(a) The sample has size 110, and it is from a non-normally distributed population with a known standard deviation of 0.45.
- z = -3.06
- t = 0.45
- It is unclear which test statistic to use.
(b) The sample has size 14, and it is from a normally distributed population with an unknown standard deviation.
- z = -1.81
- t = 0.46
- It is unclear which test statistic to use.
In scenario (a), the sample has a large size of 110, and it is from a non-normally distributed population with a known standard deviation of 0.45. In this case, we can use the z-test because of the large sample size. The z-test compares the sample mean to the hypothesized population mean in terms of the standard deviation of the sampling distribution. The formula for the z-test is:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the hypothesized population mean, σ is the known population standard deviation, and n is the sample size.
Substituting the given values, we get:
z = (1.9 - 2) / (0.45 / √110) = -3.06
Therefore, the test statistic for scenario (a) is z = -3.06.
In scenario (b), the sample has a small size of 14, and it is from a normally distributed population with an unknown standard deviation. In this case, we can use the t-test because of the small sample size and unknown population standard deviation. The t-test compares the sample mean to the hypothesized population mean in terms of the standard error of the sampling distribution. The formula for the t-test is:
t = (x - μ) / (s / √n)
where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Substituting the given values, we get:
t = (1.9 - 2) / (0.5 / √14) = -1.81
Therefore, the test statistic for scenario (b) is t = -1.81.
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Two light sources of identical strength are placed 18 m apart. An object is to be placed at a point P on a line l parallel to the line joining the light sources and at a distance d meters from it (see the figure). Locate P on l so that the intensity of illumination is minimized. Use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.(a) Find an expression for the intensity I(x) at the point P (assume k = 1.) l(x)=_____(b) If d = 18 m, find the value of x that minimizes the intensity. x= _____
The value of x that minimizes the intensity when d = 18 m is approximately x = 9.
(a) To find an expression for the intensity I(x) at point P, we'll use the fact that intensity is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.
Let's denote the distance from the first light source to point P as r1 and from the second light source as r2.
We can use the Pythagoras theorem, to find the expressions for r1 and r2:
[tex]r1 = sqrt(x^2 + d^2)[/tex]
[tex]r2 = sqrt((18-x)^2 + d^2)[/tex]
Since the light sources have identical strength and we assume k=1, the intensity I(x) at point P can be represented as:
[tex]I(x) = 1/r1^2 + 1/r2^2[/tex]
[tex]I(x) = 1/(x^2 + d^2) + 1/((18-x)^2 + d^2)[/tex]
(b) If d = 18 m, we need to find the value of x that minimizes the intensity:
[tex]I(x) = 1/(x^2 + 18^2) + 1/((18-x)^2 + 18^2)[/tex]
To find the minimum value, we can take the derivative of I(x) with respect to x and set it to 0:
[tex]dI(x)/dx = 0[/tex]
Using a calculator or software to find the derivative and solve for x, we get:
x ≈ 9
So, the value of x that minimizes the intensity when d = 18 m is approximately x = 9.
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Claim: The mean pulse rate (in beats per minute) of adult males is equal to 69 bpm. For a random sample of 173 adult males, the mean pulse rate is 68.6 bpm and the standard deviation is 10.8 bpm. Find the value of the test statistic.
The value of the test statistic is ___
the value of the test statistic is -0.4866
To find the value of the test statistic, we can use the formula for a t-test:
t = (x - μ) / (s / √n)
where x is the sample mean, μ is the population mean (69 bpm), s is the sample standard deviation, and n is the sample size (173).
Substituting the given values, we get:
t = (68.6 - 69) / (10.8 / √173)
t = -0.4 / (10.8 / 13.1529)
t = -0.4 / 0.8223
t = -0.4866 (rounded to four decimal places)
Therefore, the value of the test statistic is -0.4866
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Any help would be appreciated! Thank you Table 1: Example Sequences TAGTGACTGA TAAGTTCCGA then nd = 5 since there are 5 sites (positions 3, 4, 5, 6, 8) in the sequence in which different letters are observed and n 10 since 10 sites are considered. Therefore, the p-distance between these sequences would be p= bo = 0.50 10 In the following, we will be using the two sequences listed below one-sample z-test for one proportion
Based on the information provided, it seems that your question may be related to calculating p-distance between two sequences using the number of different sites and the total number of sites considered. In order to calculate p-distance between two sequences, we need to first identify the number of different sites (nd) and the total number of sites considered (n) in the sequences. Once we have these values, we can use the formula p = nd/n to calculate the p-distance.
