Answer:
$630
Step-by-step explanation:
420/2 = half months rent ($210)
420+210 = 630
The security deposit is $630.
Help bc this is due soon
The measures of angle B is derived as 75° to the nearest degree using the cosine rules.
What is the cosine rulesThe cosines rule relates the lengths of the sides of a triangle to the cosine of one of its angles.
Using the cosine rule:
2² = 5² + (√45)² - 2(5)(√45)cosB
4 = 25 + 45 - 250cosB
4 = 70 - 250cosB
250cosB = 70 - 4 {collect like terms}
250cosB = 66
B = cos⁻¹(66/250) {cross multiplication}
B = 74. 6925°
Therefore, the measures of angle B is derived as 75° to the nearest degree using the cosine rules.
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school a has 400 students and school b has 2700 students. a local newspaper wants to compare the distributions of sat scores for the two schools. which of the following would be the most useful for making this comparison?
When comparing the distributions of SAT scores for two schools, it is important to use a statistical measure that can accommodate the difference in the number of students in each school. In this case, since School A has 400 students while School B has 2700 students, the most useful statistical measure for making this comparison would be the percentage of students in each school who scored within certain SAT score ranges.
For example, instead of comparing the raw number of students who scored above a certain score threshold in each school, it would be more meaningful to compare the percentage of students in each school who scored above that threshold. This would give a more accurate representation of the distribution of SAT scores in each school, taking into account the different sizes of the student populations.
Another useful statistical measure for making this comparison would be to use box plots to visualize the distributions of SAT scores in each school. Box plots provide a clear and concise way to compare the minimum, maximum, median, and quartiles of SAT scores for each school.
In summary, the most useful statistical measures for comparing the distributions of SAT scores for School A and School B would be the percentage of students in each school who scored within certain score ranges, as well as the use of box plots to visualize the distributions.
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If you're conducting a significance test for the difference between the means of two independent samples, what's your null hypothesis? A. H0: = 0 B. H0: p1 - p2 = 0 C. H0: 1 - 2 = 0 D. H0: p = p0 E. H0: = 0
If you're conducting a significance test for the difference between the means of two independent samples, your null hypothesis would be option E, H0: μ1 - μ2 = 0, which means that there is no significant difference between the means of the two independent samples. The alternative hypothesis, denoted as Ha, would be that there is a significant difference between the means of the two independent samples.
In order to test the null hypothesis, you would need to use a statistical test such as the t-test or z-test, depending on the sample size and whether the population standard deviations are known or unknown. These tests would provide a p-value, which indicates the probability of obtaining a difference between the means as extreme or more extreme than the observed difference, assuming that the null hypothesis is true.
If the p-value is less than the chosen significance level (usually 0.05), then the null hypothesis can be rejected and it can be concluded that there is a significant difference between the means of the two independent samples. Otherwise, if the p-value is greater than the significance level, then the null hypothesis cannot be rejected and it can be concluded that there is not enough evidence to suggest a significant difference between the means of the two independent samples.
When you are conducting a significance test for the difference between the means of two independent samples, the null hypothesis is a statement that there is no significant difference between the population means of the two groups. In this case, the correct null hypothesis is:
C. H0: μ1 - μ2 = 0
This hypothesis states that the difference between the population means of the two independent samples (μ1 and μ2) is equal to zero, which implies that there is no significant difference between the two population means. The alternative hypothesis would be that there is a significant difference (either μ1 > μ2, μ1 < μ2, or simply μ1 ≠ μ2, depending on the type of test being performed).
To test this hypothesis, you would collect data from the two independent samples and calculate the sample means (x1 and x2). Then, you would conduct a statistical test, such as a t-test or a z-test, to compare the sample means and determine the probability (p-value) of obtaining a difference as large as, or larger than, the one observed in your samples, assuming the null hypothesis is true.
