The random variable x is known to be uniformly distributed
between 10 and 20.
a. Compute P( 10 ≤ x ≤ 15)

The confidence interval for the mean at 95% confidence level with the variance known and unknown are (101.17, 102.47), (101.06, 102.58) respectively

A.)

Confidence interval is related by the formula:

Where :

μ = population mean

t = critical value for the confidence level

s = sample standard deviation

n = sample size

The critical value for a 95% confidence level is 1.96.

C.I = 101.82 ± 1.96 * 0.81 * √(1/6)

C.I = 101.82 ± 0.648

C.I = (101.172, 102.468)

Hence, the confidence interval is (101.17, 102.47)

B.)

Assuming variance is Unknown

We use the t-distribition; The critical value at 95% confidence and (6-1) degree of freedom = 2.306

CI = μ ± t * s * √(1/n)

C.I = 101.82 ± 2.306 * 0.81 * √(1/6)

C.I = 101.82 ± 0.763

C.I = 101.057, 102.583

Hence, the confidence interval is (101.06, 102.58)

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The equation of the regression line for the relationship between job performance (X) and attitude ratings (Y) is Y = 57.124 + 0.352X.

To find the equation of the regression line, we will use a technique called simple linear regression. This method allows us to model the relationship between two variables using a straight line equation. In our case, the variables are job performance (denoted as Perf) and attitude ratings (denoted as Att).

The equation of a regression line is typically represented as: Y = a + bX

To find the equation of the regression line, we need to calculate the values of 'a' and 'b' using the given data points. Let's go step by step:

Mean of Perf (X): (59 + 63 + 65 + 69 + 58 + 77 + 76 + 69 + 70 + 64) / 10 = 66.0

Mean of Att (Y): (75 + 64 + 81 + 79 + 78 + 84 + 95 + 80 + 91 + 75) / 10 = 80.2

Perf differences:

(59 - 66.0), (63 - 66.0), (65 - 66.0), (69 - 66.0), (58 - 66.0), (77 - 66.0), (76 - 66.0), (69 - 66.0), (70 - 66.0), (64 - 66.0)

Att differences:

(75 - 80.2), (64 - 80.2), (81 - 80.2), (79 - 80.2), (78 - 80.2), (84 - 80.2), (95 - 80.2), (80 - 80.2), (91 - 80.2), (75 - 80.2)

Squared Perf differences:

(-7)², (-3)², (-1)², (3)², (-8)², (11)², (10)², (3)², (4)², (-2)²

Squared Att differences:

(-5.2)², (-16.2)², (0.8)², (-1.2)², (-2.2)², (3.8)², (14.8)², (-0.2)², (10.8)², (-5.2)²

Step 3: Calculate the sum of the squared Perf differences and the sum of the squared Att differences.

Sum of squared Perf differences:

7² + 3² + 1² + 3² + 8² + 11² + 10² + 3² + 4² + 2² = 369

Sum of squared Att differences:

5.2² + 16.2² + 0.8² + 1.2² + 2.2² + 3.8² + 14.8² + 0.2² + 10.8² + 5.2² = 734.72

Sum of Perf differences multiplied by Att differences:

(-7)(-5.2) + (-3)(-16.2) + (-1)(0.8) + (3)(-1.2) + (-8)(-2.2) + (11)(3.8) + (10)(14.8) + (3)(-0.2) + (4)(10.8) + (-2)(-5.2) = 129.8

Calculate the slope (b) using the following formula:

b = sum of Perf differences multiplied by Att differences / sum of squared Perf differences

b = 129.8 / 369 = 0.352

a = Mean of Att (Y) - b * Mean of Perf (X)

a = 80.2 - 0.352 * 66.0 = 57.124

Y = a + bX

Y = 57.124 + 0.352X

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Answer:

-1/3x + 5 = y

Step-by-step explanation:

Step-by-step explanation:

Circumference of a circle= 2πr

where r=6

and π=22/7

C=2πr

=2×22/7×6

=44/7×6

=264/7

=37.71cm

Answer:

You know that the beginning salary is $32,000, and it is raised by $1,000 per year.

a) We want to find a recursive relation, let's try to find a pattern:

S₁ = salary on the first year =  $32,000

S₂ = salary on the second year = $32,000 + $1,000 = $33,000

S₃ = salary on the third year = $33,000 + $1,000 = $34,000

and so on.

