Therefore , the solution of the given problem of average comes out to be the average price of a pizza in 2000 was about $9 (or half of $18).
Explain average.An organised collection's median value is the precise value that makes up the collection's mean. In this case, the ratio between the lowest and highest 50% of the data is a normal but rather probability measure. When finding the middle and mode, a variety of algorithms can be applied in order to identify any unusual or even amounts of values.
Here,
By dividing 72 by the interest rate, we can calculate how many years it will take an investment to double in value at a specific interest rate. This is known as the law of 72. In this instance
With an inflation rate of 3.25 percent, it takes roughly how many years for the price of a pizza to double as a result of inflation:
72 / 3.25 = 22.15 years
In light of this, we can calculate that a pizza cost $9 (half of $18) roughly 22 years ago, which corresponds to the following year:
2022 - 22 ≈ 2000
As a result of inflation running at an average of 3.25% for many years, the average price of a pizza in 2000 was about $9 (or half of $18).
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Please help! If you could please also explain so I understand how it is done. Thank you :)
Mr. Paquette has 2 pencils and 6 pens after simplification.
What is a Equation?An equation is a mathematical statement that asserts that two expressions are equal. It is a representation of a relationship between two or more variables, which can be solved to find the values of those variables that satisfy the equation. An equation typically consists of two sides separated by an equal sign (=), with expressions containing variables and constants on each side.
What is simplification?Simplification refers to the process of making something simpler or easier to understand. It involves breaking down complex or complicated information into smaller, more manageable parts, and eliminating unnecessary or redundant details.
Let's assume that Mr. Paquette has x pencils. Since he has a combined total of 8 pencils and pens, he must have (8 - x) pens.
We know that for each pencil he has, he has 3 pens. Therefore, the total number of pens he has is 3 times the number of pencils:
Total number of pens = 3x
We also know that the total number of pencils and pens is 8. Therefore, we can write an equation:
x + 3x = 8
Simplifying and solving for x, we get:
4x = 8
x = 2
So Mr. Paquette has 2 pencils and 6 pens (since he has 3 pens for each pencil).
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Identify the following: · Statistic · Significance Level · Test Statistic · P-value And then complete the 4 step process to assess this claim: In a random sample of 300 patients, 21 experience nausea. A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer’s’ disease will experience nausea. Test this claim with α= 0.05.
10%.
Identified are:StatisticSignificance LevelTest StatisticP-valueStep 1: State the null and alternative hypothesis for the claimThe null hypothesis H0: P≥0.1 (or P=0.1)The alternative hypothesis H1: P<0.1Step 2: Determine the significance levelα= 0.05Step 3: Compute the test statistic, P-value, and critical value.Test statistic is given as z= (x- μ) / (σ/√n)Where x= 21 (number of patients who experienced nausea)μ= nP = 0.1(300)= 30σ= √(n*P*(1-P)) = √(300*0.1*0.9) = 4.74z= (21-30)/(4.74/√300)= -5.87P-value= P(z<-5.87)≈0Critical value= zα= z0.05= -1.64Step 4: Make a decisionReject H0 if z≤zα else do not reject H0z= -5.87 < zα= -1.64Hence, reject the null hypothesis, which means there is evidence that the proportion of patients who experience nausea from taking the drug is less than 10%.
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what is the value of v100
Step-by-step explanation:
I'm sorry, but I cannot provide a specific answer to this question without additional context. The value of v100 could refer to many different things depending on the context.
If you could provide more information about what v100 represents or what it pertains to, I would be happy to try and help you with your question.
Pay your bills: In a large sample of customer accounts, a utility company determined that the average number of days between when a bill was sent out and when the payment was made is 30 with a standard deviation of 3 days. Assume the data to be approximately bell-shaped.
Between what two values will approximately 68% of the numbers of days be?
Approximately 68% of the customer accounts have payments made between 23 and 37 days.
How the mean and standard deviation is related: The formula for calculating the standard deviation uses the mean.
In actuality, we cannot determine a sample's standard deviation without first knowing what the sample means.
We want to calculate what two values will approximately 68% of the number of days be.
For a bell-shaped distribution, we can apply the 68-95-99.7 rule, which states that approximately 68% of the data will fall within 1 standard deviation from the mean.