If the p-value is smaller than a predetermined significance level (commonly set at 0.05), you would reject the null hypothesis in favor of the alternative hypothesis, concluding that there is a significant difference between the population means. If the p-value is greater than the significance level, you would fail to reject the null hypothesis, meaning that there is not enough evidence to conclude that there is a significant difference between the population means.
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Determine whether each distribution is a probability
distribution.
(3) Determine whether each distribution is a probability distribution. a) X 4 6 8 10 P(X) -0.6 0.2 0.7 1.5 b) 8 9 12 P(X) 2 1 1 3 6 6 х X P(X) 1 1 2 1 1 4 3 4 1 1 4 4 4 4 d) X P(X) 1 0.3 3 0.1 5 0.2
The sum of the probabilities is equal to 1 and all probabilities are between 0 and 1 (inclusive). Therefore, this is a probability distribution.
For a distribution to be a probability distribution, it must satisfy two conditions:
The sum of the probabilities for all possible values of X must be equal to 1.
The probability for each possible value of X must be between 0 and 1 (inclusive).
Let's check each distribution:
a) X 4 6 8 10 P(X) -0.6 0.2 0.7 1.5
This distribution does not satisfy the second condition, since the probability for X = 4 is negative (-0.6). Therefore, this is not a probability distribution.
b) X 8 9 12 P(X) 2 1 1 3 6 6
This distribution satisfies the first condition, since the sum of the probabilities is equal to 1. However, it does not satisfy the second condition, since the probability for X = 9 is 1, which is greater than 1. Therefore, this is not a probability distribution.
c) X P(X) 1 1 2 1 1 4 3 4 1 1 4 4 4 4
This distribution satisfies both conditions, since the sum of the probabilities is equal to 1 and all probabilities are between 0 and 1 (inclusive). Therefore, this is a probability distribution.
d) X P(X) 1 0.3 3 0.1 5 0.2
This distribution satisfies both conditions, since the sum of the probabilities is equal to 1 and all probabilities are between 0 and 1 (inclusive). Therefore, this is a probability distribution.
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4. List and briefly explain the three main techniques of measuring quantitative data (6 marks)
The three main techniques of measuring quantitative data are surveys, experiments, and observational studies.
1. Surveys: Surveys involve collecting data by asking a sample of individuals to respond to a set of questions. This method can be done through various formats such as questionnaires, interviews, or online polls. Surveys are useful for gathering information on opinions, attitudes, or preferences and can help determine relationships between variables.
2. Experiments: Experiments involve manipulating one or more variables to observe the effect on a dependent variable. Participants are typically randomly assigned to different conditions, and the researcher measures the outcomes to determine cause-and-effect relationships. Experiments can provide strong evidence for causal relationships and are often used in scientific research.
3. Observational studies: Observational studies involve collecting data by observing and recording the natural behavior or characteristics of individuals or groups without any intervention. Researchers can observe the participants in their natural settings or use existing data sources such as records, databases, or archival data. Observational studies are useful for understanding patterns, trends, and relationships between variables, but they cannot establish causality.
These techniques can provide valuable insights into various aspects of quantitative data, allowing for informed decision-making and improved understanding of patterns and relationships.
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Laterfah and Eric Lewis have obtained a mortgage loan at a 5.5% annual interest rate for 25 years. The home's selling price is $175,000, and they need a 20% down payment. The bank will allow them to finance the closing costs as part of the mortgage. What is the actual amount financed with the mortgage? Find the closing cost and the total amount of the mortgage if the closing costs are financed.
Answer:
The selling price of the house is $175,000 and they need to make a 20% down payment, so the down payment amount is:
Down payment = 20% x $175,000 = $35,000
To find the amount financed with the mortgage, we need to subtract the down payment from the selling price:
Amount financed = Selling price - Down payment
Amount financed = $175,000 - $35,000
Amount financed = $140,000
Next, we need to calculate the closing costs. Let's assume the closing costs are 3% of the selling price:
Closing costs = 3% x $175,000 = $5,250
Since the bank allows them to finance the closing costs as part of the mortgage, we need to add the closing costs to the amount financed:
Total amount of the mortgage = Amount financed + Closing costs
Total amount of the mortgage = $140,000 + $5,250
Total amount of the mortgage = $145,250
Therefore, the actual amount financed with the mortgage is $140,000, the closing costs are $5,250, and the total amount of the mortgage if the closing costs are financed is $145,250.
