Angles in a Triangle

A 95% confidence interval for the standard normal distribution is typically given by  (-1.96, 1.96) . mean that correspond to the upper and lower bounds of the interval

what is mean ?

In statistics, the standard deviation is a measure of the amount of variation or dispersion in a set of data. It is the square root of the variance and is denoted by the symbol σ (sigma). The standard deviation is expressed in the same units

In the Given question :

A 95% confidence interval for the standard normal distribution is typically given by  (-1.96, 1.96)

This means that we are 95% confident that the true value of a normally distributed variable falls within this range. The values of -1.96 and 1.96 represent the number of standard deviations from the mean that correspond to the upper and lower bounds of the interval, respectively. This interval is often used in statistical inference to estimate the population mean or proportion based on a sample.

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what 95% confidence interval for a standard normal distribution then what is the number of standard deviations from the mean that correspond to the upper and lower bounds of the interval ?

Answer:

1

Step-by-step explanation:

1.  One way to do this is converting both into improper fractions.  To do this, multiply the whole number by the denominator and add that to the numerator.

2 3/5 --> 2*5 is 10 --> 10+3 is 13.  --> 13/5

2.  This leaves us with 13/5 - 8/5

3.  Subtract the numerators

13/5 - 8/5 = 5/5

4.  Simplify.  If the numerator is the same number as the denominator, it's a whole number.

5/5 = 1

The standard deviation of the number of green M&M's in a sample of 5 is 0.91

Standard deviation is a measure of the amount of variation or dispersion of a set of data values. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean.

To solve this problem, we can use the binomial probability formula:

[tex]P(X=k) = C(n,k) * p^k * (1-p)^{(n-k)[/tex]

where:

X is the number of green M&M's in the sample

k is the number of green M&M's we are interested in (3 or 4 or at most 3 or at least 3)

n is the sample size (5)

p is the probability of getting a green M&M in one trial

We can use the given percentages to calculate the probability of getting a green M&M:

p = 0.15

Now we can calculate the probabilities for the different scenarios:

Probability that exactly three of the five M&M's are green:

[tex]P(X=3) = C(5,3) * 0.15^3 * 0.85^2 = 0.0883[/tex]

Probability that three or four of the five M&M's are green:

[tex]P(X=3\ or\ X=4) = P(X=3) + P(X=4) = C(5,3) * 0.15^3 * 0.85^2 + C(5,4) * 0.15^4 * 0.85^1 = 0.1527[/tex]

Probability that at most three of the five M&M's are green:

P(X≤3) = [tex]P(X=0) + P(X=1) + P(X=2) + P(X=3) = C(5,0) * 0.15^0 * 0.85^5 + C(5,1) * 0.15^1 * 0.85^4 + C(5,2) * 0.15^2 * 0.85^3 + C(5,3) * 0.15^3 * 0.85^2 = 0.9053[/tex]

Probability that at least three of the five M&M's are green:

P(X≥3) [tex]= P(X=3) + P(X=4) + P(X=5) = C(5,3) * 0.15^3 * 0.85^2 + C(5,4) * 0.15^4 * 0.85^1 + C(5,5) * 0.15^5 * 0.85^0 = 0.2413[/tex]

To find the expected number of green M&M's in a sample of size 5, we can use the formula:

E(X) = n * p

where E(X) is the expected value of X.

E(X) = 5 * 0.15 = 0.75

Therefore, on average we expect 0.75 green M&M's in a sample of 5.

To find the standard deviation, we can use the formula:

SD(X) = sqrt(n * p * (1-p))

SD(X) = sqrt(5 * 0.15 * 0.85) = 0.9138 (rounded to two decimal places)

Therefore, the standard deviation of the number of green M&M's in a sample of 5 is 0.91

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Answer:

The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared.

Step-by-step explanation:

Thus, this resulting binomial is called a difference of squares.

Answer:

Step-by-step explanation:

Square of the sum

square means you are multiplying something twice and looks like:  

ex.       [tex]x*x=x^{2}[/tex]

sum is the addition of numbers

so square of the sum looks like:

(a+b)²

Square of the difference

difference means subtraction of numbers

so square of the difference looks like:

(b-a)²

Answer:

The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared.

