Answer:
The population consists of the 201 people who bought tires last month.
The sample consists of the 51 people who responded.
Step-by-step explanation:
In statistical analysis, the population is a set of all values that can be considered for an experiment. It is a set of observations from which a sample is selected. The sample is a subset of the population.
In this case, it is provided that a tire company sends an email survey to all customers who purchase new tires each month.
The number of tires sold last month was N = 201.
The email survey was sent to 100 of these 201 people and 51 responded.
In this case, since the survey is done every month, the population consists of all the people who bought tires the previous month.
So, the population consists of the 201 people who bought tires last month.
The number of people who responded was, n = 51.
The sample consists of these 51 people because the data from these 51 people would be used to conduct and draw conclusion about the survey.
So, the sample consists of the 51 people who responded.
Suppose you want to make a cylindrical pen for your cat to play in (with open top) and you want the volume to be 100 cubic feet. Suppose the material for the side costs $3 per square foot, and the material for the bottom costs $7 per square foot. What are the dimensions of the pen that minimize the cost of building it
Answer:
Step-by-step explanation:
GIVEN: Suppose you want to make a cylindrical pen for your cat to play in (with open top) and you want the volume to be [tex]100[/tex] cubic feet. Suppose the material for the side costs [tex]\$3[/tex] per square foot, and the material for the bottom costs [tex]\$7[/tex] per square foot.
TO FIND: What are the dimensions of the pen that minimize the cost of building it.
SOLUTION:
Let height and radius of pen be [tex]r\text{ and }h[/tex]
Volume [tex]=\pi r^2h=100\implies h=\frac{100}{\pi r^2}[/tex]
total cost of building cylindrical pen [tex]C=3\times \text{lateral area}+7\times\text{bottom area}[/tex]
[tex]=3\times2\pi r h+7\times\pi r^2=\pi r(6h+7r)[/tex]
[tex]=\frac{600}{r}+7\pi r^2[/tex]
for minimizing cost , putting [tex]\frac{d\ C}{d\ r}=0[/tex]
[tex]\implies -\frac{600}{r^2}+44r=0 \Rightarrow r^3=\frac{600}{44}\Rightarrow r=2.39\text{ feet}[/tex]
[tex]\implies h=5.57\text{ feet}[/tex]
Hence the radius and height of cylindrical pen are [tex]2.39\text{ feet}[/tex] and [tex]5.57\text{ feet}[/tex] respectively.
What is 12/15 × 3?(fraction times whole number)
Answer:
2.4 or 2 2/5
Step-by-step explanation:
When you multiply 12/15 and 3 you get 2.4 or 2 2/5
Answer:
Exact form: 12/5
Decimal form: 2.4
Mixed number form: 2 2/5
Step-by-step explanation:
You multiply 12 by 3, and 15 by 1, then simplify your answer.
PLEASE HURRY NOW: Find the area of the rectangle below.
12 square units
24 square units
10 square units
20 square units
SurveyMonkey is a popular website for hosting online surveys and from time to time they will also create surveys based on a subject they are interested in answering questions about. Given the widespread adoption of dating sites and apps, they wanted to learn how people feel about them. It was found that out of 400 responses, 100 have used or are currently using online dating services. The average age of those that were using online dating services was 35 years old with a sample standard deviation of 12 years. What distribution should you use to compute a confidence interval for the average age of people using online dating services
Answer:
So we use the t-distribution to compute a confidence interval for the average age of people using online dating services
Step-by-step explanation:
Confidence interval for a mean.
We have to decide between the t-distribution and the z-distribution.
T-distribution: We use the sample standard deviation.
Z-distribution: We use the population standard deviation.
In this problem:
The average age of those that were using online dating services was 35 years old with a sample standard deviation of 12 years.
So we use the t-distribution to compute a confidence interval for the average age of people using online dating services
A magazine is considering the launch of an online edition. The magazine plans to go ahead only if it is convinced that more than 15% of current readers would subscribe. The magazine contacted a simple random sample of 400 current subscribers, and 67 of those surveyed expressed interest. What should the company do? Test appropriate hypotheses and state your conclusion.
Answer:
They should go on to launch the online edition
Step-by-step explanation:
Total surveyed = 400
A = accept if 15% above subscribes
B = reject if subscribers are less than 15%
on the survey it is clearly stated that 67 expressed interest.
lets get 16% of total survey
= 15% x 400
= 60.
Since number of subscribers that showed interest is greater than number 15%
Hence the company can go ahead to launch the online edition
Accept A
Basic Computation: Setting Hypotheses Suppose you want to test the claim that a population mean equals 30 .