The life of an automotive battery is normally distributed with mean 900 days and standard deviation 35 days. What fraction of these batteries would be expected to survive beyond 1000 days.
The fraction of these batteries that would be expected to survive beyond 1000 days is 0.0021 or approximately 0.21%.
To find the fraction of automotive batteries that would be expected to survive beyond 1000 days, we need to use the information given about the mean and standard deviation of the battery life.
We know that the mean (average) battery life is 900 days, and the standard deviation is 35 days. This means that the distribution of battery life follows a normal curve, with most batteries falling within a range of values centered around the mean.
To find the fraction of batteries that would survive beyond 1000 days, we need to calculate the z-score for this value. The z-score represents the number of standard deviations that a value is from the mean.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the value we are interested in (1000 days), μ is the mean (900 days), and σ is the standard deviation (35 days).
Plugging in these values, we get:
z = (1000 - 900) / 35 = 2.86
We can use a z-score table or calculator to find the proportion of values beyond this z-score.
From the z-score table, we can see that the area beyond a z-score of 2.86 is 0.0021. This means that only 0.21% of automotive batteries would be expected to survive beyond 1000 days.
Therefore, the fraction of these batteries that would be expected to survive beyond 1000 days is 0.0021 or approximately 0.21%.
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In which shapes does the measure of
∠
K
=
40
°
∠
K
=
40
°
?
Select the shapes you want to choose.
Answer:
Step-by-step explanation:
Let y = 5x2 Find the change in y, Δy when x = 4 and Ax 0. 2 Find the differential dy when x = 4 and dx = 0. 2
The differential [tex]dy=2[/tex] when [tex]x = 4[/tex] and [tex]dx = 0. 2.[/tex]
To find the change in [tex]y,Δy[/tex] , when x changes from, we can use the [tex]4 to 4 +Δx = 4 + 0.2 = 4.2[/tex] formula:
[tex]Δy = y(x + Δx) - y(x)[/tex]
where[tex]y(x) = 5x^2.[/tex]
So, plugging in[tex]x = 4[/tex] and[tex]x + Δx = 4.2[/tex] , we get:
[tex]Δy = y(4.2) - y(4)[/tex]
[tex]= 5(4.2)^2 - 5(4)^2[/tex]
[tex]= 44.2[/tex]
Therefore, the change in y is [tex]44.2[/tex] when x changes from [tex]4 to 4.2.[/tex]
To find the differential dy when [tex]x = 4[/tex] and [tex]dx = 0.2,[/tex] we can use the formula:
[tex]dy = f'(x) × dx[/tex]
where the derivative of y with respect to x, which is:
[tex]f'(x) = 10x[/tex]
Plugging in [tex]x = 4[/tex] we get:
[tex]= 2[/tex]
Therefore, the differential [tex]dy = 2[/tex] when [tex]x = 4[/tex] and [tex]dx = 0.2.[/tex]
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2. Plot the point (-3, 2,-2)
x
y +
the following points have been plotted on the cartesian plan:
(-3, 2) and
(-3, -2). The above represent coordinates.
What are coordinates ?
A coordinate system in geometry is a system that employs one or more integers, or coordinates, to define the position of points or other geometric components on a manifold such as Euclidean space.
A coordinate system is a framework for specifying the relative positions of objects in a specific region, such as an area on the earth's surface or the whole earth's surface. A geographic coordinate system determines locations on the world by using a three-dimensional spherical surface.
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) Which can replace the missing value so that the relation is still a function?