Step-by-step explanation:

Thus, this resulting binomial is called a difference of squares.

Answer:

Step-by-step explanation:

Square of the sum

square means you are multiplying something twice and looks like:  

ex.       [tex]x*x=x^{2}[/tex]

sum is the addition of numbers

so square of the sum looks like:

(a+b)²

Square of the difference

difference means subtraction of numbers

so square of the difference looks like:

(b-a)²

Betty consumed a total of 148.294 milliliters of liquid with her breakfast.

We can change the volumes of the coffee and orange juice to milliliters so that all of the measurements are in the same unit:

4 ounces =  4 * 29.5735, = 118.294 ml.

8 ounces = 8 x 29.5735, = 236.588 ml.

Half of Betty's 4-ounce glass of orange juice, or:

1/2 * 118.294 mls = 59.147 mls

She consumed 25% of her 8-ounce coffee, which is :

0.25 * 236.588 mls = 59.147 mls.

So, in total, Betty consumed:

30 mls of cough medicine + 59.147 mls of orange juice + 59.147 mls of coffee = 148.294 mls

Therefore, Betty consumed a total of 148.294 milliliters of liquid with her breakfast.

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The angles and percentages have been calculated as shown below.

A pie chart for the information about the different make of cars is shown below.

For the total number of car makes, we have the following:

Total make of cars = 12 + 8 + 8 + 2 + 10

Total make of cars = 40.

Next, we would determine the angles and percentage as follows;

Toyota

12/40 × 360 = 108°

12/40 × 100 = 30%.

Nissan

8/40 × 360 = 72°

8/40 × 100 = 20%.

Datsun

8/40 × 360 = 72°

8/40 × 100 = 20%.

Volvo

2/40 × 360 = 18°

2/40 × 100 = 5%.

Peugeot

10/40 × 360 = 90°

10/40 × 100 = 25%.

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The maximum height at the moment is 191.0625 ft.

What is the projectile's height?

Let's find out the projectile's maximum height now that we know what it is. The highest vertical location along the object's flight is considered to be its maximum height. The range of the bullet is defined as its horizontal displacement.

Here, we have

Given: The equation for a projectile's height versus time is h(t)=-16t²+Vt+h. A tennis ball machine serves a ball vertically into the air from a height of 2 feet, with an initial speed of 110 feet per second.

The equation for a projectile's height versus time is

h(t) = -16t²+Vt+h

h = 2

V = 110

Substitute these values into the function

h(t) = -16t²+110t+2

Take the first derivative

h'(t) = -32t + 110

Equate the derivative with zero h'(t) = 0

0 = -32t + 110

110 = 32t

t = 110/32

t = 3.4375

Find the maximum height at the moment

t = 3.4375 (s)

h(3.4375) = -16(3.4375)² + 110(3.4375) + 2

h(3.4375) = -189.0625 + 378.125 + 2

h(3.4375) = 191.0625 ft

Hence, the maximum height at the moment is 191.0625 ft.

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The number that does not belong to the set of numbers is given as follows:

B. Three

Numbers are classified as even or odd depending on whether they are divisible by 2 or not, as follows:

For this problem, we have that the number five does not belong to the set of even numbers, as the numbers two, four and six do.

The options are:

A.Two

B.Three

C.Four

D.Six​

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The value of t= -1.946; p = 0.029; Reject the null hypothesis; there is strong evidence to suggest that the highlighters last less than 14 hours.

To test the hypothesis that the highlighters last less than 14 hours, we will use a one-sample t-test. The null hypothesis for this test is that the mean continuous writing time for the highlighters is equal to or greater than 14 hours. The alternative hypothesis is that the mean continuous writing time for the highlighters is less than 14 hours.

In this problem, we are given that x = 13.6 hours and s = 1.3 hours. The sample size is n = 40. Substituting these values into the formula for the test statistic, we get:

t = (13.6 - 14) / (1.3 / √(40)) = -1.946

The p-value for the test can be found using a t-distribution table or a statistical software program. The p-value is the probability of observing a t-value as extreme as the one we calculated, assuming the null hypothesis is true. In this problem, the p-value is 0.029.