(a) State the null hypothesis. ___________________________
(b) State the alternate hypothesis if you have no information regarding how the population mean might differ from 30. _________________
(c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be greater than 30. _______________
(d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may not be as large as 30. ______________
Answer:
Part a
Null hypothesis: [tex]\mu =30[/tex]
Part b
Alternative hypothesis: [tex]\mu \neq 30[/tex]
Part c
Alternative hypothesis: [tex]\mu > 30[/tex]
Part d
Alternative hypothesis: [tex]\mu < 30[/tex]
Step-by-step explanation:
Previous concepts
A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".
The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".
The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".
Solution to the problem
For this case we want to test if the true mean is equal to 30 or no.
Part a
Null hypothesis: [tex]\mu =30[/tex]
Part b
Alternative hypothesis: [tex]\mu \neq 30[/tex]
Part c
Alternative hypothesis: [tex]\mu > 30[/tex]
Part d
Alternative hypothesis: [tex]\mu < 30[/tex]
if a line with a slope of 6 crosses the y-axis at (0,-4), what is the equation of the line?
Answer:
y = 6x-4
Step-by-step explanation:
The slope intercept form of a line is
y = mx+b where m is the slope and b is the y intercept
We know the slope is 6 and the y intercept is -4
y = 6x + -4
y = 6x-4
Slope-intercept form: y = mx + b (m = slope and b = y-intercept)
Slope (m) = 6
y-intercept (b) = -4
Put into its final form.
y = 6x - 4
Best of Luck!
The domain of the function is
The range of the function is
Answer:
Domain is where the function is defined. In other words, all valid values of x that allows us to find y.
Range is values of y for the function.
Step-by-step explanation:
Here is the example of function [tex]y = \sqrt{x}[/tex], so domin is between 0 and positive infinity. x cannot be negative.
The range is from 0 (because that's the lowest value of y) to positive infinity
Answer:
all nonzero real numbers for both of them
Step-by-step explanation:
e2021
a field is shaped like a rectangle with a semicircle at the end. What is the area of the field? 100m 50 m
The area of a rectangle combined with a semi-circle can be found by adding the rectangular area calculated via Length x Width to the semi-circular area calculated by 0.5 x π x r². The total area in this case would be approximately 5981.75 m².
Explanation:The area of a field that is shaped like a rectangle with a semi-circle at one end can be found by summing the area of the rectangle and the area of the semi-circle. The area of a rectangle is given by the formula Length x Width. So in this instance, the rectangle's area would be 100m x 50m = 5000 m². The area of a semi-circle is given by the formula 0.5 x π x r², where r is the radius of the semi-circle. Given one side of the rectangle is along the diameter of the semi-circle, the radius of the semi-circle would be half the width of the rectangle, i.e., 25m. So the area of the semi-circle would be 0.5 x π x 25m² = 0.5 x 3.1416 x 625 = 981.75 m² approximately. Therefore, the total area of the field would be 5000 m² + 981.75 m² which is 5981.75 m² approximately.
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Circle P has a circumference of approximately 75 inches
What is the approximate length of the radius, /? Use 3.14
for . Round to the nearest inch.
O
O
O
12 inches
24 inches
Answer:
5 inches.
Step-by-step explanation:
We know that the circumference of a circle is equivalent to πr² or πd.
We need to know the radius given the circumference.
C = πr²
75 = 3.14(r^2) (Divide both sides by 3.14)
23.88535031847134 = r^2 (Take the square root of both sides)
r ≈ 4.88726409338
r ≈ 5 inches.
-1/7/-5/9
Step1:
Step2:
Step3:
Answer:
9/35
Step-by-step explanation:
-1/7 ÷ -5/9
Copy dot flip
-1/7 * -9/5
Multiply the numerators
-1*-9 = 9
Multiply the denominators
7*5 = 35
9/35
Step-by-step explanation:
[tex]\frac{-\frac{1}{7} }{-\frac{5}{9} }[/tex]
Invert the fraction in the denominator and multiply it by the fraction in the numerator.
[tex](-\frac{1}{7} )(-\frac{9}{5} )\\\\\frac{9}{35}[/tex]
An automobile parts supplier owns a machine that produces a cylindrical engine part. This part is supposed to have an outside diameter of three inches. Parts with diameters that are too small or too large do not meet customer requirements and must be rejected. Lately, the company has experienced problems meeting customer requirements. The technical staff feels that the mean diameter produced by the machine is off target. In order to verify this, a special study will randomly sample 100 parts produced by the machine. The 100 sampled parts will be measured, and if the results obtained cast a substantial amount of doubt on the hypothesis that the mean diameter equals the target value of three inches, the company will assign a problem-solving team to intensively search for the causes of the problem.
a. The parts supplier wishes to set up a hypothesis test so that the problem-solving team will be assigned when the null hypothesis is rejected. Set up the null and alternative hypotheses for this situation.
Answer:
a) The null hypothesis is
H₀: μ₀ = 3 inches
And the alternative hypothesis is
Hₐ: μ₀ ≠ 3 inches
b) Check Explanation.
c) The $3000 is obviously the cost of a type I error.