{(3, 7), (0, −2), (____, −3), (−2, 1), (1, 4)}
A. 0 B. 1 C. 2 D. 3
The missing value so that the relation is still a function is {(3, 7), (0, −2), (1, −3), (−2, 1), (1, 4)} (option b)
In this case, let's first check if the given set is a function without the missing value. We can do this by checking if each input value appears only once in the set. If we look at the set {(3, 7), (0, −2), (−2, 1), (1, 4)}, we can see that each input value appears only once, which means that the set is a function.
Now, we need to determine which value can replace the missing value so that the set remains a function. Let's consider each option one by one.
If we replace the missing input value with 1, we get {(3, 7), (0, −2), (1, −3), (−2, 1), (1, 4)}. Here, we can see that each input value maps to a unique output value, which means that the set is still a function.
Therefore, option B is the correct answer.
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A plane intersects a rectangular pyramid horizontally as shown. Describe the cross-section. Responses A rectangle B circlecircle C triangletriangle D trapezoid
The description of the cross section tells us that it is a rectangle
How to describe the cross sectionRectangles are four-sided, two-dimensional shapes that boast two sets of paralleled, opposite sides with identical lengths. All four corner angles measure at ninety degrees and the opposing sides always have the same length.
The area can be calculated by multiplying its length and width, while the perimeter is found by adding all four side measurements together. Furthermore, the perpendicular diagonals of a rectangle will bisect one another and yield equal measurement when fully extended.
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Fâ-statistics computed using maximum likelihoodâ estimators:
A.
can be used to test joint hypotheses.
B.
do not follow the standard F distribution.
C.
are not meaningful since the entire regression R² concept is hard to apply in this situation.
D.
cannot be used to test joint hypotheses.
A. can be used to test joint hypotheses.
In statistical analysis, F-statistics are used to compare the fit of two nested models, typically to test joint hypotheses. Maximum likelihood estimators are a popular method for estimating the parameters of a statistical model by maximizing the likelihood function. They are widely used in various fields due to their desirable properties, such as being consistent and asymptotically efficient.
When F-statistics are computed using maximum likelihood estimators, they can still be employed to test joint hypotheses. This involves comparing the difference in the log-likelihoods between two nested models, one being a restricted model and the other being an unrestricted model. The test statistic, in this case, follows an F distribution under the null hypothesis, which states that the restrictions imposed on the model are valid.
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helppppp please thank you.
Answer:
68 squer meter
Step-by-step explanation:
it is irregular shape so u have to give section as i draw it then rename it as A1 and A2
A1 = L×W
=13m × 2m
= 26m2
A2= L × W
=7m × 6m
= 42m2
so after weget each area then we will add them b/c we need the total area of the figur not the section
let At = area of totalAt = A1 + A2
= 26m2 + 42m2
=68m2 good luck..
82% of 300 boys polled said that they liked to play outdoors. How many boys liked to play outdoors?
Answer:
246 boys like to play outdoors
Step-by-step explanation:
82% = 0.82
0.82 x 300 = 246
Answer:
246 boys like to play outdoors
Step-by-step explanation:
82% = 0.82
0.82 x 300 = 246
if ABC measures 122,what does ADC measure
Answer: If ABC measures 122 degrees and ADC is an inscribed angle that intercepts the same arc as a central angle ABC, then ADC measures half of ABC which is 61 degrees
Step-by-step explanation:
Answer:
61 degrees
Step-by-step explanation
Angle extended to the circumference is half the angle at the centre
ADC = ½ ABC = ½ × 122 = 61
A new park is designed to contain a circular garden. The garden has a diameter of 50 m. Use 3.14 for π.
If the gardener wants to outline the garden with fencing, how many meters will the gardener need to outline the garden?
The circumference of the circle-shaped garden is 157 meters.
The gardener will need 157 meters of fencing to outline the garden.
We have,
The circumference of a circle is given by the formula C = πd, where d is the diameter.
Using this formula,
C = πd
C = 3.14 × 50
C = 157 meters
Therefore,
The circumference of the circle is 157 meters.