We already can see that the recursive relation is: "the salary of the previous year plus $1,000", this can be written as:

Sₙ = Sₙ₋₁ + $1,000

Such that S₁ = $32,000

b) Your salary in the fifth year is S₅

Let's construct it:

S₃ = $34,000

S₄ = $34,000 + $1,000 = $35,000

S₅ = $35,000 + $1,000 = $36,000

Your salary on the fifth year is $36,000

c) When we have a recursive relation like:

Aₙ = Aₙ₋₁ + d

The sum of the first N elements is given by:

Sum(N) = N*(2*A₁ + (N - 1)*d)/2

Then the sum of your salary for the first 20 years is:

S(20) = 20*(2*$32,000 + (20 - 1)*$1,000)/2

S(20) = $830,000

Step-by-step explanation:

2.) -20× + 24

maaf kalo salah

Answer:

x = y = 2√2

Step-by-step explanation:

Find the diagram attached

To get the unknown side x and y, we need to use the SOH CAH TOA identity

Opposite side = x

Adjacent = y

Hypotenuse = 4

Sin theta = opposite/hypotenuse

sin 45 = x/4

x = 4 sin 45

x = 4 * 1/√2

x = 4 * 1/√2 * √2/√2

x = 4 * √2/√4

x = 4 * √2/2

x = 2√2

Similarly;

cos theta = adjacent/hypotenuse

cos 45 = y/4

y = 4cos45

y = 4 * 1/√2

y = 4 * 1/√2 * √2/√2

y = 4 * √2/√4

y = 4 * √2/2

y = 2√2

Step-by-step explanation:

54, 54, 72

[tex]180 - 2 \times 54 = \\ = 180 - 108 = \\ = 72[/tex]

Answer:

15 ÷ 17

Step-by-step explanation:

Since our hypotenuse side is 17 cm, our opposite side is 8 cm and our adjacent side is 15 cm. Since the acute angle is x degrees,

From trigonometric ratios, cosx° = adjacent/hypotenuse

= 15 cm/17 cm

= 15/17

= 15 ÷ 17

The null and alternative hypotheses for testing the claim that the proportion of people who own cats is significantly different from 90% at the 0.02 significance level are:

H0: p = 0.9 (proportion of cat owners is 90%)

H1: p ≠ 0.9 (proportion of cat owners is not equal to 90%)

Based on a sample of 500 people, where 82% owned cats, we can conduct a hypothesis test to determine the p-value at the 0.02 significance level. The p-value is the probability of obtaining a sample proportion as extreme as the observed proportion (82%) assuming the null hypothesis is true.

The p-value for this test is the probability of observing a sample proportion as different from 90% as 82%. Since the p-value is not provided in the question, it needs to be calculated based on the sample data and the assumed null distribution.

If the p-value is less than 0.02, we would reject the null hypothesis and conclude that the proportion of cat owners is significantly different from 90%. However, if the p-value is greater than or equal to 0.02, we would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in the proportion of cat owners from 90%.

Without the calculated p-value, we cannot make a definitive conclusion about rejecting or failing to reject the null hypothesis.


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Answer:

For [tex]x = 3[/tex] the value of [tex]f(x) = 2\cdot x^{2}+9[/tex] will be the same of [tex]g(x) = 3^{x}[/tex].

Step-by-step explanation:

To determine for which value of [tex]x[/tex], we need to apply the following identity ([tex]f(x) = g(x)[/tex]) and solve numerically the resulting expression:

[tex]2\cdot x^{2}+9 = 3^{x}[/tex]

[tex]3^{x}-2\cdot x^{2}-9=0[/tex] (1)

A quick approach is using graphic tool and looking for the value of [tex]x[/tex] such that  [tex]3^{x}-2\cdot x^{2}-9=0[/tex]. The result of the analysis is included below in the attached image. We find the following result:

[tex]x = 3[/tex]

For [tex]x = 3[/tex] the value of [tex]f(x) = 2\cdot x^{2}+9[/tex] will be the same of [tex]g(x) = 3^{x}[/tex].

Answer: I really don’t know

Step-by-step explanation: I need help with this as well. I’m really sorry if you were looking for a real answer

Answer:

3= (87÷h)

Step-by-step explanation:

The question is to pick among the option the solutions that will be equal to 3= (87÷h)

Therefore the most suitable and correct option is

3×9= (87÷h)÷9

27= (87÷h)÷9

Divide by 9

3= (87÷h)

Hence the answer is 3=(87÷h)

a )

First of all we need to find the value of x ,

because the angles are written in terms of the variable x .

______________________________________

STV angle & SUV angle have same measure because both of them are the front angle of SV arc .