Step-by-step explanation:
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and is always about the absence of significant difference between two proportions being compared. It usually maintains that random chance is responsible for the outcome or results of any experimental study/hypothesis testing. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis takes the other side of the hypothesis; that there is indeed a significant difference between two proportions being compared. It usually confirms the the theory being tested by the experimental setup. It usually maintains that other than random chance, there are significant factors affecting the outcome or results of the experimental study/hypothesis testing. It usually contains the signs ≠, < and > depending on the directions of the test.
For this question, we want to verify that the mean diameter produced by the machine is really off-target.
Note that, that target is 3 inches.
Hence, the null hypothesis will be that that there is no difference between the mean diameter of the 100 cylinders sampled and the target of 3 inches.
And the alternative hypothesis will confirm the concerns of the technical staff that there is a significant difference between the mean diameter of the 100 cylinders sampled and the target diameter of 3 inches. That is, the mean diameter is off-target.
Mathematically,
The null hypothesis is
H₀: μ₀ = 3 inches
And the alternative hypothesis is
Hₐ: μ₀ ≠ 3 inches
b) A type I error involves rejecting the null hypothesis and accepting the alternative hypothesis when in reality, the null hypothesis is true. It involves saying there is significant evidence to show that the mean diameter of the sampled cylinders is indeed different from the targeted 3 in. (that is, the mean diameter of sampled cylinders is off-target), so they try to fix the issue when in reality, there is actually no significant difference between the mean diameter of sampled cylinders and the targeted 3 inches.
While a type II error involves failing to reject the null hypothesis when in reality it should have been rejected.
It entails not rejecting the null hypothesis and making conclusions based on the null hypothesis, when in reality, the alternative hypothesis should have been accepted together with its conclusion.
In this one the firm would conclude that they do not need to change anything as there is no significant difference between the mean diameter of the sampled cylinders and the targeted 3 inches, when in reality, there is a significant difference between the mean diameter of the sampled cylinders and the targeted 3 inches diameter.
c) The $3000 is obviously the cost of a type I error.
This is because the type I error is the one that involves tryin to fix the problem that did not even exist in reality.
Hope this Helps!!!
Answer:
Step-by-step explanation:
An Automobile Parts supplier owns a machine that produces a cylindrical engine part. The part is supposed to have an outside diameter of 3 inches in order for customers requirements to be met.
Lately, the company has been producing cylindrical engine parts with diameters that are either < 3inches or > 3inches. Technical staff feel that the mean diameter produced by the machine, from a random sample of 100 engine parts is not equal to 3 inches. The parts supplier wishes to set up a hypothesis test so that the problem-solving team can swing into action if or when the Null Hypothesis is rejected.
The null hypothesis is:
100 Cylindrical Engine Parts have a mean diameter = 3 inches
The alternative hypothesis is:
100 Cylindrical Engine Parts have a mean diameter that is NOT equal to 3 inches
[You can input the "not equal to" sign]
If the null hypothesis is rejected after statistical procedures, the supplier will proceed to employ the services of the problem-solving team!
A process is producing a particular part where the thickness of the part is following a normal distribution with a µ = 50 mm and σ = 5 mm. If a random sample of 25 parts were taken, what is the probability that this selected sample has an average thickness greater than 53?
Answer:
0.13% probability that this selected sample has an average thickness greater than 53
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 50, \sigma = 5, n = 25, s = \frac{5}{\sqrt{25}} = 1[/tex]
What is the probability that this selected sample has an average thickness greater than 53?
This is 1 subtracted by the pvalue of Z when X = 53. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{53 - 50}{1}[/tex]
[tex]Z = 3[/tex]
[tex]Z = 3[/tex] has a pvalue of 0.9987
1 - 0.9987 = 0.0013
0.13% probability that this selected sample has an average thickness greater than 53
|-1 1/5| and |-3/5| from least to greatest
Answer:
3/5 then 11/5
Step-by-step explanation:
These symbols "I I" represent absolute values.
11/5 equals to 2 1/5
3/5 is less than 2 1/5
my question is on the picture ^
Answer:
C
Step-by-step explanation:
If the line is parallel that means 1.2 has to be the same. And passing through (5,0) means a plus five.
A staff member at UF's Wellness Center is interested in seeing if a new stress reduction program will lower employees high blood pressure levels. Twenty people are selected and have their blood pressure measured. Each person then participates in the stress reduction program. One month after the stress reduction program, the blood pressure levels of the employees were measured again. Did the program reduce the average blood pressure level? The 95% confidence interval was (5.6, 10.2). What can we expect will be the p-value for a two sided test using this data? Group of answer choices The p-value should be smaller than 0.95 The p-value should be larger than 0.95 The p-value should be smaller than 0.05. The p-value should be higher than 0.05.
