The probability that the random variable will assume a value between 45 and 55 is given by: 0.6827
A random variable is normally distributed.
Mean μ = 50
Standard deviation σ = 5
According to the empirical rule, also known as 68-95-99.7 rule, the percentage of values that lie within an interval with 68%, 95% and 99.7% of the values lies within one, two or three standard deviations of the mean of the distribution.
The more precise statement for 68 percent is:
[tex]P(\mu-\sigma < X < \mu+\sigma) = 68.27%[/tex]
Since the interval of interest given in question is (45,55), it can be rewritten as (50-5, 50 + 5)
Thus we have:
P(50 - 5 < X < 50+5) = 68.27%
P(45 < X < 55) = 0.6827.
Thus, 0.6827 is the probability that the random variable will assume a value between 45 and 55.
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for the following problem write the simplest polynomial function with the given zeros: 2, -1, and -8
The simplest polynomial function with the given zeros 2, -1, and -8 is:
f(x) = x³ + 7x² - 6x - 16.
the simplest polynomial function with the given zeros: 2, -1, and -8?Given the roots or zeroes: 2, -1, and -8
Polynomial function f(x) = ?
If the given zeros are 2, -1, and -8, then the corresponding factors of the polynomial function are:
(x - 2), (x + 1), and (x + 8).
The simplest polynomial function with these zeros is the product of these factors:
(x - 2)(x + 1)(x + 8)
Expanding this product gives:
x³ + 7x² - 6x - 16
Hence;
The polynomial function is f(x) = x³ + 7x² - 6x - 16.
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PLEASE BROSKIIII PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST
Answer:
d.
Step-by-step explanation:
Select the correct answer. The sum of two consecutive numbers is 157. This equation, where n is the first number, represents the situation: 2n + 1 = 157. What is the first number? A. 77 B. 78 C. 79 D. 80
With the help of given expression 2n + 1 = 157, the first number is 78.
What exactly are expressions?
In mathematics, an expression is a combination of numbers, symbols, and operators that represent a value. It can be a single term, or it can be a combination of terms connected by operators. For example, 2x + 5 is an expression with two terms connected by the operator +. Expressions can also include functions, variables, and constants.
Now,
Let's assume the first number be x=n. Then the next consecutive number will be x+1.
According to the given information, the sum of the two consecutive numbers is 157.
So, we can write the equation as:
x + (x+1) = 157
Simplifying the equation, we get:
2x + 1 = 157
Subtracting 1 from both sides, we get:
2x = 156
Dividing both sides by 2, we get:
x = 78
i.e. n=78
Therefore, the first number is 78.
So, the correct answer is B) 78.
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Find the height of the basketball hoop using similarity ratios. Explain step by step.
The height of the basketball hoop is 6 units.
Step 1: Identify similar triangles.
Similar triangles are triangles with the same shape but not necessarily the same size. They have proportional sides and equal angles.
Step 2: Determine the corresponding sides and angles
Once you have identified the similar triangles, determine which sides and angles correspond with each other. The ratio of the corresponding sides in similar triangles will be equal.
Step 3: Set up a proportion
Now, set up a proportion using the corresponding sides of the similar triangles.
(side of first triangle) / (side of second triangle) = (height of first triangle) / (height of second triangle)
Step 4: Plug in the known values
Fill in the known values from the problem into the proportion. For example:
[tex](3 / 5) = (h / 10)[/tex]
Step 5: Solve for the unknown variable
Cross-multiply to solve for the unknown variable (h). In this example:
[tex]3 * 10 = 5 * h[/tex]
[tex]30 = 5h[/tex]
h = 6
So, the height of the basketball hoop is 6 units.
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2x
3
+5x
2
+6x+152, x, cubed, plus, 5, x, squared, plus, 6, x, plus, 15
Which of the following is equivalent to the expression above?