Then, for a mean of 30 and a standard deviation of 7, we can calculate the two values as:
[tex]X_1=u+z_1.n=30-1.7=30-7=23\\X_2=u+z_2.n=30+1.7=30+7=37,[/tex]
Approximately 68% of the customer accounts have payments made between 23 and 37 days.
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Can some on solve this?
The length οf the missing side is 10 cm.
What is a triangle?A triangle is a clοsed, twο-dimensiοnal geοmetric shape that has three straight sides and three angles. It is οne οf the mοst basic shapes in geοmetry and is used extensively in mathematics, science, and engineering.
Tο sοlve fοr x, we can use the Pythagοrean theοrem which states that in a right triangle, the square οf the hypοtenuse (the side οppοsite the right angle) is equal tο the sum οf the squares οf the οther twο sides.
In this case, we can see that the hypοtenuse is 10 cm and οne οf the οther sides is x cm. The remaining side is then (10 - x) cm (since the triangle is isοsceles).
Applying the Pythagorean theorem, we get:
x² + (10 - x)² = 10²
x² + 100 - 20x + x² = 100
2x² - 20x = 0
2x(x - 10) = 0
So x = 0 or x = 10. However, x cannot be 0 since it would not form a triangle. Therefore, the only valid solution is x = 10.
Therefore, the length of the missing side is 10 cm.
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create an art work from home in any theme that will alllow you to aply youre learnings on angle pairs and lines. The art work should be made of composed od corguent traingles
u have to pay me to do it for you thanks
Writing Suppose the relationship between x and y is proportional. When x is 29, y is 275. 5. Find the constant of proportionality of y to x. Use the constant of proportionality to find x when y is 408. 5. Use pencil and paper. Explain how you can tell a relationship that is proportional from a relationship that is not proportional
If x and y have a proportional relationship, we may write: y = kx where k is the proportionality constant. To calculate k, we may utilise the supplied data point (x=29, y=275.5) and plug .
The following values into the equation: 275.5 = k(29) (29) When we solve for k, we get: k = 275.5/29 ≈ 9.5 As a result, the proportionality constant is 9.5. We can utilise this constant to obtain x for y=408.5 by plugging y=408.5 and k=9.5 into the equation and solving for x: 408.5 = 9.5x x = 408.5/9.5 ≈ 43 As a result, when y equals 408.5, x is around 43. By charting the data and seeing if the points fall on a straight line flowing through the origin, we may determine if a connection is proportional (0,0). If the points are lost.
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3 A cake is made of three round layers. The diameters and heights are in the ratio 5:3:2 The cake's bottom layer has a diameter of 30cm and a height of 10cm. Work out the following, giving your answers to the nearest whole number. cm (1 mark ) a
The diameter of the middle layer is 18 cm and the height is 9 cm. The diameter of the top layer is 12 cm and the height is 5 cm.
The ratios of the diameters and the heights are 5:3:2. Therefore, to calculate the diameters and heights of the other two layers, we need to take the bottom layer's diameter and height as our reference values.
The diameter of the bottom layer is 30 cm and the height is 10 cm.
To find the diameter and height of the middle layer, we multiply the bottom layer's values by 3/5 and then 3/2 respectively. So, the diameter of the middle layer is (30 x 3/5) cm = 18 cm and the height is (10 x 3/2) cm = 9 cm.
To find the diameter and height of the top layer, we multiply the bottom layer's values by 2/5 and then 1/2 respectively. So, the diameter of the top layer is (30 x 2/5) cm = 12 cm and the height is (10 x 1/2) cm = 5 cm.
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A circus has 15 performers, of which 5 are clowns. What is the probability that a randomly selected performer will be a clown?
Write your answer as a fraction or whole number
the probability of selecting a clown is 1/3, which is the answer to the problem.
The probability of an event happening is defined as the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the total number of performers is 15, and the number of clowns is 5. Therefore, the probability of selecting a clown is:
P(clown) = number of clowns / total number of performers
= 5 / 15
= 1/3
Probability is a branch of mathematics that deals with the study of random events and their likelihood of occurrence. It is defined as the measure of the likelihood or chance of an event occurring.
The probability of an event is expressed as a number between 0 and 1, where 0 means that the event is impossible, and 1 means that the event is certain to happen. An event with a probability of 0.5 (or 50%) is considered to be equally likely to happen or not happen.