In the example provided in Table 1, we have two sequences (TAGTGACTGA and TAAGTTCCGA) and we can see that there are 5 sites in which different letters are observed. Therefore, nd = 5. The total number of sites considered is 10, so n = 10. Using the formula p = nd/n, we can calculate the p-distance between these sequences as p = 5/10 = 0.50.
If you have two sequences of your own and want to calculate the p-distance between them, you can follow the same process. Count the number of different sites (nd) and the total number of sites considered (n) in the sequences, and then use the formula p = nd/n to calculate the p-distance.
As for the one-sample z-test for one proportion, this is a statistical test that is used to determine whether a sample proportion is significantly different from a known population proportion. The test involves calculating a z-score, which is a measure of how many standard deviations the sample proportion is away from the population proportion. If the z-score is large enough (i.e., falls in the rejection region), we can reject the null hypothesis and conclude that the sample proportion is significantly different from the population proportion. However, this may not be directly related to your original question about p-distance between sequences, so please let me know if you need more information on this topic.
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A certain star is 5.6×10^2 light years from the Earth. One light year is about 5.9×10^12 miles. How far from the earth (in miles) is the star?
the star is approximately [tex]3.304*10^{15}[/tex] miles from the Earth.
What is distance ?
As its name implies, any distance formula outputs the distance (the length of the line segment). In coordinate geometry, there is a number of formulas for finding distances, such as the separation between two points, the separation between two parallel lines, the separation between two parallel planes, etc.
First, we can convert the distance of one light year to miles:
1 light year = [tex]5.9*10^{12}[/tex] miles
Next, we can use dimensional analysis to convert the distance of the star from light years to miles:
[tex]5.6*10^{2} light years * (5.9*10^{12} miles/1 light year) = 3.304*10^{15} miles[/tex]
Therefore, the star is approximately [tex]3.304*10^{15}[/tex] miles from the Earth.
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Write an exponential function to model each situation then solve. Find each amount after the specified time.
3. A Ford truck that sells for $52,000 depreciates 18% each year for 8 years.
4. Rewrite the integral 2∫0 y^3∫0 y^2∫0 f(x, y, z) dz dx dy as an iterated integral in the order dxdydz. [6 points)
The iterated integral in the order dxdydz for the given integral 2∫0 y³∫0 y²∫0 f(x, y, z) dz dx dy is ∫0 1∫0 y²∫0 y³ 2f(x,y,z) dx dz dy.
Here, we integrate over x from 0 to z³/y³, then over z from 0 to y², and finally over y from 0 to 1. The order dxdydz is used to compute the integral by breaking it down into smaller parts and evaluating each part separately.
By changing the order of integration, we can simplify the process of integration and make it easier to compute.
This change in order helps us to evaluate the integral in a more organized manner, allowing us to identify any patterns or relationships between the variables. Therefore, the iterated integral in the order dxdydz is a useful tool in solving complex integrals.
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(1 point) Use integration by parts to evaluate the integral. (n(2x))dz = +C
To evaluate this integral using integration by parts, we need to choose two functions, $u$ and $dv$, such that their product is equal to $n(2x)$:
u=n(2x), dv=dz
Then, we can write the integral as:
∫ n(2x)dz = ∫ udv
Using the integration by parts formula:
∫ udv = uv − ∫ vdu
we can find the integral:
\begin{align*}
\int n(2x) dz &= \int u dv \
&= uv - \int v du \
&= n(2x) z - \int z dn(2x) \
&= n(2x) z - \int n'(2x) \cdot 2 dz \
&= n(2x) z - 4\int n'(2x) dz \
&= n(2x) z - 4n(2x) + C
\end{align*}
where $n'(2x)$ is the derivative of $n(2x)$ with respect to $2x$, and $C$ is the constant of integration. Therefore, the integral of $n(2x)$ with respect to $z$ is:
∫ n(2x)dz = n(2x)z − 4n(2x) + C
where $C$ is an arbitrary constant.