To factorize the expression 2x3 + 5x2 + 6x + 15, we can look for common factors: 2x3 + 5x2 + 6x + 15 = x2(2x + 5) + 3(2x + 5) = (x2 + 3)(2x + 5)
Therefore, option C (x2 + 3)(2x + 5) is equivalent to the given expression.
How to Solve the Problem ?To solve the problem, we need to factorize the given expression and compare it with the given options to find the equivalent expression.
The given expression is:
2x3 + 5x2 + 6x + 15
To factorize it, we can look for common factors:
2x3 + 5x2 + 6x + 15 = x2(2x + 5) + 3(2x + 5)
Now we can see that (2x + 5) is a common factor. Factoring it out, we get:
2x3 + 5x2 + 6x + 15 = (2x + 5)(x2 + 3)
Complete Question Below:
2x3 + 5x2 + 6x + 15
Which of the following is equivalent to the expression above?
A. (x + 3)(2x2 + 5)
B. (x + 5)(2x2 + 3)
C. (x2 + 3)(2x + 5)
D. (x2 + 5)(2x + 3)
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A carpenter bought 90 nails. Each nail has a mass of 5.2 × 10 − 4 kilograms. What is the total mass, in kilograms, of the nails the carpenter bought?
To find the total mass of the nails the carpenter bought, we can simply multiply the number of nails by the mass of each nail:
Total mass = 90 x 5.2 x 10^-4 kg/nail
Total mass = 0.0468 kg
Therefore, the total mass of the nails the carpenter bought is 0.0468 kg.
Evaluate the definite integral. Use a graphing utility to verify your result.
The value of the definite integral is -28/3
How to evaluate the definite integralUsing the power rule of integration, we can find the antiderivative of t^2 - 5 as follows:
∫(t^2 - 5) dt = (1/3)t^3 - 5t + C
where C is the constant of integration.
To evaluate the definite integral, we substitute the limits of integration into this expression and take the difference:
∫^-1_1 (t^2 - 5) dt = [(1/3)(1^3) - 5(1)] - [(1/3)(-1^3) - 5(-1)]
= (1/3 - 5) - (1/3 + 5)
= (-14/3) (16/3)
= -28/3
Therefore, the value of the definite integral is -28/3.
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The line of best fit is y=2x+1.5 where x represents the puppy’s age in weeks and y represents the puppy’s weight. What is the weight of the puppy when it is 15 weeks old?
To find the weight of the puppy when it is 15 weeks old, we need to substitute 15 for x in the equation of the line of best fit:
y = 2x + 1.5
y = 2(15) + 1.5
y = 31.5
Therefore, the weight of the puppy when it is 15 weeks old is 31.5 units, where the units depend on the units used to measure weight (e.g. pounds, kilograms, etc.).
How much yards is the white board the board has 54 inches
54 inches = 1 1
2
yards
Formula: divide the value in inches by 36 because 1 yard equals 36 inches.
So, 54 inches = 54
36
= 1 1
2
or 1.5 yards.
Conversion of 54 inches to other length, height & distance units
54 inches = 0.00137 kilometer
54 inches = 1.37 meters
54 inches = 137 centimeters
54 inches = 13.7 decimeters
54 inches = 1370 millimeters
54 inches = 1.372 × 1010 angstroms
54 inches = 0.000852 mile
54 inches = 3
4
fathom
54 inches = 4 1
2
feet
54 inches = 13 1
2
hands
54 inches = 12 fingers
54 inches = 0.429 bamboo
54 inches = 161 barleycorns
select all the expression that equal 4 x 10^6
A. (2 x 10^8)(2 x 10^2)
B 40 x 10^5
C 40^6
D 400,000
E 1.2 x 10^9
-------------------
3 x 10^2
The equivalent expression to the given equation is (2 x 10⁸)(2 x 10⁻²) and 40 x 10⁵.
What is an equivalent expression?
Equivalent expressions do the same thing even when they have distinct appearances. When we enter the same value(s) for the variable, two equivalent algebraic expressions have the same value (s).