The gardener will need 157 meters of fencing to outline the garden.
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Solve for x.
120*
T
67
R
S
(5x + 21)
According to the figure of a circle, x is equal to 17
How to solve for xThe total arc length in a circle is equal to 360 degrees
hence arc QR + arc RS + arc QS = 360 degrees
Where
arc QR = 120
arc RS = 2 * 67 = 134
arc QS = 5x + 20
plugging in the values
120 + 134 + 5x + 21 = 360
5x = 360 - 120 - 134 - 21
5x = 85
x = 17
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Find the area of the figure.
Answer: Area, A, is x times y.
Step-by-step explanation:
in a paired analysis we first take the difference of each pair of observations, and then we do inference on these difference. (True or False)
True, in a paired analysis, we first calculate the difference for each pair of observations and then perform inference on these differences.
The difference between each pair of observations is taken, and then statistical inference is performed on these differences. This type of analysis is often used when the data are collected in pairs, such as before-and-after measurements or measurements on matched subjects.
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: Let G be the dihedral group D(4):
G = ,
and let H be the subset {1, b}. Prove that H is
not a normal subgroup of G. Show that
multiplication of the left coset
(aH)(bH) = abH.
To prove that H is not a normal subgroup of G, we need to show that there exists an element g in G such that gHg^-1 is not a subset of H.
First, note that the left cosets of H in G are {1, b} and {a, ab}. Let g = a. Then we have:
gHg^-1 = a{1, b}a^-1 = {a, ab}
Since {a, ab} is not a subset of H, we have shown that H is not a normal subgroup of G.
Now, let's show the multiplication of the left coset
(aH)(bH) = {a, ab}{b, bb} = {ab, abb, b, bb}
To simplify this expression, we can use the fact that b^2 = 1 and ab = ba^-1. Then, we have:
(aH)(bH) = {ab, abb, b, bb} = {ba^-1, baa^-1, b, 1} = {b, a, ba, 1} = (abH)
Therefore, (aH)(bH) = abH.
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Find f(g(x)) if
f(x) = x² + 2x
g(x) = x - 9
Answer: f(g(x)) = (x - 9)² + 2(x - 9)
= x² - 18x + 81 + 2x - 18
= x² - 16x + 81
Step-by-step explanation:
Consider a normal population distribution with the value of σ known. (a) what is the confidence level for the interval x ± 2. 81σ/ n ? (round your answer to one decimal place. )
The confidence level is 1 - α = 1 - 0.005 = 0.995 or approximately 99.5%.
We can use the formula for a confidence interval for a population mean, which is:
[tex]x ± z(α/2) * σ/√n[/tex]
where x is the sample mean, σ is the population standard deviation, n is the sample size, and z(α/2) is the critical value from the standard normal distribution corresponding to the desired confidence level (α).
In this case, the interval is x ± 2.81σ/√n, which is equivalent to z(α/2) = 2.81.
To find the confidence level, we need to solve for α. We can do this by finding the area in the tails of the standard normal distribution that corresponds to z(α/2) = 2.81. Using a standard normal table or a calculator, we find that the area in the right tail is 0.0025, so the area in both tails is 0.005.
Therefore, the confidence level is 1 - α = 1 - 0.005 = 0.995 or approximately 99.5%.
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Here is a box plot that summarizes data for the time, in minutes, that a fire department took to respond to 100 emergency calls.
Select all the statements that are true, according to the box plot.
a) Most of the response times were under 13 minutes.
b) Fewer than 30 of the response times were over 13 minutes.
c) More than half of the response times were 11 minutes or greater.
d) There were more response times that were greater than 13 minutes than those that were less than 9 minutes.
e) About 75% of the response times were 13 minutes or less.
According to the given box plot:
Most of the response times were under 13 minutes is true
Fewer than 30 of the response times were over 13 minutes is true
More than half of the response times were 11 minutes or greater is true
About 75% of the response times were 13 minutes or less is true
In box plot the difference between a dataset's first and third quartiles is used to determine the interquartile range (IQR), a measure of variability.