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Express 5.39393939394... as a rational number, in the form p/qwhere p and q are positive integers with no common factors.Previous problem - LIST Next Problem 9. (5 points) Express 5.39393939394... as a rational number, in the form where p and q are positive integers with no common factors. p = and q =
The representation of 5.39393939394 in a rational number form is equal to p /q = 534 / 99.
Let us consider 'x' to express the decimal number.
This implies,
x = 5.3939393939...
Multiply both the side of the equation by 100 we get,
⇒ 100x = 539.39393939...
Subtracting expression of x from the expression of 100x, we get,
⇒ 99x = 534
Dividing both sides of the expression by 99, we get,
⇒ x = 534/99
Since 534 and 99 have no common factors other than 1.
534 and 99 are both positive integers.
Expressed the repeating decimal 5.3939393939... as a rational number in the form p/q .
Where p = 534 and q = 99.
This implies ,
p /q = 534 / 99.
Therefore, the expression to represents the given decimal number in a rational number form is equal to 5.3939393939... = 534/99.
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In an experimental taste test, a random sample of 200 middle school-aged children were given two different cookies, one was the name brand of Oreo and the other was the generic brand. Let’s suppose that of the 200 students sampled, 161 were able to identify which cookie was the Oreo brand and which cookie was the generic brand. You want a 98% confidence interval for the proportion of students that can identify the brands.
Let’s suppose you want the margin of error to be within 2 percentage points. How many middle schoolers would you have to sample in order to make this happen?
To get a 98% confidence interval for the number of students who can recognize the brands with a margin of error of 2 percentage points, we need to survey 1077 middle school-aged kids.
What is confidence interval?A confidence interval is a range of values that, with a certain degree of certainty, is likely to contain the real value of a population parameter. An unknown value, like a population mean or proportion, can be estimated using this statistical concept using a sample from the population.
Apply the following calculation to find the sample size required to achieve a margin of error of 2 percentage points with a 98% confidence interval:
[tex]n = \dfrac{[Z^2 \times p \times (1-p)]} { E^2}[/tex]
where:
n = sample size
Z = z-score associated with the desired level of confidence (98%)
p = estimated proportion of success (we'll use 0.5 since we don't have any prior information)
E = desired margin of error (0.02)
Plugging in the values,
[tex]n = \dfrac{(2.33)^2 \times 0.5 \times (1 - 0.5)} { (0.02)^2}[/tex]
n ≈ 1076.29
Rounding up, we get a sample size of 1077.
Therefore, we need to sample 1077 middle school-aged children in order to obtain a 98% confidence interval for the proportion of students that can identify the brands, with a margin of error of 2 percentage points.
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In order to achieve a 98% confidence interval with a 2% margin of error, we would need to sample 361 middle school-aged children.
To calculate the sample size needed to achieve a 2% margin of error with 98% confidence interval, we need to use the following formula:
n = (Zα/2[tex])^2\times p \times(1 - p) / E^2[/tex]
Where:
n = sample size
Zα/2 = critical value for the desired confidence level (in this case, 98% which corresponds to a Z-value of 2.33)
p = estimated proportion of students who can identify the Oreo brand cookie, based on the initial sample of 200 students (p = 161/200 = 0.805)
E = desired margin of error (in this case, 0.02)
Plugging in the values, we get:
[tex]n = (2.33)^2\times 0.805 \times (1 - 0.805) / 0.02^2[/tex]
n = 360.39
Rounding up to the nearest whole number, we get a sample size of 361 students.
Therefore, in order to achieve a 98% confidence interval with a 2% margin of error, we would need to sample 361 middle school-aged children.
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List 3 possible numbers that will round to the nearest tenth and become:
1. 9.4
2. 10.1
Thank you
Answer:
1) 9.36, 9.37, 9.41
2) 10.09, 10.11, 10.12
Answer:
1. 9.35, 9.36, 9.37
2. 10.05, 10.06, 10.07
Step-by-step explanation:
any number 5 or higher can allow the number to round up
4 or lower the number stays the same
The west wall of a square room has a length of 13 feet. What is the perimeter of the room? A. There is not enough information B. 169 C. 52 D. 48
The perimeter of the square room having a west wall of the length of 13 feet is 52 feet. Thus, the right answer is option C which says 52.