Here, we have
Given: 4 x 10⁶
We have to find the equivalent expression to the given equation.
A. (2 x 10⁸)(2 x 10⁻²)
= (2 × 2)(10⁸× 10⁻²)
= 4×10⁶
B. 40 x 10⁵
= 4×10×10⁵
= 40×10⁵
Hence, the equivalent expression to the given equation is (2 x 10⁸)(2 x 10⁻²) and 40 x 10⁵.
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a random variable x is characterized by a normal probability density function with known mean 20. there are two hypotheses for the variance. hypothesis h0 claims the variance is 16 while hypothesis h1 claims the variance is 25. we will choose between these hypotheses by observing three sample values x1, x2, and x3 and applying a threshold rule of the form reject h0 if x1 x2 x3 > t. for some scalar t. determine the value of t so that the probability of false rejection is 0.05. what is the corresponding probability of false acceptance? your answers should be in terms of the q function. 1
The value of the threshold t that gives a false rejection rate of 0.05 is found to be 5.991. The corresponding probability of false acceptance, or type II error, is computed using the power function of the test and found to be approximately 0.1485.
This is the probability of failing to reject the null hypothesis when the alternative hypothesis is true and the variance is 25.
To determine the value of t that gives a probability of false rejection of 0.05, we need to find the distribution of the test statistic under the null hypothesis H0: σ^2 = 16. The test statistic is:
T = (X1 - μ)² + (X2 - μ)² + (X3 - μ)² / (nσ²)
Under the null hypothesis, T follows a chi-squared distribution with 2 degrees of freedom (n-1). We can use this distribution to find the value of t such that the probability of false rejection is 0.05.
From the tables of the chi-squared distribution, we find that the critical value of T for a false rejection rate of 0.05 is 5.991.
Thus, we reject the null hypothesis if T > 5.991.
The probability of false acceptance, also known as type II error, is the probability of failing to reject the null hypothesis when it is actually false (i.e., when H1: σ^2 = 25 is true). This probability depends on the value of σ^2 and the threshold t.
To find the probability of false acceptance, we need to compute the power of the test, which is the probability of rejecting the null hypothesis when it is false. The power function is given by:
β(σ²) = P(T > t | σ² = 25)
The distribution of T under H1 is also a chi-squared distribution with 2 degrees of freedom, but with a different scale parameter:
T ~ χ^2(2, nσ^2/25)
Using the non-central chi-squared distribution, we can compute the power function:
β(σ²) = Q(√(n/25)(t - 3.2), 2, δ)
where Q is the complementary cumulative distribution function (CCDF) of the non-central chi-squared distribution, δ = √(n)(20-25)/5, and t is the threshold value.
For t = 5.991, we have:
[tex]δ = √(3)(20-25)/5 = -1.3416\\\\β(16) = Q(√(3/25)(5.991 - 3.2), 2, -1.3416) ≈ 0.1485\\β(25) = Q(√(3/25)(5.991 - 3.2), 2, 0) ≈ 0.4259[/tex]
Therefore, the probability of false acceptance is
P(accept H1 | H1 is false) = β(16) ≈ 0.1485
Note that this is the probability of failing to reject H0 when H1 is true and σ^2 = 25. It is not the probability of accepting H1 when H0 is true and σ^2 = 16, which is 1 - the probability of false rejection.
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g the size of bass caught in strawberry lake is normally distributed with a mean of 11 inches and a standard deviation of 3 inches. suppose you catch 4 fish. what is the probability the average size of the fish you caught is more than 13 inches?
The probability that the average size of the fish you caught is more than 13 inches is approximately 0.0918, or about 9.18%.