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are used to represent an unknown quantity in a mathematical expression.
Answer:
Variables
Step-by-step explanation:
A variable is a quantity that can be changed and is not fixed. Variables are essential as they form a major algebra component. We usually use “x” and “y” to express an unknown integer.
answer the question below
The time it takes to empty the tank is given as follows:
1.85 minutes.
How to obtain the volume of a cylinder?The volume of a cylinder of radius r and height h is given by the equation presented as follows, which the square of the radius is multiplied by π and the height.
V = πr²h.
The parameters for this problem are given as follows:
r = 0.7 ft, as the radius is half the diameter.h = 3 ft.Hence the volume of the cylinder in ft³ is given as follows:
V = π x 0.7² x 3
V = 4.62 ft³.
Oil is drained at a rate of 2.5 ft³ per minute, hence the time it takes to drain the oil is given as follows:
4.62/2.5 = 1.85 minutes.
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Students in the high school choir sing one of four voice parts.
sopranos 5
altos 22
basses 8
tenors 5
What is the probability that a randomly selected singer will be a soprano?
The probability of selecting a soprano is 0.125 or 12.5%.
What distinguishes a probability from an odds ratio?The percentage of times you anticipate seeing a certain occurrence across many trials is the chance that it will happen. Probabilities are always in the 0 to 1 range. The probability of an event happening divided by the likelihood that it won't happen is how odds are calculated.
Probability 0 is equivalent to chances 0. Odds less than 1.0 are represented by probabilities between 0 and 0.5. The chances of 1.0 are the same as a probability of 0.5. Consider it like this: 50% of the time, a coin will land on its head. The chances are "50:50," which translates to 1.0.
The total number of singers are:
5 + 22 + 8 + 5 = 40
Now, the probability of selecting a soprano is:
P(soprano) = 5 / 40
P(soprano) = 1/8 or 0.125
Therefore, the probability of selecting a soprano is 0.125 or 12.5%.
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Express x^2 + 12x + 11 in the form (x+p)^2 + q
To convert a quadratic expression [tex]ax^2 + bx + c[/tex] to the form [tex](x+p)^2 + q[/tex], where p=b/2a and [tex]q=(b^2-4ac)/4a[/tex], complete the square by adding and subtracting [tex](b/2)^2[/tex] and simplify the resulting expression.
To express the quadratic expression [tex]x^2 + 12x + 11[/tex] in the form [tex](x+p)^2 + q[/tex], we need to complete the square by adding and subtracting a suitable constant term.
First, we take half of the coefficient of x (which is 12) and square it, giving [tex](12/2)^2 = 36[/tex]. We then add and subtract this value inside the bracket:
[tex]x^2 + 12x + 11 = x^2 + 12x + 36 - 36 + 11[/tex]
Now we can group the first three terms and write them as a square of a binomial:
[tex]x^2 + 12x + 36 = (x + 6)^2[/tex]
Simplifying the remaining terms, we have:
[tex]x^2 + 12x + 11 = (x + 6)^2 - 25[/tex]
Therefore, we have expressed the quadratic expression [tex]x^2 + 12x + 11[/tex] in the form [tex](x+p)^2 + q[/tex], where p = 6 and q = -25.
In general, to express a quadratic expression of the form [tex]ax^2 + bx + c[/tex] in the form [tex](x+p)^2 + q[/tex], where a, b, and c are constants and p and q are to be determined, we can follow these steps:
Take half of the coefficient of x (which is b/2) and square it, giving [tex](b/2)^2[/tex].
Calculate this value inside the bracket:
[tex]ax^2 + bx + c = ax^2 + bx + (b/2)^2 - (b/2)^2 + c[/tex]
Group the first three terms and write them as a square of a binomial:
[tex]ax^2 + bx + (b/2)^2 = a(x + b/2a)^2[/tex]
Simplify the remaining terms:
[tex]ax^2 + bx + (b/2)^2 + c - (b/2)^2 = a(x + b/2a)^2 + (4ac-b^2)/4a[/tex]
We can simplify the expression (4ac-b^2)/4a by recognizing that it is the discriminant of the quadratic equation [tex]ax^2 + bx + c = 0[/tex], which is [tex]b^2 - 4ac[/tex]. Therefore, we have:
[tex]ax^2 + bx + c = a(x + b/2a)^2 + (b^2 - 4ac)/4a[/tex]
We can rewrite this in the form [tex](x+p)^2 + q[/tex] by identifying p and q:
p = b/2a, and [tex]q = (b^2 - 4ac)/4a[/tex].