A square is a 2-Dimensional shape. It is a quadrilateral having 4 equal sides and 4 equal angles of 90°. Perimeter refers to the sum of the length of the boundary of a given structure.
Perimeter of square = 4s
where s is the side of a square
Given, that it is a square room, thus, the length of the west wall is equal to the side of the square room
the side of the square room = 13 feet
therefore, perimeter = 4 * 13 = 52 feet
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help with questions please asapFind the derivative of the function. y = x-9 + x -9 -6 +x - 1 y'(x) = -9x -8 - 6x-5 5-1 * Need Help? Read It Watch It Find the derivative of the function. y = 9x5 – 4x3 + 5x – 8 y' = 45x4 – 12
The derivative of the function y = [tex]x^{-9}+x^{-6}+x^{-1}[/tex] is given by y' = [tex]-9x^{-10}-6x^{-7}-\ln x[/tex] and the derivative of the function y = 9x⁵ - 4x³ + 5x - 8 is given by 45x⁴ - 12x² +5.
The function given is,
y = [tex]x^{-9}+x^{-6}+x^{-1}[/tex]
In the above function, independent variable is 'x' and dependent variable is 'y'.
Differentiating the above function with respect to 'x' we get,
dy/dx = d/dx [tex](x^{-9}+x^{-6}+x^{-1})[/tex]
y'(x) = [tex]-9x^{-9-1}-6x^{-6-1}-\ln x=-9x^{-10}-6x^{-7}-\ln x[/tex]
Another function is,
y = 9x⁵ - 4x³ + 5x - 8
Differentiating the above function with respect to 'x' we get,
y' = 9*5x⁴ - 4*3x² + 5 - 0 = 45x⁴ - 12x² +5
Hence the derivative functions are [tex]-9x^{-10}-6x^{-7}-\ln x[/tex] and 45x⁴ - 12x² +5.
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a manufacturer is looking for ways to increase production of the number of items produced at their factories. it is known that the average number of items produced per week is is 914 items. after opening two new factories in the past six months, the manufacturer believes the average number of items produced per week has increased. what are the hypotheses? fill in the blanks with the correct symbol (
The hypotheses for the manufacturer's belief that the average number of items produced per week has increased after opening two new factories are: Null Hypothesis (H0): The average number of items produced per week is the same as before and has not increased after opening two new factories, and Alternative Hypothesis (H1): The average number of items produced per week has increased after opening two new factories.
Null Hypothesis (H0): The average number of items produced per week is the same as before and has not increased after opening two new factories.
Alternative Hypothesis (H1): The average number of items GV7B NHYUJMIK,L per week has increased after opening two new factories.
In hypothesis testing, the null hypothesis (H0) is the assumption that there is no significant difference or effect, while the alternative hypothesis (H1) is the opposite, suggesting that there is a significant difference or effect.
In this case, the null hypothesis (H0) states that the average number of items produced per week is the same as before, meaning that the opening of two new factories did not result in an increase in production.
On the other hand, the alternative hypothesis (H1) suggests that the average number of items produced per week has increased after opening two new factories, indicating a significant effect of the new factories on production.
The manufacturer believes that the average number of items produced per week has increased, hence the alternative hypothesis (H1) is formulated to reflect this belief.
Therefore, the hypotheses for the manufacturer's belief that the average number of items produced per week has increased after opening two new factories are: Null Hypothesis (H0): The average number of items produced per week is the same as before and has not increased after opening two new factories, and Alternative Hypothesis (H1): The average number of items produced per week has increased after opening two new factories.
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0.0001776 can also be expressed as:a) 1.776 x 10^-3b) 1.776 x 10^-4c) 17.72 x 10^4d) 1772 x 10^5e) 177.2 x 10^7
Answer:
b) 1.776 × 10^-4
Step-by-step explanation:
a) 1.776 × 10^-3 is 0.001776 (3 places to the right)
b) 1.776 × 10^-4 is 0.0001776 (4 places to the right)
And the rest of the options go with a different number so there is no way they can be expressed as 0.0001776.
Therefore, the correct answer is b.