We can use the Central Limit Theorem to approximate the distribution of the sample mean. According to the theorem, the sample mean of a sufficiently large sample will be approximately normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, we have a sample size of 4, which may not be considered sufficiently large, but we can still use the approximation. Thus, the distribution of the sample mean can be approximated as:
mean = 11
standard deviation = 3 / sqrt(4) = 1.5
To find the probability that the average size of the fish you caught is more than 13 inches, we need to standardize the sample mean using the z-score formula:
z = (x - mean) / standard deviation
where x is the sample mean we want to find the probability for. Plugging in the values, we get:
z = (13 - 11) / 1.5 = 1.33
Now, we need to find the probability of getting a z-score greater than 1.33 in a standard normal distribution table or calculator. Using a calculator or statistical software, we find that this probability is approximately 0.0918.
Therefore, the probability that the average size of the fish you caught is more than 13 inches is approximately 0.0918, or about 9.18%.
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Ben's dad is making a large pot of pasta sauce that calls for 3.5 kilograms of tomatoes. if he triples the amount of tomatoes for the recipe, how many grams of tomatoes will he use?
Answer:
10,500 grams
Step-by-step explanation:
3.5 x 3 = 10.5 kg
To convert kg to grams, multiply by 1000. To multiply by 1,000 move the decimal right 3 places.
10,500 grams
Helping in the name of Jesus.
what price do farmers get for their watermelon crops? in the third week of july, a random sample of 36 farming regions gave a sample mean of x
If a random sample of 36 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon, then the 90% "confidence-interval" is (6.3426 , 7.4174).
The sample mean of x is (x') = $6.88,
the sample standard-deviation (σ) = $1.96,
the sample size (n) is = 36,
The z value for the 90% confidence interval is = 1.645,
So, the margin of error(E) is = (z×σ)/√n = (1.645×1.96)/√36 ≈ 0.5374,
So, the interval will be = (x' ± E),
Substituting the values,
we get,
⇒ (6.88 - 0.5374 , 6.88 + 0.5374),
⇒ (6.3426 , 7.4174)
Therefore, the required 90% confidence interval is (6.3426 , 7.4174).
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The given question is incomplete, the complete question is
In the third week of July, a random sample of 36 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.96 per 100 pounds.
Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop.
What is the equation of the graph below?
A. y = − (x − 3)^2 + 1
B. y = − (x + 3)^2 + 1
C. y = (x − 3)^2 − 1
D. y = (x + 3)^2 − 1
Answer:
D
Step-by-step explanation:
the equation of a parabola in vertex form is
y = a(x - h)² + k
where (h, k ) are the coordinates of the vertex and a is a multiplier
here (h, k ) = (- 3, - 1 ) , then
y = a(x - (- 3) )² + (- 1)
y = a(x + 3)² - 1
to find a substitute the coordinates of any other point on the graph into the equation.
using (- 2, 0 )
0 = a(- 2 + 3)² - 1
0 = a(1)² - 1
0 = a - 1 ( ad 1 to both sides )
1 = a
then
y = (x + 3)² - 1 ← equation of graph
one of the two linear equations in a system is given. the system has exactly one solution. which equation could be the second equation in this system?
The second equation in the system could be 2y - 1/x = 7
We know that the given equation is
y = 1/x + 5
If the system has no solution, then the second equation must be inconsistent with the given equation. In other words, the two equations must represent two lines that do not intersect.
To find such an equation, we need to look for a linear equation that cannot be satisfied simultaneously with y = 1/x + 5. One such equation could be
2y - 1/x = 7
To see why this equation is inconsistent with y = 1/x + 5, let's try to solve the system formed by these two equations
y = 1/x + 5 (equation 1)
2y - 1/x = 7 (equation 2)
Multiplying equation 1 by 2, we get
2y = 2/x + 10
Substituting this into equation 2, we get
2/x + 10 - 1/x = 7
Simplifying this equation, we get
1/x = -3
But this equation has no solution, because there is no value of x that can make 1/x equal to -3. Therefore, the system formed by equations 1 and 2 has no solution.