The interquartile range (IQR) depicts the range of values centred on the data's median. Because it is more resistant to outliers than other measures of variability like the range or the standard deviation, it is valuable in statistics.
Observations that are more than 1.5 times the IQR from the adjacent quartile are considered probable outliers and can be located using the IQR.
Box plots and the IQR are frequently combined to compare and summarise data from several groups or samples.
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Which system of equations has no solution? A y=x+6 , y=−x+2 B y=−3x+1 , y=4x+6 C y=−4x−3 , y=−4x−5
Answer:
The system of equations y = x + 6 and y = -x + 2 has no solution.
To see why, we can set the equations equal to each other and solve for x: x + 6 = -x + 2 2x = -4 x = -2
However, if we substitute x = -2 back into the original equations, we get: y = x + 6 = -2 + 6 = 4 y = -x + 2 = -(-2) + 2 = 4
So we end up with the same value for y in both equations, which means that the system has a unique solution of (-2, 4). Therefore, the answer is that neither system of equations listed in the search results has no solution.
Step-by-step explanation:
The University Grille on Commonwealth Avenue just released the findings from a three year-study of students’ salad orders to determine the popularity of Caesar and Ranch dressing. In this study, the ordering habits of 3000 students who have ordered salads were analyzed. 185 of these students never ordered any dressing on their salads. 2100 of the students ordered Caesar dressing, but never ordered Ranch. What is the probability that a randomly-selected student from this survey ordered Ranch?
The probability that a randomly-selected student from this survey ordered Ranch is approximately 0.2383.
We have,
Let R be the event that a student ordered Ranch dressing.
We want to find P(R), the probability that a randomly-selected student from the survey ordered Ranch.
Out of the 3000 students surveyed, 185 never ordered any dressing, so the remaining 3000 - 185 = 2815 students ordered some kind of dressing. Of these, 2100 ordered Caesar but not Ranch, so the remaining
2815 - 2100 = 715 students ordered Ranch or both dressings.
Now,
P(R) is the proportion of students who ordered Ranch or both dressings out of the total number of students surveyed:
P(R) = 715 / 3000 = 0.2383 (rounded to four decimal places)
Thus,
The probability that a randomly-selected student from this survey ordered Ranch is approximately 0.2383.
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The germination rate for bush bean seeds from a particular company is 92% (e. 92% of seeds planted and tended according to the directions will
sprout). Seeds are sold in varying smaller-sized size packets as well as in bulk. Assume that the selection of seeds for packets is random and all seeds
are independent of one another. Let X be the number of seeds that sprout.
A nursery buys bush bean seeds in bulk. If they plant exactly 1000 seeds, how many should they expect to sprout?
Answer: 920
If you consider all bulk lots of 1000 seeds, what is the standard deviation of the number of seeds that will sprout?
Answer:
If they plant exactly 1000 seeds, what is the probability that they get between 950 and a 1000 sprouts (inclusive)?
Answer:
If they plant exactly 1000 seeds, what is the probability that between 90% and 95% (not inclusive) of seeds sprout?
Answer:
How many seeds should they plant if they want to have a 5% chance of getting less than or equal to 1000 sprouts?
Answer
Check
The probability of between 90% and 95% (not inclusive) of seeds sprouting is indeed about 0.231.
Using the given information, we can model X, the number of seeds that sprout, as a binomial random variable with n = 1000 and p = 0.92.
To find the expected number of seeds that will sprout, we can use the formula for the expected value of a binomial distribution: E(X) = np. Therefore, E(X) = 1000 * 0.92 = 920.
To find the standard deviation of the number of seeds that will sprout, we can use the formula for the standard deviation of a binomial distribution: SD(X) = sqrt(np(1-p)). Therefore, SD(X) = sqrt(1000 * 0.92 * 0.08) = 8.05.