Angela's annuity pays $600 per month for 5 years at 9 % per year compoundedmonthly. Becky's annuity pays $300 per month for 10 years at 9 % compoundedmonthly.What is the total payment for Angela's and Becky's annuities?Do both annuities have the same beginning value? Show your calculations.In your therefore statements, include how changing the number of years affectsthe annuity.
The total payment of Angela's and Becky's annuities is $31,271.38, under the condition that Angela's annuity pays $600 per month for 5 years at 9 % per and Becky's annuity pays $300 per month for 10 years at 9 % compounded monthly.
To evaluate the total payment for Angela's and Becky's annuities, we can perform the formula for the future value of an annuity
[tex]FV = PMT * [(1 + r)^n - 1] / r[/tex]
Here FV = future value of the annuity,
PMT = monthly payment,
r= monthly interest rate,
n = number of months.
In case of Angela's annuity
PMT = $600
r = 9% / 12 = 0.0075
n = 5 years × 12 months/year = 60 months
[tex]FV = $600 * [(1 + 0.0075)^{60 - 1}] / 0.0075[/tex]
= $44,772.64
In case of Becky's annuity
PMT = $300
r = 9% / 12 = 0.0075
n = 10 years × 12 months/year = 120 months
[tex]FV = $300 * [(1 + 0.0075)^{120 - 1}] / 0.0075[/tex]
= $44,772.64
Then, both annuities have the same future value of $44,772.64.
Now, to evaluate the present value of each annuity, we can perform the formula for the present value of an annuity
[tex]PV = PMT * [1 - (1 + r)^{-n}] / r[/tex]
Here
PV = present value of the annuity.
In case of Angela's annuity
[tex]PV = $600 * [1 - (1 + 0.0075)^{-60}] / 0.0075[/tex]
= $31,271.38
In case of Becky's annuity
[tex]PV = $300 * [1 - (1 + 0.0075)^{-120}] / 0.0075[/tex]
= $31,271.38
Hence, both annuities have the same present value of $31,271.38.
Changing the number of years will affect the annuity in two ways:
1) It affects the future value and by that it will also affects the present value.
2) Increasing the number of years increases not only the future value but also the present value of an annuity.
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Gary will toss a fair coins 3 times . What is the probability that “ heads” will occur more the once?
Answer:
the chances of you getting heads every time is 1/2 * 1/2 * 1/2, or 1/8.
Step-by-step explanation:
If you flip a coin, the chances of you getting heads is 1/2. This is true every time you flip the coin so if you flip it 3 times, the chances of you getting heads every time is 1/2 * 1/2 * 1/2, or 1/8.
Answer:
8
Step-by-step explanation:
2^3=8
The triangles are similar. Find the value of Z.
Answer:9
Step-by-step explanation:
28/10 =25.2/z
28/10=2.8
25.2/9=2.8
Z=9
Answer:9
Step-by-step explanation:
28/10 =25.2/z
28/10=2.8
25.2/9=2.8
Z=9
in 2009, the population of Hungary was approximated by P=9.906(0.997)t, where P is in millions and t is in years since 2009. Assume the trend continues.
(a) What does this model predict for the population of Hungary in the year 2011? Round your answer to two decimal places.
(b) How fast (in people/year) does this model predict Hungary's population will be decreasing in the year 2011? Give your answer to the nearest thousand.
(a) Rounding this to two decimal places gives us an estimate of Hungary's population in 2011 as 9.71 million.
(b) Rounding this to the nearest thousand gives us an estimate of -21 people per year, which means the population is decreasing by about 21 people per year in 2011 according to this model.
(a) To find the population of Hungary in 2011, we need to substitute t=2 (since 2011 is two years after 2009) into the given formula:
[tex]P = 9.906(0.997)^t[/tex]
[tex]P = 9.906(0.997)^2[/tex]
P ≈ 9.706
Rounding this to two decimal places gives us an estimate of Hungary's population in 2011 as 9.71 million.
(b) To find how fast Hungary's population is decreasing in 2011, we need to take the derivative of the given function with respect to t:
[tex]dP/dt = 9.906ln(0.997)(0.997)^t[/tex]
Now we substitute t=2 to get the rate of change in 2011:
[tex]dP/dt = 9.906 ln(0.997)(0.997)^2 \approx -0.0208[/tex]
Rounding this to the nearest thousand gives us an estimate of -21 people per year, which means the population is decreasing by about 21 people per year in 2011 according to this model.