[tex]STV angle \: = SUV angle \: \: = \frac{SV \: arc}{2} \\ [/tex]

So :

[tex]3x - 5 = 2x + 15[/tex]

Add sides 5

[tex]3x - 5 + 5 = 2x + 15 + 5[/tex]

[tex]3x = 2x + 20[/tex]

Subtract sides minus 2x

[tex]3x - 2x = 2x + 20 - 2x[/tex]

Collect like terms

[tex]x = 2x - 2x + 20[/tex]

[tex]x = 20[/tex]

Thus the measure of angle T equals :

[tex]measure \: of \: angle \: T = 3x - 5 \\ [/tex]

Now just need to put the value of x which we found :

[tex]measure \: of \: angle \: T \: = 3 \times (20) - 5 \\ [/tex]

[tex]measure \: of \: angle \: T \: = 60 - 5[/tex]

[tex]measure \: of \: angle \: T \: = 55°[/tex]

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b )

angle S & angle V are also have same measure because they both are the front angles to the TU arc .

And we need to find the value of x again in this part exactly like we did for a .

[tex]angle \: \: S = angle \: \: V[/tex]

As the question told :

[tex]angle \: \: S = 3x[/tex]

and ,

[tex]angle \: \: V = x + 16[/tex]

Thus :

[tex]3x = x + 16[/tex]

Subtract sides minus x

[tex]3x - x = x + 16 - x[/tex]

Collect like terms

[tex]2x = x - x + 16[/tex]

[tex]2x = 16[/tex]

Divide sides by 2

[tex] \frac{2x}{2} = \frac{16}{2} \\ [/tex]

Simplification

[tex]x = 8[/tex]

So ;

[tex]measure \: \: of \: \: angle \: \: S = 3x[/tex]

[tex]measure \: \: of \: \: angle \: \: S = 3(8)[/tex]

[tex]measure \: \: of \: \: angle \: \: S = 24°[/tex]

And we're done.

(i) True.

(ii) False.

The first statement is true.

It is a well-known fact that for any random variable X, the covariance of X with itself is equal to the variance of X. This can be easily shown by applying the definition of covariance and variance and using the fact that the correlation between X and X is always 1.

The second statement is false. The mean of an exponential random variable is equal to 1/λ, where λ is the rate parameter. On the other hand, the standard deviation of an exponential random variable is equal to 1/λ as well. These two values are not equal, unless λ=1. Therefore, the statement is false.

In summary, the first statement is true, while the second statement is false. The covariance of a random variable with itself is equal to its variance, but the mean and standard deviation of an exponential random variable are not equal unless λ=1.

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Answer:

The approximate age of the bone is approximately 11552 years.

Step-by-step explanation:

The current proportion of carbon-14 with respect to its original amount is defined by following formula:

[tex]\frac{y}{a} = e^{-0.00012\cdot t}[/tex] (1)

Where:

[tex]y[/tex] - Current amount of carbon-14, no unit.

[tex]a[/tex] - Initial amount of carbon-14, no unit.

[tex]t[/tex] - Time, in years.

If we know that [tex]\frac{y}{a} = 0.25[/tex], then the approximate age of the bone is:

[tex]t = -8333.333\cdot \ln \frac{y}{a}[/tex]

[tex]t\approx 11552.453\,yr[/tex]

The approximate age of the bone is approximately 11552 years.

The explicit formula for the beetle population after n weeks can be determined using the given data. The formula is Pn = 4 + (n - 0) * ((67 - 4) / (7 - 0)), where Pn represents the population after n weeks. It will take 28 weeks for the beetle population to reach 256.

The linear growth model assumes that the beetle population increases by a fixed amount each week. To find the explicit formula, we start by calculating the growth rate per week. We know that in 7 weeks, the population increased from 4 to 67. The change in population is 67 - 4 = 63, and the change in weeks is 7 - 0 = 7. Therefore, the growth rate per week is (67 - 4) / (7 - 0) = 9.

Using this growth rate, we can express the population after n weeks using the formula Pn = 4 + (n - 0) * 9. This simplifies to Pn = 4 + 9n. Now, to determine how many weeks it takes for the population to reach 256, we substitute Pn = 256 into the formula. Solving for n, we get 256 = 4 + 9n. By rearranging the equation, we find 9n = 252, and dividing both sides by 9 yields n = 28. Therefore, it will take 28 weeks for the beetle population to reach 256.

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Answer:

bbv

Step-by-step explanation:

Recurring decimal is decimal representation of a number whose digits are periodic and infinite. Proved algebraically that the recurring decimal 0.178 can be written as the fraction 59/330 below.

Given number in the decimal form is [tex]0. 1 \overline 7 \overline 8[/tex]

Suppose the number is equal to the x,

[tex]x=0. 1 \overline 7 \overline 8[/tex]

Recurring decimal is decimal representation of a number whose digits are periodic and infinite.