To make a decision about the null hypothesis, we compare the p-value to the significance level, which is typically set at 0.05. If the p-value is less than the significance level, we reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis.

In this problem, the p-value is less than 0.05, so we reject the null hypothesis. This means there is strong evidence to suggest that the highlighters last less than 14 hours. We can conclude that the manufacturer's claim that their highlighters can write continuously for 14 hours is not supported by the sample data.

Hence the correct option is (c).

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Complete Question:

Solve the problem. Suppose a consumer product researcher wanted to find out whether a highlighter lasted less than the manufacturer's claim that their highlighters could write continuously for 14 hours. The researcher tested 40 highlighters and recorded the number of continuous hours each highlighter wrote before drying up. Test the hypothesis that the highlighters wrote for less than 14 continuous hours. Following are the summary statistics:

x =13.6 hours,

s =1.3 hours

Report the test statistic, p-value, your decision regarding the null hypothesis, and your conclusion about the original claim. Round all values to the nearest thousandth.

a)  z = 1.946; p = 0.029; Reject the null hypothesis; there is strong evidence to suggest that the highlighters last less than 14 hours.

b) t = -1.946; p = 0.029; Fail to reject the null hypothesis; there is not strong evidence to suggest that the highlighters last less than 14 hours. o

c)  t= -1.946; p = 0.029; Reject the null hypothesis; there is strong evidence to suggest that the highlighters last less than 14 hours.

d) z = 1.946; p = 0.974; Fail to reject the null hypothesis; there is not “strong evidence to suggest that the highlighters last less than 14 hours.

Okay, let's break this down step-by-step:

From the table, we know the x-values are 1, 3, 5, 7, and 9.

And the y-values are 4, 6, 8, ?, 12.

So at x=7, there is a missing y-value that we need to determine.

To find this, we need to look at the linear relationship between the points. Some clues:

1) There are 5 points total.

2) The x-values increase by 2 each time.

3) The y-values also seem to be increasing by 2 each time.

So the linear relationship is: y = mx + b (where m is the slope and b is the y-intercept)

In this case, the slope is 2 (because y increases by 2 for every increase of 1 in x).

And the y-intercept is 4 (because when x = 1, y = 4).

So the equation is: y = 2x + 4

Plugging in x = 7:

y = 2(7) + 4

y = 14 + 4

y = 18

Therefore, when x = 7, the second coordinate in the scatterplot is 18.

Rounded to the tenths place: 1.8

Does this help explain the steps? Let me know if you have any other questions!

the probability that Juan will catch at least 10 passes out of 15 is approximately 0.987.

The binomial distribution, which models the number of successful trials (catches) in a certain number of independent trials (passes thrown), can be used to solve this problem.

The likelihood of not catching a pass is 0.2, but the likelihood of catching one is 0.8. The likelihood of catching at least 10 passes out of 15 can be calculated as follows:

P(X >= 10) equals P(X = 10) plus P(X = 11). + ... + P(X = 15)

where X is how many of the 15 passes were intercepted.

The probability of catching precisely k passes out of n can be calculated using the binomial distribution formula:

P(X = k) = (n choose k) × p²k × (1 - p)²(n-k)

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n distinct items.

Plugging in the values for n = 15, p = 0.8, and k = 10, 11, 12, 13, 14, and 15, we get:

P(X >= 10) = P(X = 10) + P(X = 11) + ... + P(X = 15)

= (15 choose 10) × 0.8²10 × 0.2²5 + (15 choose 11) × 0.8²11 × 0.2²4 + ... + (15 choose 15) × 0.8²15 × 0.2²0

≈ 0.987

Therefore, the probability that Juan will catch at least 10 passes out of 15 is approximately 0.987.

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Answer:

For 25- to 34-year-olds who worked full time, year round, higher educational attainment was associated with higher median earnings.

Step-by-step explanation:

This pattern was consistent for each year from 2010 through 2020. For example, in 2020, the median earnings of those with a master’s or higher degree were $69,700, some 17 percent higher than the earnings of those with a bachelor’s degree ($59,600). In the same year, the median earnings of those with a bachelor’s degree were 63 percent higher than the earnings of those who completed high school ($36,600).

Answer: The probability of drawing a blue marble and then a yellow marble is 1/12 or 0.0833.