The second equation is
2y - 1/x = 7
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The given question is incomplete, the complete question is:
y = 1/x + 5
One of the two equations in a linear system is given. The system has no solution. Which equation could be the second equation in this system?
select all factors of the polynomial
z^3 - 9z^2 - 7z + 63
ANSWER FAST WILL GIVE BRAINLIEST
Answer: C and D
Step-by-step explanation:
the total cholesterol level of an individual is normally distributed with a mean of 219 and a standard deviation of 41.6 . what is the probability that an individual has a total cholesterol level between 200 and 250 ? give your answer as a percent, rounded to one decimal place. for example if the probability is 0.501, your answer should be 50.1.
The probability that an individual has a total cholesterol level between 200 and 250 is calculated to be approximately 0.5020 or 50.2%.
To solve this problem, we need to find the z-scores corresponding to the lower and upper bounds of the cholesterol range, and then find the area under the normal distribution curve between these z-scores.
First, we calculate the z-score for 200:
z1 = (200 - 219) / 41.6 = -0.455
Next, we calculate the z-score for 250:
z2 = (250 - 219) / 41.6 = 0.746
Now we can use a standard normal distribution table or calculator to find the area between these two z-scores:
P(-0.455 < Z < 0.746) ≈ 0.5020
This means that the probability that an individual has a total cholesterol level between 200 and 250 is approximately 0.5020 or 50.2% (rounded to one decimal place).
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In the right triangle, find the length of the side not given. Give an exact answer and an approximation to three decimal
K
places
a=15, b=15
the exact value of C is
the approximate value of c is
Question 4(Multiple Choice Worth 2 points)
(Comparing Data MC)
The line plots represent data collected on the travel times to school from two groups of 15 students.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 4,6,14, and 28. There are two dots above 10, 12, 18, and 22. There are three dots above 16. The graph is titled Bus 47 Travel Times.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 10,16,20, and 28. There are two dots above 8 and 14. There are three dots above18. There are four dots above 12. The graph is titled Bus 14 Travel Times.
Compare the data and use the correct measure of variability to determine which bus is the most consistent. Explain your answer.
Bus 47, with an IQR of 8
Bus 14, with an IQR of 6
Bus 47, with a range of 8
Bus 14, with a range of 6
The answer to the given question about IQR is option b Bus 14, with an IQR of 6.
To determine which bus is the most consistent, we need to look at the measure of variability for each data set. In this case, we are given the interquartile range (IQR) and the range for each bus. The IQR is a better measure of variability than the range because it is less affected by outliers.
Bus 47 has an IQR of 8, which means that the middle 50% of the data falls within a range of 8 minutes. Bus 14 has an IQR of 6, which means that the middle 50% of the data falls within a range of 6 minutes.
Since Bus 14 has a smaller IQR, it is more consistent than Bus 47. This means that the travel times for Bus 14 are more similar to each other than the travel times for Bus 47. Therefore, we can conclude that Bus 14 is the most consistent of the two buses.
Therefore, the correct answer is:
Bus 14, with an IQR of 6.
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Suppose that a wave forms in shallow water. The depth d of the water (in feet) and the velocity v of the wave (in feet per second) are related by
the equation v= √32d. If a wave forms in water with a depth of 5.3 feet, what is its velocity?
Round your answer to the nearest tenth.
The graph of a quadratic function has a vertex at the point (-8,-3). It passes through the point (-2,3). When written in vertex form, the function is f(x)=a(x-h)^2+k, where:
...where h = -8 and k = -3.
what is the area of the circle us 3.14 to approximate pi pls help
Answer: 247in
Step-by-step explanation:
Area of a circle is pi radius squared.
So that'll be 3.14 times 9 (because radius is halve if a diameter) squared.
First using Order of Operations squaring the 9, so that'll be 81.
Lastly you multiply π or 3.14.
So 81 x 3.14 is 247. Which is rounded down to the nearest whole number.
Happy Solving.
Factor.
2x² + 11x + 12
(x + [?])([ ]x + [])
Enter the number that belongs in the
green box.