As the number 78 is the recurring number. Thus the recurring number can be written as,

[tex]x=0.1787878.....[/tex]                      .......equation 1.

Suppose this is equation number 1.

Multiply the above equation with 100 both the sides,

[tex]100\times x=100\times0. 1 787878....[/tex]

[tex]100x=100\times0.1787878...[/tex]

[tex]100x=17.87878...[/tex]

Subtract the above equation from equation number 1. Thus,

[tex]\begin{aligned}\ 100x-x&=17.87878-0.1787878\\ 99x&=17.7\\ \end[/tex]

Solve for x ,

[tex]x=\dfrac{17.7}{99} [/tex]

Multiply with 10 in both numerator and denominator,

[tex]x=\dfrac{177}{990} \\ x=\dfrac{59}{330} \\[/tex]

Hence proved algebraically that the recurring decimal 0.178 can be written as the fraction 59/330

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Answer:

x = 93.7ft or x = 94ft

Step-by-step explanation:

Hope that helps :)

Answer:

Let X be the random variable representing the amount of beer poured by the filling machine. Since X follows a normal distribution with mean μ = 12.10 and standard deviation σ = 0.05, we can use the standard normal distribution to find the probability that a bottle contains between 12.00 and 12.06 ounces.

First, we need to standardize the values 12.00 and 12.06 by subtracting the mean and dividing by the standard deviation:

z1 = (12.00 - 12.10) / 0.05 = -2 z2 = (12.06 - 12.10) / 0.05 = -0.8

Now we can use a standard normal distribution table to find the probability that a standard normal random variable Z is between -2 and -0.8:

P(-2 < Z < -0.8) = P(Z < -0.8) - P(Z < -2) ≈ 0.2119 - 0.0228 ≈ 0.1891

So, the probability that a bottle contains between 12.00 and 12.06 ounces of beer is approximately 0.1891.

Step-by-step explanation:

Answer:

y = 2/3x - 6

Step-by-step explanation:

Use the slope intercept equation, y = mx + b

Plug in the slope and given point, then solve for b

y = mx + b

-4 = 2/3(3) + b

-4 = 2 + b

-6 = b

Plug in the slope and b into the equation

y = 2/3x - 6

So, the equation of the line is y = 2/3x - 6

The approximation of the integral ∫cos(x³ + 10) dx using composite Simpson's rule with n = 3 is 0.126. When approximating the integral using Romberg integration, R₃,₃ gives an approximation of order h⁶.

To calculate the approximation using composite Simpson's rule, we divide the interval of integration into subintervals and apply Simpson's rule to each subinterval. The formula for Simpson's rule is:

S = h/3 * (f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(xₙ))

where h is the step size and n is the number of subintervals. In this case, we have n = 3, so we divide the interval into three equal subintervals, and the step size is h = (b - a) / n = (π - 0) / 3 = π/3.

Evaluating the function cos(x³ + 10) at the points x₀ = 0, x₁ = π/3, x₂ = 2π/3, and x₃ = π, we get:

f(x₀) = cos((0)³ + 10) = cos(10) ≈ -0.8391

f(x₁) = cos((π/3)³ + 10) = cos(π³/27 + 10) ≈ -0.4586

f(x₂) = cos((2π/3)³ + 10) = cos(8π³/27 + 10) ≈ -0.8391

f(x₃) = cos((π)³ + 10) = cos(π³ + 10) ≈ -0.3473

Using the Simpson's rule formula, we can now calculate the approximation:

S ≈ π/3 * (f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃))

 ≈ π/3 * (-0.8391 + 4(-0.4586) + 2(-0.8391) + 4(-0.3473))

 ≈ 0.126

To calculate the order of approximation using Romberg integration, we use the formula:

Rₙ,ₖ = Rₙ₋₁,ₖ₋₁ + (Rₙ₋₁,ₖ₋₁ - Rₙ,ₖ₋₁) / (4ₖ - 1)

where Rₙ,ₖ represents the Romberg approximation at level n and column k. The order of approximation is determined by the highest power of h in the error term. In this case, we have R₃,₃, so the order is h⁶.

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The simplified ratio for 0.875 is 7:8 .

To express the ratio for 0.875, we need to convert the decimal value to a ratio form.

0.875 can be written as 875 / 1000 because the decimal value is equivalent to the fraction obtained by dividing the numerator by the denominator.

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 125 in this case.

Dividing 875 and 1000 by 125, we get:

875 / 125 = 7

1000 / 125 = 8

So, the simplified ratio for 0.875 is 7:8 .

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