Note that this value is very small and may not be significant in practice.
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HURRY PLEASE HELPPPPP
The equivalent of the given Fahrenheit temperature in degree Celsius is 30 °C.
Calculating temperature in degree CelsiusFrom the question, we are to determine the given Fahrenheit temperature in degree Celsius
From the given information,
The given Fahrenheit temperature is 86 °F.
From the given formula
C = 5/9 (F - 32)
To use this formula to determine the equivalent of a Fahrenheit temperature in degree Celsius, we will substitute the Fahrenheit temperature into the formula and simplify.
C = 5/9 (F- 32)
Substitute F = 86
C = 5/9 (86 - 32)
C = 5/9 (54)
C = 5/9 × 54
C = 5 × 6
C = 30 °C
Hence,
The temperature is 30 °C.
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This question consists of ten multiple choice questions. Each question is worth 2 marks, Select the correct answer by circling the appropriate choice for each question.
(a) A parameter is best described by:
(a) a feature of a sample
(b) a sample summary
(c) a feature of the population
(d) an unknown value of the sample
(c) a feature of the population. The correct answer is feature of population.
A parameter is a characteristic or feature of the entire population being studied, not just a sample. It is a numerical value that summarizes a population's distribution. In contrast, a sample summary is a characteristic or feature of a sample, such as the mean or standard deviation, that provides information about the sample but not the entire population.
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A soft drink dispensing machine is said to be out of control if the variance of the contents exceeds 1.15 deciliters. A random sample of 25 drinks from this machine is studied and the sample variance is computed to be 2.03 deciliters. Assume that the contents are approximately normally distributed. Construct a 90% lower confidence bound on σ^2. (Round your answer to 2 decimal places)
The 90% lower confidence bound for the population variance (σ²) is approximately 3.52 deciliters
To construct a 90% lower confidence bound on σ^2, we'll need to use the chi-square (χ²) distribution and the following terms:
1. Sample variance (s²): 2.03 deciliters
2. Sample size (n): 25
3. Degrees of freedom (df): n - 1 = 24
4. Confidence level: 90%
Step 1: Find the chi-square value for the given confidence level.
For a 90% lower confidence bound, we need to find the chi-square value corresponding to the lower 10% tail (α = 0.10) and the given degrees of freedom (24). Using a chi-square table or calculator, we find the χ² value to be 13.85.
Step 2: Compute the lower confidence bound for σ².
Now, we will use the formula for the lower confidence bound for σ²:
Lower bound = [(n - 1) * s²] / χ²
Lower bound = (24 * 2.03) / 13.85
Lower bound = 48.72 / 13.85
Lower bound ≈ 3.52 deciliters
So, the 90% lower confidence bound for the population variance (σ²) is approximately 3.52 deciliters.
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Find the area of this semi-circle with diameter,
d
= 98cm.
Give your answer rounded to 2 DP.
Answer:
3769.57 cm²
Step-by-step explanation:
Diameter = 98 cm
Radius = 98/2 = 49 cm
Area of Semi Circle = πr²/2
= 3.14 × 49 × 49/2
= 3769.57 cm²
So 3769.57 cm² is the answer
2. Determine the volume of the solid obtained by rotating the region enclosed by y = Vr, = y = 2, and r = 0 about the c-axis.
The volume of the solid obtained by rotating the region enclosed by y = √(r), y = 2, and r = 0 about the c-axis is (64/3)π cubic units.
The region enclosed by y = √(r), y = 2, and r = 0 is a quarter-circle in the first quadrant with a radius of 4.
To find the volume of the solid obtained by rotating this region about the c-axis, we can use the disk method.
Consider an element of the solid at a distance r from the c-axis with thickness dr.
When this element is rotated about the c-axis, it generates a disk with radius r and thickness dr.
The volume of this disk is [tex]\pi r^2[/tex] dr.
Integrating this expression over the range of r from 0 to 4, we get:
[tex]V = \int[0,4] \pi r^2 dr[/tex]
[tex]= \pi [(4^3)/3 - 0][/tex]
= (64/3)π.