To find the probability that between 950 and 1000 seeds will sprout (inclusive), we can use the cumulative distribution function of the binomial distribution. P(950 <= X <= 1000) = P(X <= 1000) - P(X <= 949) = binom.dist(1000, 0.92, TRUE) - binom.dist(949, 0.92, TRUE) ≈ 0.991.
To find the probability that between 90% and 95% (not inclusive) of seeds sprout, we need to find the values of k such that P(0 <= X <= k) = 0.95 - 0.90 = 0.05. We can use a normal approximation to the binomial distribution with mean np = 920 and standard deviation sqrt(np(1-p)) = 8.05. The standardized value for k is (k - np) / sqrt(np(1-p)), which we can find using the standard normal distribution table or a calculator. We get z ≈ 1.645. Solving for k, we get k = np + z * sqrt(np(1-p)) ≈ 940. Therefore, the probability that between 90% and 95% (not inclusive) of seeds sprout is P(X <= 939) - P(X <= 920) ≈ 0.231.
To find the number of seeds they should plant if they want to have a 5% chance of getting less than or equal to 1000 sprouts, we can use the inverse cumulative distribution function of the binomial distribution. We need to find the value of n such that P(X <= 1000) = 0.95. We can start with a guess of n = 1200 and use the binomial distribution function to calculate P(X <= 1000) for different values of n until we get a value close to 0.95. We can also use a normal approximation to the binomial distribution with mean np and standard deviation sqrt(np(1-p)) to get an estimate for n. We get z ≈ 1.645 as before, so we can solve for np to get np ≈ 977. Solving for n, we get n ≈ 1061. Therefore, they should plant 1061 seeds if they want to have a 5% chance of getting less than or equal to 1000 sprouts.
Check:
The expected value of X is indeed 920.
The standard deviation of X is indeed 8.05.
The probability of getting between 950 and 1000 sprouts (inclusive) is indeed about 0.991.
The probability of between 90% and 95% (not inclusive) of seeds sprouting is indeed about 0.231.
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Perform the division algorithm to find the quotient q and the remainder r, and show that the values that you found are indeed correct by expressing a as a = d · q + r, when
(a) 19 is divided by 7.
(b) 39 is divided by 4.
(c) 45 is divided by 3.
(d) -18 is divided by 5.
(e) −44 is divided by 10.
(f) -111 is divided by 11. (g) 8 is divided by 3.
Performing the division algorithm to find the quotient q and the remainder r, and putting them in the formula for:
(a) 19 is divided by 7. q = 2, r = 5(b) 39 is divided by 4. q = 9, r = 3(c) 45 is divided by 3. q = 15, r = 0(d) -18 is divided by 5. q = -3, r = -3(e) −44 is divided by 10.q = -4, r = -4(f) -111 is divided by 11. q = -10, r = -1(g) 8 is divided by 3. q = 2, r = 2.A division algorithm is one that, given two numbers N and D (the numerator and denominator, respectively), computes the quotient and/or remainder of Euclidean division. Some are done by hand, while others are done by digital circuit designs and software.
There are two types of division algorithms: slow division and quick division. Each cycle of slow division algorithms yields one digit of the final quotient. Slow division examples include restoring, non-performing restoring, non-restoring, and SRT division. Fast division methods begin with a close approximation to the eventual quotient and produce twice as many digits on each repetition. This category includes the Newton-Raphson and Goldschmidt algorithms.
By division algorithm
a = d.q + r
a = dividend
d = divisor
q = quotient
r = remainder
a) 19 is divided by 7.
a = d.q + r
19 = 7.2 + 5
So, q = 2, r = 5
(b) 39 is divided by 4.
a = d.q + r
39 = 4.9 + 3
So, q = 9, r = 3
(c) 45 is divided by 3.
a = d.q + r
45 = 3.15 + 0
So, q = 15, r = 0
(d) -18 is divided by 5.
a = d.q + r
-18 = 5.(-3) + (-3)
So, q = -3, r = -3
(e) −44 is divided by 10.
a = d.q + r
-44 = 10.(-4) + -4
So, q = -4, r = -4
(f) -111 is divided by 11.
a = d.q + r
-111 = 11.(-10) + -1
So, q = -10, r = -1
(g) 8 is divided by 3.
a = d.q + r
8 = 3.2 + 2
So, q = 2, r = 2.
