanation:
Find the volume of the hemisphere. Round to the nearest tenth.
Could someone please help me?
Answers
Step-by-step explanation: I hope this helps.
Answer:
why does the vertical line test tell us whether the graph of a relation represents a function? when a vertical line intersects the graph of a relation more than once, it indicates that for that input there is more than one output, which means the relation is not a function. when a vertical line intersects the graph of a function more than once, it indicates that for that output there is more than one input, which means the function is not a relation. when a vertical line intersects the graph of a function more than once, it indicates that for that input there is more than one output, which means the function is not a relation. when a vertical line intersects the graph of a relation more than once, it indicates that for that output there is more than one input, which means the relation is not a function.
Answers
When a vertical line intersects the graph of a function more than once, it indicates that for that output there is more than one input, which means the function is not a relation. (option b).
In mathematics, functions are an essential concept used to describe relationships between sets of numbers. A function is a set of ordered pairs, where each input corresponds to exactly one output.
Conversely, if every vertical line intersects the graph at most once, then the graph represents a function. This is because there is only one output for each input, satisfying the definition of a function.
It's important to note that not all relations are functions. A relation is a set of ordered pairs, while a function is a relation where each input corresponds to exactly one output.
Therefore, if a vertical line intersects a graph more than once, it indicates that there is more than one output for that particular input, and the relationship is not a function.
Hence the option (b) is correct.
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Mr. Smith keeps track of his students' homework completion. He keeps track of how many boys and
girls do not complete their homework. He puts students who don't complete their homework into two categories: first-time offenders and repeat offenders. He uses a table to keep track of the results.
Boys-Girls-Total-First-Time Offenders & Repeat Offenders
a. In one month 36 girls and 12 boys did not do their homework for the first time. 12 girls and 30
boys did not do their homework again. Put these figures in your table.
b.How many students did not complete all of their homework assignments this month?
c. What percentage of the students who did not complete their homework were boys who were
First-Time Offenders?
d. Are boys or girls more likely to not complete their homework? Explain your reasoning.
Answers
Answer:
The question seems to be incomplete. It mentions that Mr. Smith keeps track of his students' homework completion and how many boys and girls do not complete their homework. It also mentions that he puts students who don't complete their homework into two categories: first-time offenders and repeat offenders, and uses a table to keep track of the results[1][2]. However, it does not provide enough information to answer a specific question.
If you have a specific question related to this topic, please provide more details so that I can assist you better.
help asap assignment closes soon!
Answers
The approximate surface area of the cone is equal to 35 ft²
How to calculate surface area of a cone?In Mathematics, the surface area (SA) of a cone can be calculated by using this mathematical expression:
Surface area (SA) of a cone = πr(l + r)
Where:
l represents the slant height of the cone.r represents the radius of the cone.Note: Radius = diameter/2 = 2/2 = 1 feet.
By substituting the given parameters into the formula for the surface area (SA) of a cone, we have the following;
Surface area (SA) of a cone = 3.14 × (1 + 10)
Surface area (SA) of a cone = 3.14 × (11)
Surface area (SA) of a cone = 34.54 ≈ 35 ft²
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the number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is:
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Thus, the number of 6-digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is 20.
What is the number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated?There are 1200 6-digit numbers that can be formed using the digits 0, 1, 2, 5, 7, and 9.Using the formula for a 2-digit number, namely A and B, the number must satisfy A - B = 0,11 or -11.
There is a pattern, therefore, there are 100 numbers that satisfy each. It's because the numbers 0, 1, 2, 5, 7, and 9 are such that the sum of the odd digits minus the sum of the even digits is always either 0, 11, or -11. This is only possible if there is an even number of odd digits or an even number of even digits.
There are 6C3 ways to pick 3 odd digits from {1,5,7,9} and 3C3 ways to pick 3 even digits from {0,2}. By taking the product of these values, we can find the total number of possibilities:[tex]6C3 × 3C3 = 20 × 1 = 20[/tex]
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-16-5x=-16. please help
Answers
Answer:
x = 0
Step-by-step explanation:
-16-5x=-16. please help
-16 - 5x = -16
5x = -16 + 16
5x = 0
x = 0 : 5
x = 0
------------------
check
-16 - 5 × 0 = -16 (remember PEMDAS)
-16 = -16
the answer is good
Answer:
x = 0
Step-by-step explanation:
- 16- 5x = - 16
+ 16 + 16
- 5x = 0
x = 0
what does 56 3/4 ( 1 3/8 + 1 1/5) equal
Answers
Answer:
23,381/160 or 146 21/160
Step-by-step explanation:
56 3/4 = 227/4
1 3/8 = 11/8
1 1/5 = 6/5
227/4( 11/8 + 6/5)
227/4(55/40 + 48/40)
227/4( 103/40) = 23,381/160 or 146 21/160
PLS HELP
One pint of paint covers 100 square feet. What is the least amount of pints of paint needed to paint the walls of a room in the shape of a rectangular prism with a length of 16 feet, a width of 14 feet, and a height of 11 feet?