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Use Alternating Series Test to determine whether the following series are convergence[infinity]Σn=1 (-1)^n+1 (n^2/n^3+4).please show work as I am studying for my final
To use the Alternating Series Test, we need to check that the terms of the series are decreasing in absolute value and approach 0 as n approaches infinity.
Let's start by looking at the absolute value of the terms:
|(-1)^n+1 (n^2/n^3+4)| = n^2/(n^3+4)
To show that this is decreasing, we can look at the ratio of consecutive terms:
[n+1]^2 / [(n+1)^3 + 4] * [(n^3+4) / n^2] = (n^3 + 3n^2 + 2n) / [(n^3+4)(n+1)]
Since the numerator is increasing and the denominator is increasing faster, this ratio is less than 1 for all n >= 1. Therefore, the terms are decreasing in absolute value.
Next, let's show that the terms approach 0. As n approaches infinity, the denominator of each term approaches infinity faster than the numerator, so the entire fraction approaches 0. Since the sign of each term alternates, this means the series converges.
Therefore, using the Alternating Series Test, we can conclude that the series:
[infinity]Σn=1 (-1)^n+1 (n^2/n^3+4)
converges.
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Find the x intercepts of the graph. ( enter your answer as a comma separated list. Use n as an integer constant.) y=sin pix over 2 plus 1
X=
The x-intercept of the graph are - 1 and 3.
What is x-intercept?The intersection of a function's graph and the x-axis is known as the x-intercept in mathematics. The function has a root or a solution for the equation f(x) = 0 at this time since the y-coordinate of the graph is zero.
The zero or function's root are other names for the x-intercept. The x-value for which the function f(x) equals zero is known as the zero x value.
When a function has an x-intercept, we set f(x) = 0 and then solve for x. The value(s) of x that are obtained provide the x-coordinate(s) of the point(s) where the function's graph crosses the x-axis.
The x-intercept of the graph is obtained when the value of the y-coordinate is 0.
Thus, from the graph we see that the x-intercept is at the points -1 and 3.
Hence, the x-intercept of the graph are - 1 and 3.
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What are the absolute Maximum and Minimum of f(t) = 7 cos t, −3π/2 ≤ t ≤ 3π/2?
The absolute Maximum of f(t) = 7 cos t, −3π/2 ≤ t ≤ 3π/2 is 7 and the absolute Minimum of f(t) = 7 cos t, −3π/2 ≤ t ≤ 3π/2 is -7.
The function f(t) = 7 cos(t) has a period of 2π, which means that it repeats itself every 2π units. In the given interval, the function takes its maximum and minimum values at the endpoints of the interval and at the critical points where the derivative of the function is zero.
The critical points of f(t) in the interval −3π/2 ≤ t ≤ 3π/2 are t = −π/2, π/2, and 3π/2, where the derivative of the function f(t) is zero:
f'(t) = -7 sin(t) = 0
This occurs when sin(t) = 0, which implies that t = −π/2, π/2, and 3π/2.
Therefore, the absolute maximum and minimum of f(t) occur at the endpoints of the interval and the critical points, and are as follows:
Absolute maximum: f(π/2) = 7
Absolute minimum: f(−3π/2) = -7
So, the absolute maximum value of f(t) is 7, which occurs at t = π/2, and the absolute minimum value of f(t) is -7, which occurs at t = −3π/2.
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A polynomial P is given. P(x) = x3 – 5x2 + 4x – 20
(a) Factor P into linear and irreducible quadratic factors with real coefficients. P(x) =
(b) Factor P completely into linear factors with complex coefficients. P(x)=
(a) To factor P into linear and irreducible quadratic factors with real coefficients, we can start by looking for any rational roots using the rational root theorem.
The possible rational roots of P are ±1, ±2, ±4, ±5, ±10, ±20.
We can see that P(-1) = 0, so x + 1 is a factor of P. Using long division or synthetic division, we can find that P(x) = (x + 1)(x^2 - 6x + 20).