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Which function is represented by the graph?
The equation represented on the graph obtained from equation of a sinusoidal function is the option; y = coa(x - π/4) - 2
What is a sinusoidal function?A sinusoidal function is a periodic function that repeats at regular interval and which is based on the cosine or sine functions.
The coordinate of the points on the graph indicates that we get;
The period, T = π/4 - (-7·π/4) = 8·π/4 = 2·π
Therefore, B = 2·π/(2·π) = 1
B = 1
The amplitude, A = (-1 - (-3))/2 = 2/2 = 1
The vertical shift, D = (-1 + (-3))/2 = -4/2 = -2
The vertical shift, D = -2
The horizontal shift is the amount the midline pint is shifted relative to the y-axis
The points on the graph indicates that the peak point close to the y-axis is shifted π/4 units to the right of y-axis, therefore, the horizontal shift, C = π/4
cos(0) = 1 which is the peak point value of the trigonometric ratio, in the function which indicates that the trigonometric function of the equation for the graph is of the form, y = A·cos(B·(x - C) + D
Plugging in the above values into the sinusoidal function equation of the form; y = A·cos(B·(x - C) + D
We get;
A = 1, B = 1, C = π/4, and D = -2
The function representing the graph is therefore;
y = cos(x - π/4) - 2
The correct option is therefore;
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Find a power series representation for the function. Determine the radius of convergence, R. (Give your power series representation centered at x = 0.)
f(x) = ln(2 − x)
To find a power series representation for f(x) = ln(2-x), we can use the formula for the power series expansion of ln(1+x):
ln(1+x) = Σ (-1)^(n+1) * (x^n / n)
We can use this formula by setting x = -x/2 and multiplying by -1 to get:
ln(2-x) = ln(1 + (-x/2 - 1)) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))
Therefore, the power series representation for f(x) is:
f(x) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))
The radius of convergence of this series can be found using the ratio test:
lim |a_n+1 / a_n| = lim |(-1)^(n+2) * (x^(n+2) / (n+2) * 2^(n+3)) * (n+1) / (-1)^(n+1) * (x^(n+1) / (n+1) * 2^(n+2))|
= lim |x / 2 * (n+1) / (n+2)| = |x/2|
Therefore, the radius of convergence is R = 2. The power series representation centered at x = 0 is:
f(x) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))
To find a power series representation for the function f(x) = ln(2 - x), we can first rewrite the function as f(x) = ln(1 + (1 - x)). Now, we'll use the Taylor series formula for ln(1 + u) centered at x = 0:
ln(1 + u) = u - (1/2)u^2 + (1/3)u^3 - (1/4)u^4 + ... + (-1)^n(1/n)u^n + ...
In our case, u = (1 - x), so we can substitute it into the formula:
f(x) = (1 - x) - (1/2)(1 - x)^2 + (1/3)(1 - x)^3 - (1/4)(1 - x)^4 + ... + (-1)^n(1/n)(1 - x)^n + ...
This is the power series representation of the function f(x) = ln(2 - x) centered at x = 0.
Now, let's find the radius of convergence (R) using the ratio test:
lim (n -> ∞) |(-1)^{n+1}(1/n+1)(1 - x)^{n+1}| / |(-1)^n(1/n)(1 - x)^n|
Simplify the expression:
lim (n -> ∞) |(n/n+1)(1 - x)|
The limit depends on the value of (1 - x). To ensure convergence, the limit should be less than 1:
|(1 - x)| < 1
This inequality holds for -1 < (1 - x) < 1, which implies that the interval of convergence is 0 < x < 2. Therefore, the radius of convergence, R, is 1.
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