Therefore, the quadratic expression [tex]ax^2 + bx + c[/tex] can be expressed in the form [tex](x+p)^2 + q[/tex], where p = b/2a and [tex]q = (b^2 - 4ac)/4a[/tex].
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Make 'h' the subject of the following equation:
k = 3h - 1
The mathematical expression that would make "h" the subject of the equation is h= k-1/3
What is an equation?To begin an equation is a mathematical expression that relates variables by using the = symbol. Due to this, the letters of variables involved can be changed to create equivalent equations.
How to make the variable h the subject of the equation?The equation given is written is k= 3h - 1. To make h the subject of formula 3h= k - 1 Divide both sides by the coefficient of h which in this case is 33h/3= k-1/3h= k - 1/3.
Hence the value of h in the equation is k - 1/3
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Find the range of the data set: 10, 18, 21, 28, 12, 20. A. 10 B. 12 C. 18 D. 20
Answer:
18
Step-by-step explanation:
The range of the data set is 18, which is calculated by subtracting the smallest value (10) from the largest value (28).
Decrease £14132.77 by 14.5%
Give your answer rounded to 2 Dp
Answer:
900
Step-by-step explanation:
because 14132.77/14.5%
Suppose that, for a certain mathematics class, the scores are normally distributed with a mean of 75 and a standard deviation of 8. The teacher wishes to give A's to the top 5% of the students and F's to the bottom 5%. The next 15% in either direction will be given B's and D's, with the other students receiving C's. Find the bottom cutoff for receiving an A grade.
The bottom cutoff for receiving an A grade is 88.16.
What is the statistical application of the empirical rule?A statistical concept known as the empirical rule, often called the 68-95-99.7 rule, asserts that for a normal distribution: Around 68% of the data falls within one standard deviation of the mean.
The data is within two standard deviations of the mean for around 95% of the time. The data is within three standard deviations of the mean for about 99.7% of the time.
To find the bottom cutoff we need to find 5% of the students that score higher.
Using the z-score for 5% distribution we have:
z = 1.645
Formula for converting a z-score to a raw score:
z = (x - μ) / σ
x = z * σ + μ
x = 1.645 * 8 + 75
x = 88.16
Hence, the bottom cutoff for receiving an A grade is 88.16.
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Which sign makes the statement true?
Answer: <
It is "<" because 8 1/2 as a decimal is 8.50 and the other decimal is 8.375. 8.50 is greater than 8.375.
Your welcome!
=^..^=
Answer:<
Step-by-step explanation:
8.375= [tex]8\frac{375}{1000} =8\frac{3}{80} \\[/tex]
1/2=40/80
40/80>3/80
so,
Simplify the expression
Answer:
2 Is the Ans
Step-by-step explanation:
As per the Picture....
could you please help me thank you and I think the values of X might be wrong thank you .
and I am sorry for a little bit of points it's because I don't have a lot I only have one hundred.
The area of the triangle is 5√11 cm².
What are angles?
A point where two lines meet produces an angle.
The breadth of the "opening" between these two rays is referred to as a "angle". It is depicted by the figure.
Radians, a unit of circularity or rotation, and degrees are two common units used to describe angles.
By connecting two rays at their ends, one can make an angle in geometry. The sides or limbs of the angle are what are meant by these rays.
The limbs and the vertex are the two main parts of an angle.
The common terminal of the two beams is the shared vertex.
According to our question-
b = base of the triangle = 2(√6² - 5²) = 2(√36 - 25) = 2√11 cm
h = height of the triangle = 5 cm
area of the triangle = 1 / 2 × 2√11 × 5
area of the triangle = 5√11
area of the triangle = 5√11 cm²
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Caitlyn sells xylophones and yo-yos at the county fair. She wants to sell more than 30 toys. She sells the
xylophones for $3. 00 and the yo-yos for $0. 50. She needs to sell more than $40. 00 worth of toys in order to
earn a profit.
a) Write a system of inequalities (where is the number of xylophones and y is the number of yo-yos).