Step-by-step explanation:

There are 16 marbles in the bag. To calculate the probability of drawing a blue marble and then a yellow marble, we can use the formula:

P(blue and yellow) = P(blue) x P(yellow|blue)

The probability of drawing a blue marble on the first draw is 5/16. After one blue marble has been drawn, there are 15 marbles left in the bag, so the probability of drawing a yellow marble on the second draw, given that the first marble was blue, is 4/15.

P(blue and yellow) = (5/16) x (4/15) = 1/12 or 0.0833

Answer:

----------------------

If the system of linear equations has infinitely many solutions, it means the two lines overlap.

In other words, each point of one of the lines also belongs to the second line.

Choices a, b, c  give us one or no solutions and therefore not the answer.

Choice d is reflecting the infinitely many solutions and hence is the correct one.

Answer:

To convert kilometers per hour to miles per hour, we need to multiply by 0.62. Therefore, to find the maximum speed that Hannah can drive without exceeding the speed limit of 130 kilometers per hour, we can multiply 130 by 0.62:

130 km/h * 0.62 = 80.6 mph

Since Hannah needs to stay within the speed limit, the maximum whole-number speed she can drive is 80 mph, which is option D.

Step-by-step explanation:

Answer: 9.32 inches.

Step-by-step explanation:

Answer:

This shows the step by step process of rhetorical reduction of the question given

Emilio can find that the critical value t* for a 90% confidence level and 11 degrees of freedom is approximately 1.796.

To construct a t interval for the mean lifespan with 90% confidence, Emilio needs to use a t-distribution with n-1 degrees of freedom. The confidence interval for the population is given by:

confidence interval = x ± t × (s·x/√n)

Where x is the sample mean, s·x is the sample standard deviation, n is the sample size, and t is the critical value of the t-distribution.

Since the sample size is n=12, the degrees of freedom for the t-distribution will be (n-1) = 11. To find the critical value t* for a 90% confidence level and 11 degrees of freedom, Emilio can use a t-distribution table or a statistical software.

Using a t-distribution table or calculator, Emilio can find that the critical value t* for a 90% confidence level and 11 degrees of freedom is approximately 1.796.

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Answer:

C is correct

Step-by-step explanation:

Answer:

the value of B is 3

Step-by-step explanation:

We can start by factoring out the greatest common factor of the polynomial, which is 2x:

2x(6x3 + 15x2 + 2x + 5)

Now, we can factor the expression inside the parentheses by grouping:

2x[(6x3 + 2x) + (15x2 + 5)]

2x[2x(3x + 1) + 5(3x + 1)]

2x(2x + 5)(3x + 1)

Comparing this expression to the given expression:

Ax(Bx2+1)(2x+5)

We see that A = 2, B = 3, and the factor (2x + 5) is the same in both expressions. Therefore, the value of B is 3.

The value of the inequality is g≤2.5

In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.

The given inequality is 4g≤10

To solve the inequality, divided bo sides of the inequality by the coefficient of g which is 4

This gives the value

4g/g≤10/4

⇒ that g = 2.5

Therefore the value of the inequality is g≤2.5

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Answer:

13.56 mm²

Step-by-step explanation:

surface area = 2 circle area + the side

circle = π r² = 3.14

2 of them = 6.28

side = circumference * 1 = 2πr *1 = 6.28

total 6.28*2 = 13.56 mm²

The person is looking at each peak on a leg of a right triangle. The hypotenuse is the distance between the mountains.

The distance is determined by

x^2+(x+5)^2=25^2

x^2+x^2+10x+25=625

2x^2+10x-600=0

x^2+5x-300=0

(x+20)(x-15)=0

only positive root is x=15

x+5=20

The vista point is 15 miles from one mountain peak and 20 miles from the other.

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The 95% confidence interval of the mean pulse rate for adult females is 68.2 bpm < μ < 76.4 bpm

For a 95% confidence interval, the Z-score is 1.96. Plugging in the values we have for the sample mean, sample standard deviation, and sample size, we get:

Confidence interval = 75.8 ± (1.96 × (3.7 / √50))

Simplifying the expression, we get:

Confidence interval = 71.5 bpm < μ < 80.2 bpm

This means that we can be 95% confident that the true population mean pulse rate for adult females falls within this range.