Answers
To find the surface area of the rectangular prism, we need to calculate the area of each of the six sides and then add them together. The area of each of the four walls is the product of the length and height, which gives us:
16 feet × 11 feet = 176 square feet (for the two longest walls)
14 feet × 11 feet = 154 square feet (for the two shorter walls)
The area of the ceiling and floor is the product of the length and width, which gives us:
16 feet × 14 feet = 224 square feet (for the ceiling)
16 feet × 14 feet = 224 square feet (for the floor)
To find the total surface area, we add up the areas of all six sides:
176 + 176 + 154 + 154 + 224 + 224 = 1104 square feet
Therefore, we need at least 1104/100 = 11.04 pints of paint to cover the walls of the room. Rounding up to the nearest whole pint, we need 12 pints of paint.
Points-
Pls answer this correctly!
Answers
After 60 minutes in the refrigeration, the can of soda will be approximately 42°F.
What is refrigeration?
We are given the following information:
[tex]T_{a}[/tex] = 35°F (temperature of the refrigerator)
To = 73°F (initial temperature of the can of soda)
T = 58°F (temperature of the can of soda after 10 minutes)
We can use these values to find the decay constant k:
T = [tex]T_{a}[/tex] + (To - [tex]T_{a}[/tex])[tex]e^{-kt}[/tex]
58 = 35 + (73 - 35)[tex]e^{-10k}[/tex]
23 = 38[tex]e^{(-10k)}[/tex]
ln(23/38) = -10k
k ≈ 0.0725 (rounded to the nearest thousandth)
Now we can use this value of k to find the temperature of the can of soda after 60 minutes:
T = [tex]T_{a}[/tex] + (To - [tex]T_{a}[/tex])[tex]e^{-kt}[/tex]
T = 35 + (73 - 35)[tex]e^{(-0.0725*60x^{2} )}[/tex]
T ≈ 42°F (rounded to the nearest degree)
Therefore, after 60 minutes in the refrigerator, the can of soda will be approximately 42°F.
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Can somebody please help me with this Geometry asap. In advance, thank you so much.
Answers
6.61 is the value of x in the circle.
What does the circle's tangent look like?
A line that only has one point of intersection with a circle is said to be tangent to the circle. Point of contact is the term used to describe the location where the circle and the tangent meet. A straight line that touches a circle at point P is said to be tangent to the circle at P. As the radius connects the circle's centre to point P, the tangent is perpendicular to that radius.
(39x - 4) = 254
39x = 254 + 4
39x = 258
x = 258/39
x = 6.61
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suppose we want to test whether there is a difference between the room rates of luxury hotels in new york versus los angeles. we collect data from 20 random luxury hotels in new york and 30 random luxury hotels in l.a. what are the degrees of freedom associated with a pooled-variance two-sample t-test or confidence interval? type your answer here
Answers
The degrees of freedom associated with a pooled-variance two-sample t-test or confidence interval in the given case would be 48.
When a pooled-variance two-sample t-test or confidence interval is to be conducted on two groups, the degrees of freedom is calculated using the following formula: df = n1 + n2 - 2, where n1 and n2 are the sample sizes of the two groups being compared.
In the given scenario, data is collected from 20 random luxury hotels in New York and 30 random luxury hotels in Los Angeles. Therefore, the degrees of freedom for a pooled-variance two-sample t-test or confidence interval would be:df = 20 + 30 - 2df = 48. Hence, the degrees of freedom associated with a pooled-variance two-sample t-test or confidence interval are 48.
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how do you solve for x when the equation is 3x+15=45
Answers
Answer:
x = 10
Step-by-step explanation:
To solve for x, we need to get x by itself.
Whatever we do to one side, we do it to the other side to keep it balanced.
3x + 15 = 45
- 15 = -15 (Subtract 15 from both sides)
3x = 30 (Divide both sides by 3)
3 = 3
x = 10
What happens when you use each of the following on a inequality number line?
Answers
Answer: If you add the same number to both sides of an inequality, the inequality remains true. If you subtract the same number from both sides of the inequality, the inequality remains true. If you multiply or divide both sides of an inequality by the same positive number, the inequality remains true.