Answer:
x + y > 30
3x + 0.5y > 40.00
Step-by-step explanation:
Let x be the number of xylophones and y be the number of yo-yos sold.
The total number of toys sold must be greater than 30, so we have:
x + y > 30
The total amount earned must be greater than $40.00 to earn a profit, so we have:
3x + 0.5y > 40.00
Therefore, the system of inequalities is:
x + y > 30
3x + 0.5y > 40.00
Jewellery Libby makes and sells jewellery. Libby is making a bracelet. She uses white beads and blue beads in the ratio 1:3 Libby uses 6 white beads for the bracelet. She has 10 blue beads. How many m
Libby requires an additional 8 blue beads to make a bracelet combing white beads and blue beads in a ratio of 1:3.
What is the ratio?The ratio refers to the relative size of one value or quantity compared to another.
Ratios are expressed in percentages, fractions, or decimals and can also be expressed in the standard ratio forms (:) to show the comparative relationship one value bears to another.
The ratio of white beads to blue beads = 1:3
The sum of ratios = 4 (1 + 3)
The number of white beads = 6
The total number of beads = 24 (6 ÷ ¹/₄)
The number of blue beads required = 18 (24 - 6)
The number of blue beads Libby has = 10
The additional number of blue beads required = 8 (18 - 10)
Thus, based on the ratio of white beads to blue beads, Libby requires additional 8 blue beads to make a bracelet.
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Question Completion:How many more blue beads does Libby require?
please address those questions please type
1, You have 7 identical cookies to distribute to 4 kids.
a, How many ways can you distribute the cookies?
2, how many three colors flag can you make with the colors white, red and blue.
The sοlutiοn οf the given prοblem οf unitary cοmes οut tο be 1.20 and 2.27 .
What is an unitary methοd?By cοmbining the infοrmatiοn οbtained using this nanοsectiοn technique with all variable additiοnal data frοm twο individuals whο used a specific strategy, the task can be cοmpleted. Simply put, this mean that, if the desired οutcοme materialises, either the specified entity will be knοwn οr, actually, the cοlοur frοm bοth enοrmοus prοcesses will be skipped. Fοr fοrty pens, a refundable charge οf Rupees ($1.01) might be required.
Here,
There are several apprοaches tο the issue οf giving 4 children each 7 identical cοοkies, but οne pοpular strategy is tο use a stars and bars strategy.
Cοnsider 7 stars as the 7 biscuits and 3 bars as the 3 partitiοns separating the 4 children. Then, any cοnfiguratiοn οf stars and bars represents a pοtential cοοkie dispersal.
=> (10 + 3) chοοse 3 = 120 / 6 = 7 + 3 chοοse 3 = 20
The fοur children can therefοre share the cοοkies in 20 variοus ways. The number οf ways tο create a three-cοlοr flag is as fοllοws because we have three cοlοur οptiοns and can repeat cοlοurs in variοus ways:
=> 3 x 3 x 3 = 27
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Complete question:
Review Exercise
Acoin and a dice, labeled with letters A, B, C, D, E and F are tossed simultaneously. Find the
probability of getting a head and a vowel.
Answer:
Independent events and their probability
Carmen buys candy that costs 7 $ per pound. She will spend more than 56$ on candy. What are the possible numbers of pounds she will buy?
Use p for the number of pounds Carmen will buy.
Write your answer as an inequality solved for p.
write as an Inequality
We can say that after answering the offered question Hence, p > 8 is the equation inequality expressing the range of feasible values for p.
What is equation?An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.
Let's use the letter "p" to indicate how many pounds Carmen will spend.
Candy is $7 for every pound.
Carmen's sweets expenditure is expressed as follows:
Cost per pound times the weight of the candy purchased is 7p.
Carmen will undoubtedly spend more than $56 on sweets, therefore we may say:
More than $56 was spent on candy
Using the following equation in place of the amount spent on candy:
7p > $56
When we multiply both sides by 7, we get:
p > 8
So, any figure more than 8 is a possibility for the amount of pounds Carmen will purchase. We can express this as follows:
p > 8
Hence, p > 8 is the inequality expressing the range of feasible values for p.
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The volume, in cubic centimetres, of a rectangular box can be modelled by the polynomial expression 2x3−11x2−41x+140. Determine an equation to represent the area of the base of the box if the height, in centimetres, is given by 2x−5. [APP 3 marks]
The equation for area of the base of the box will be x² - 6x + 28.