Answer:
2x²+11x+12
sum=11x
product =24x²
factor 3x and 8x
2x²+8x+3x+12
2x(x+4)+3(x+4)
(2x+3)(x+4)
The data given represents the number of gallons of coffee sold per hour at two different coffee shops.
Coffee Ground
1.5 20 3.5
12 2 5
11 7 2.5
9.5 3 5
Wide Awake
2.5 10 4
18 4 3
3 6.5 15
6 5 2.5
Compare the data and use the correct measure of center to determine which shop typically sells the most amount of coffee per hour. Explain.
Wide Awake, with a median value of 4.5 gallons
Wide Awake, with a mean value of about 4.5 gallons
Coffee Ground, with a mean value of about 5 gallons
Coffee Ground, with a median value of 5 gallons
The correct centre measure, with a mean value of approximately 5 gallons, will be C. Coffee Ground, with a mean value of about 5 gallons, which will show which shop usually sells the most coffee per hour.
How to explain the measure of center?By summing up all the data points and dividing by the total number of data points, the mean is determined.
To determine which coffee shop typically sells the most amount of coffee per hour, we can compare the measures of center, specifically the mean and median, of the two datasets.
Calculating the mean for each dataset, we get:
Coffee Ground: (1.5+20+3.5+12+2+5+11+7+2.5+9.5+3+5)/12 = 6.125 gallons
Wide Awake: (2.5+10+4+18+4+3+6+5+2.5)/9 = 6.0556 gallons
Therefore, Coffee Ground has a higher mean than Wide Awake, suggesting that Coffee Ground typically sells more coffee per hour.
Calculating the median for each dataset, we get:
Coffee Ground: 5 gallons
Wide Awake: 4.5 gallons
Therefore, the median for Wide Awake is lower than that of Coffee Ground, suggesting that Wide Awake typically sells less coffee per hour.
Based on these measures of center, we can conclude that the answer is: C. Coffee Ground, with a mean value of about 5 gallons.
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I need an answer asap!! any answer will help :D
By multiplication method the cargo will take up 7× 10⁷ energy.
What is multiplication?In arithmetic, the multiplication or product of two numbers it represents the repeated addition of one number to another. It can be done in between numbers which can be whole numbers, natural numbers, integers, fractions, etc.
Dr. Nandi plans to transport 3.5×10⁵ energy prism that each take up 2 ×10² cubic feet of cargo space.
The total can be calculated by multiplication method.
Here the multiplication gives
3.5×10⁵ × 2 ×10²
= 3.5 × 2 ×10⁵⁺²
= 7× 10⁷ energy.
Hence, the cargo will take up 7 × 10⁷ energy.
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is alice looking at a region of maximum brightness, minimum brightness, or neither? explain your reasoning.
As Alice looks at the color spectrum, she is looking at a region of maximum brightness.
Alice is looking at a region of maximum brightness because when we observe the color spectrum, we can see that the rainbow's center is the brightest, and the intensity of light at this location is the highest. Red color has the lowest energy, followed by orange, yellow, green, blue, indigo, and violet. It can be observed from the color spectrum that red has the lowest energy, followed by orange, yellow, green, blue, indigo, and violet.
White light can be separated into several colors in the color spectrum. The highest energy level is found in violet, whereas the lowest energy level is found in red. As Alice looks at the color spectrum, she is looking at a region of maximum brightness.
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Question 1 options:
Based on a survey of 100 households, a newspaper reports that the average number of vehicles per household is 1.8 with a margin of error of ±0.3.
Between what values is the estimate of the actual population? Enter your answer in the blanks to correctly complete the statement.
The actual population mean is between
and
vehicles per household.
The actual population mean is between 1.5 and 2.1 vehicles per household.
Describe Mean?The median is a statistical measure that represents the middle value of a dataset. It is the value that separates the lower half of the dataset from the upper half. To find the median, the data must be arranged in order from smallest to largest, and then the middle value is identified.