To factor x^2 - 6x + 20, we can use the quadratic formula: x = (6 ± sqrt(36 - 4(1)(20))) / 2 x = 3 ± sqrt(-11)
Since the discriminant is negative, the quadratic factor x^2 - 6x + 20 is irreducible over the real numbers. Therefore, the factored form of P with real coefficients is: P(x) = (x + 1)(x^2 - 6x + 20)
(b) To factor P completely into linear factors with complex coefficients, we can use the same rational root theorem and find the same possible rational roots as before. However, this time we can also consider complex roots of the form a + bi, where a and b are real numbers and i is the imaginary unit.
Using synthetic division, we can find that P(-2 + 2i) = 0, so x - (-2 + 2i) = x + 2 - 2i is a factor of P. Similarly, we can find that x + 2 + 2i is also a factor. Using long division or synthetic division again, we can find that P(x) = (x + 1)(x + 2 - 2i)(x + 2 + 2i). Therefore, the factored form of P with complex coefficients is: P(x) = (x + 1)(x + 2 - 2i)(x + 2 + 2i)
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Question 8. Use the 45-45-90 Triangle Theorem to find the length of the hypotenuse. m∠C = 45 degrees
a = 1.5 in
Question 9. What is the vocabulary term for segment a? What is the area of the polygon? Round to the nearest tenth.
a = 2 √(3)
s = 4 yd
For question 8, since m∠C = 45 degrees and a = 1.5 in, we can use the 45-45-90 Triangle Theorem to find the length of the hypotenuse. In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of each leg. Therefore, the length of the hypotenuse is 1.5 * √2 = 2.12 inches (rounded to two decimal places).
For question 9, if the polygon is a regular hexagon with side length s = 4 yds and apothem a = 2√(3), then the area of the hexagon can be found using the formula for the area of a regular polygon: A = (1/2) * P * a, where P is the perimeter of the polygon and a is the apothem. The perimeter of the hexagon is P = 6s = 6 * 4 = 24 yds. Therefore, the area of the hexagon is A = (1/2) * P * a = (1/2) * 24 * 2√(3) = 24√(3) square yards, or approximately 41.6 square yards when rounded to the nearest tenth.
Street light failures in a town occur at an average rate of 3 units every week. What is the probability of no street light 1 point failures next week? * 0.1494 0.4286 0 0.0498
The probability of no street light failures next week can be calculated using the Poisson distribution formula.
The Poisson probability formula is: P(x) = (e^(-λ) * λ^x) / x!, where x is the number of occurrences, λ is the average rate, and e is the base of the natural logarithm (approximately 2.71828).
In this case, we want to find the probability of no street light failures (x = 0) next week, given the average rate of failures is 3 units per week (λ = 3).
Step 1: Plug in the values into the formula:
P(0) = (e^(-3) * 3^0) / 0!
Step 2: Evaluate the exponential and factorial parts:
e^(-3) ≈ 0.0498
3^0 = 1
0! = 1
Step 3: Calculate the probability:
P(0) = (0.0498 * 1) / 1 = 0.0498
So, the probability of no street light failures next week is 0.0498 or 4.98%.
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the national health statistics reports described a study in which a sample of 342 one-year-old baby boys were weighed. their mean weight was 24.4 pounds with standard deviation 5.3 pounds. a pediatrician claims that the mean weight of one-year-old boys is greater than 25 pounds. do the data provide convincing evidence that the pediatrician's claim is true? use the a
No, the data does not provide convincing evidence that the pediatrician's claim is true.
To determine if the data provide convincing evidence that the pediatrician's claim is true, we need to conduct a hypothesis test. Our null hypothesis is that the true mean weight of one-year-old boys is equal to or less than 25 pounds, while our alternative hypothesis is that the true mean weight is greater than 25 pounds.
Using the sample data given, we can calculate the test statistic as follows:
t = (24.4 - 25) / (5.3 /^(342)) = -1.69
We can then compare this test statistic to the critical value of a t-distribution with 341 degrees of freedom (since we are estimating one parameter, the mean). At an alpha level of 0.05 and with one tail (since our alternative hypothesis is one-sided), the critical value is 1.646.
Since our test statistic (-1.69) is less than the critical value (1.646), we fail to reject the null hypothesis. In other words, the data do not provide convincing evidence that the pediatrician's claim is true. We cannot conclude that the mean weight of one-year-old boys is greater than 25 pounds based on this sample.
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