What is Volume ?
In physics and mathematics, volume is a measure of the amount of three-dimensional space occupied by an object or a region. It is often measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet.
The volume (V) of a rectangular box is given by the product of its length (l), width (w), and height (h):
V = lwh
Given that the volume is modelled by the polynomial expression 2x³- 11x² - 41x + 140, we can write:
V = 2x³ - 11x² - 41x + 140
And since the height is given by 2x - 5, we can substitute this expression into the formula for volume:
V = lwh = (lw)(2x - 5)
So we need to find an equation for the area of the base (lw). We can do this by rearranging the formula for volume:
lw = V / (2x - 5)
Substituting the expression for V, we get:
lw = (2x³ - 11x² - 41x + 140) / (2x - 5)
We can simplify this expression by polynomial long division or synthetic division, which gives:
lw = x² - 6x + 28
Therefore, the equation for the area of the base of the box is:
A = x² - 6x + 28, where A is the area in square centimeters, and x is the length in centimeters.
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When r is not significantly different from 0, the best predictor of y is the ________ of the data values of y
The best predictor of y when r is not significantly different from 0 is the mean of the data values of y, which can be calculated using the formula (∑y)/n.
The best predictor of y when r is not significantly different from 0 is the mean of the data values of y. This can be calculated using the formula:
mean = (∑y)/n
Where ∑y is the sum of all the y values in the data set and n is the number of y values in the data set.
For example, if the data set is comprised of five y values of -2, -1, 0, 1, and 2, we would have ∑y = -2 + -1 + 0 + 1 + 2 = 0. The number of y values, n, is 5. Therefore, the mean of the data would be 0/5 = 0.
The mean of the data is a useful predictor of y when r is not significantly different from 0 because it is an average of all the values in the data set. This means that it takes into account all the data points and can give a reliable prediction of what the value of y is likely to be. Since r is not significantly different from 0, this means that there is no strong relationship between the two variables, and so the mean of the data can be used to provide an accurate prediction of y.
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8x=__when x= 3
pls tell me
Answer:
24
Step-by-step explanation:
(14) If z1 = 2-3i and z2=3+i, find
(a) z1
a) 2 - 3i.
Given that z1 = 2-3i and z2=3+i. We are to find a) z1. Let's learn the concept of complex numbers and solve the problem. The complex number is a combination of a real number and an imaginary number expressed as a + bi, where a and b are real numbers, and i is an imaginary number. The given complex numbers are,z1 = 2-3iz2 = 3+i(a) z1. To find z1, it is already given as 2-3i. Therefore, the answer is 2 - 3i. Hence, a) z1 is 2 - 3i.
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A fair de is rolled sx times. What is the probabiity that it comes up 4 at least once? Wrte your answer as a froction or a decimal, rounded to four decimal places.
0.4103
Answer:GivenA fair die is rolled six times.The probability that it comes up 4 at least once is to be determined.We have to find the probability that it comes up 4 atleast once which means that it can come up more than once as well.Therefore we have to consider all the cases i.e 1,2,3,4,5,6Case 1 : Number of times that it comes up 4 is 1The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(1/6) * (5/6)^5Case 2 : Number of times that it comes up 4 is 2The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C2 * (1/6)^2 * (5/6)^4)Case 3 : Number of times that it comes up 4 is 3The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C3 * (1/6)^3 * (5/6)^3)Case 4 : Number of times that it comes up 4 is 4The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C4 * (1/6)^4 * (5/6)^2)Case 5 : Number of times that it comes up 4 is 5The remaining number of times can be any of the remaining 5 numbersTherefore the probability of the event is(5C5 * (1/6)^5 * (5/6)^1)Now we will add all the cases to get the probability that it comes up 4 atleast once.P(A) = (1/6) * (5/6)^5 + (5C2 * (1/6)^2 * (5/6)^4) + (5C3 * (1/6)^3 * (5/6)^3) + (5C4 * (1/6)^4 * (5/6)^2) + (5C5 * (1/6)^5 * (5/6)^1)On solving we get P(A) = 0.4103Therefore the required probability is 0.4103 or 41.03% (approximately)Answer : 0.4103
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