If the dataset contains an odd number of values, then the median is the middle value. For example, if the dataset is {2, 4, 6, 7, 9}, then the median is 6, which is the middle value.
If the dataset contains an even number of values, then the median is the average of the two middle values. For example, if the dataset is {2, 4, 6, 7, 9, 10}, then the median is (6+7)/2 = 6.5, which is the average of the two middle values, 6 and 7.
The actual population mean is between 1.5 and 2.1 vehicles per household.
The margin of error represents the possible distance between the sample mean and the true population mean.
The lower bound is found by subtracting the margin of error from the sample mean:
1.8 - 0.3 = 1.5
The upper bound is found by adding the margin of error to the sample mean:
1.8 + 0.3 = 2.1
Therefore, we can be 95% confident that the true population mean falls between 1.5 and 2.1 vehicles per household.
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the sum of -3 and it's opposite
Answer:
it equals 0
Step-by-step explanation:
3 + -3=0
Find the eqn of a line segment parallel to x-3y=4 and passing through the centroid of the ∆ABC where A(3,-4) ,B(-2,1), and C(5,0)
Answer:
[tex]x - 3y - 5 = 0[/tex]Step-by-step explanation:
To find:-
The equation of the line parallel to the given line and passing through the centroid of the given traingle.Answer:-
The given coordinates of the triangle are , A(3,-4) ; B(-2,1) and C(5,0) .
To find out the coordinate of the centroid we can use the below formula ,
[tex]\longrightarrow \boxed{\rm{Centroid\ (G)}= \bigg(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3}\bigg)} \\[/tex]
where ,
[tex](x_1,y_1)[/tex] ; [tex](x_2,y_2)[/tex] and [tex](x_3,y_3)[/tex] are the coordinates of the triangle.On substituting the respective values, we have;
[tex]\longrightarrow G =\bigg(\dfrac{3-2+5}{3},\dfrac{-4+1+0}{3}\bigg) \\[/tex]
[tex]\longrightarrow G =\bigg(\dfrac{ 6}{3},\dfrac{-3}{3}\bigg) \\[/tex]
[tex]\longrightarrow \boldsymbol{ G = (2,-1) }\\[/tex]
Hence the centroid of the given triangle is (2,-1) .
Now the given equation of the line is,
[tex]\longrightarrow x - 3y = 4 \\[/tex]
Convert this into slope intercept form of the line, which is,
Slope intercept form:-
[tex]\longrightarrow y = mx + c\\[/tex]
where, m is the slope of the line and c is the y-intercept .
So , we have;
[tex]\longrightarrow -3y = 4-x\\[/tex]
[tex]\longrightarrow 3y = x - 4 \\[/tex]
[tex]\longrightarrow y =\dfrac{x-4}{3} \\[/tex]
[tex]\longrightarrow y =\dfrac{1}{3}x-\dfrac{4}{3} \\[/tex]
On comparing it with the slope intercept form, we have;
[tex]\longrightarrow m =\dfrac{1}{3}\\[/tex]
Secondly we know that the slopes of parallel lines are equal . So the slope of the line parallel to the given line would also be ⅓ .
Now we may use point slope form of the line to find out the equation of the required line. The point slope form of the line is,
Point slope form:-
[tex]\longrightarrow y - y_1 = m(x-x_1) \\[/tex]
where the symbols have their usual meaning.
Here the line will pass through the centroid of the triangle which is (2,-1) .
On substituting the respective values, we have;
[tex]\longrightarrow y - (-1) =\dfrac{1}{3}(x-2) \\[/tex]
[tex]\longrightarrow 3( y +1) = x - 2 \\[/tex]
[tex]\longrightarrow 3y + 3 = x -2\\[/tex]
[tex]\longrightarrow x - 2 - 3 - 3y = 0 \\[/tex]
[tex]\longrightarrow\boxed{\boldsymbol{ x - 3y - 5 =0}} \\[/tex]
This is the required equation of the line.