Now let's construct a confidence interval for adult males. We are given that the sample mean pulse rate for adult males is 72.3 bpm, and the sample standard deviation is 4.0 bpm. Using the same formula and Z-score as before, we can calculate the confidence interval as follows:

Confidence interval = 72.3 ± (1.96 × (4.0 / √50))

Simplifying the expression, we get:

Confidence interval = 68.2 bpm < μ < 76.4 bpm

This means that we can be 95% confident that the true population mean pulse rate for adult males falls within this range.

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Answer:

m∠A = 50°

Step-by-step explanation:

The ∑ of all interior angles of any given triangle is 180°. Set all angle measurements equal to 180:

[tex](x + 30) + (2x - 10) + 70 = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](x + 2x) + (30 - 10 + 70) = 180\\(3x) + (90) = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, subtract 90 from both sides of the equation:

[tex]3x + 90 = 180\\3x + 90 (-90) = 180 (-90)\\3x = 180 - 90\\3x = 90[/tex]

Next, divide 3 from both sides of the equation:

[tex]3x = 90\\\frac{(3x)}{3} = \frac{(90)}{3}\\ x = \frac{90}{3}\\ x = 30[/tex]

Plug in 30 for x in the given angle measurement of A and simplify:

[tex]m\angle{A} = 2x - 10\\m\angle{A} = 2(30) - 10\\m\angle{A} = (60) - 10\\m\angle{A} = 50[/tex]

50° is your answer for m∠A.

~

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Answer:

m∠A = 50°

Step-by-step explanation:

The ∑ of all interior angles of any given triangle is 180°. Set all angle measurements equal to 180:

[tex](x + 30) + (2x - 10) + 70 = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](x + 2x) + (30 - 10 + 70) = 180\\(3x) + (90) = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, subtract 90 from both sides of the equation:

[tex]3x + 90 = 180\\3x + 90 (-90) = 180 (-90)\\3x = 180 - 90\\3x = 90[/tex]

Next, divide 3 from both sides of the equation:

[tex]3x = 90\\\frac{(3x)}{3} = \frac{(90)}{3}\\ x = \frac{90}{3}\\ x = 30[/tex]

Plug in 30 for x in the given angle measurement of A and simplify:

[tex]m\angle{A} = 2x - 10\\m\angle{A} = 2(30) - 10\\m\angle{A} = (60) - 10\\m\angle{A} = 50[/tex]

50° is your answer for m∠A.

~

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We can use the coordinates of point P (-5, 5) to find the values of the trigonometric functions for the angle formed between the positive x-axis and the line passing through the origin and point P.

First, we can find the distance from the origin to point P using the Pythagorean theorem:

r = sqrt((-5)^2 + 5^2) = sqrt(50)

Next, we can use the fact that the sine of an angle is equal to the y-coordinate of a point on the unit circle and the cosine of an angle is equal to the x-coordinate of a point on the unit circle. Since the radius of the unit circle is 1, we need to divide the x and y coordinates of point P by sqrt(50) to get the values for sine and cosine:

sin(θ) = y/r = 5/sqrt(50) = (5/10)sqrt(2) = (1/2)sqrt(2)

cos(θ) = x/r = -5/sqrt(50) = -(5/10)sqrt(2) = -(1/2)sqrt(2)

Next, we can use the fact that the tangent of an angle is equal to the sine of the angle divided by the cosine of the angle:

tan(θ) = sin(θ)/cos(θ) = (1/2)sqrt(2)/(-(1/2)sqrt(2)) = -1

Similarly, we can use the reciprocal identities to find the values of the other three trigonometric functions:

csc(θ) = 1/sin(θ) = sqrt(2)

sec(θ) = 1/cos(θ) = -sqrt(2)

cot(θ) = 1/tan(θ) = -1

Therefore, the exact values of the six trigonometric functions for the angle formed by the line passing through the origin and point P (-5, 5) are:

sin(θ) = (1/2)sqrt(2)

cos(θ) = -(1/2)sqrt(2)

tan(θ) = -1

csc(θ) = sqrt(2)

sec(θ) = -sqrt(2)

cot(θ) = -1

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