Answer:
The p-value should be smaller than 0.05.
Step-by-step explanation:
We know that the 95% confidence interval for the difference of means is (5.6, 10.2). As the lower bound is larger than 0, we are 95% confident that the reduction program had had a effect in the blood pressure level.
Then, as the program had a significant effect in the blood pressure level, we know that the null hypothesis (that states that the blood pressure levels would no change significantly) is then rejected.
If the significance level is 0.05, according to the confidence of the interval, to reject the null hypothesis, the p-value had to be lower than the significance level. In conclusion, the p-value should be lower than 0.05.
What is 5.9 as a fraction.
THIS WILL HELP A LOT
Answer:
59/10
Step-by-step explanation:
5+9/10
50/10 + 9/10
=59/10
Answer:
it 59/10
Step-by-step explanation:
what fractions added equal 14/15
Answer:
There are 3 equivalent fractions 28 /30, 42/45,56/6
(y+6)^2-(y-2)^2
I got 16y+26 but it is wrong?
Answer:
16y + 32
Step-by-step explanation:
Expand each term.
(y+6)² - (y-2)²
= (y+6)(y+6) - (y-2)(y-2)
= y² + 12y + 36 - (y² - 4y + 4)
Subtract the second group by changing each term's signs
= y² + 12y + 36 - y² + 4y - 4
Collect like terms
= 16y + 32
If VX=15 and WX=7, what is UX? Write your answer as a whole number, improper fraction, or as a decimal rounded to the nearest hundredth.
Given:
Given that the length of VX is 15.
The length of WX is 7.
We need to determine the altitude UX of the given triangle.
Altitude UX:
The value of the altitude UX can be determined using the altitude rule theorem.
Applying the theorem, we have;
[tex]\frac{left}{altitude}=\frac{altitude}{right}[/tex]
Substituting left = UX, altitude = VX and right = VW in the above formula, we get;
[tex]\frac{UX}{VX}=\frac{VX}{XW}[/tex]
Substituting the values, we get;
[tex]\frac{UX}{15}=\frac{15}{7}[/tex]
Multiplying both sides of the equation by 15, we have;
[tex]UX=\frac{15 \times 15}{7}[/tex]
[tex]UX=\frac{225}{7}[/tex]
[tex]UX=32.14[/tex]
Thus, the length of UX is 32.14 units.
Answer: Its 32.14
Step-by-step explanation:
There are 157 college students interviewed about their work schedules. 85 of them work during the day. 43 of them work nightshift. 28 students work both. How many work old dayshift?
Step-by-step explanation:
No of students work during day shift = 85
No of students work during night shift = 43
No of students working both shift = 28
So first we will divide 28 by 2 = 14
Then it means there were 14 students working in day shift and 14 students working in night shift
Total number of students work during day shift = 85 + 14 = 99
Elias buys a vintage concert shirt for $20 at a resale shop. He estimates that the shirt will increase in value by 15% per year. Which recursive formula can you use to find out how much (in dollars) the shirt will be worth in n years?
Answer:
Step-by-step explanation:
You can use the simple one.
20*0.15*n year= answer
answer+20
or Compound
20*0.15 to de power of the year(s)
Please show step by step solution
It is known that the population variance equals 484. With a 0.95 probability, the sample size that needs to be taken if the desired margin of error is 5 or less is
25
74
189
75
Answer:
b) 74
The sample size that needs to be taken if the desired margin of error is 5 or less is 74
Step-by-step explanation:
Explanation:-
Given population variance σ² = 484
σ = √484
σ = 22
The level of significance ∝=0.95
The z-score of 0.95 level of significance = 1.96
Given Margin of error = 5
we know that the margin of error is determined by
[tex]M.E = \frac{Z_{\alpha }S.D }{\sqrt{n} }[/tex]
cross multiplication, we get
[tex]\sqrt{n} = \frac{Z_{\alpha }S.D }{M.E}[/tex]
[tex]\sqrt{n} = \frac{1.96 X22 }{5} = 8.624[/tex]
Squaring on both sides, we get
n = (8.624) ²
n = 74.37 ≅74
Conclusion:-
The sample size that needs to be taken if the desired margin of error is 5 or less is 74
A zoologist is studying four very closely related feline species. She wishes to compare their gestation periods. An observational study is conducted where the zoologist observes the gestation period of 50 randomly selected felines in each species. A one-way ANOVA test is run to test the hypothesis that the average gestation period is the same for each of the four species. A level of significance of 0.05 is used, and a P-value of 0.01 is calculated. Based on this result, the conclusion that can be drawn is that:
a. all of the average gestation periods are the same across all four species
b. at least some, but not all, of the gestation periods across all four species are the same
c. all of the average gestation periods across all four species are different
d. not all of the average gestation periods are the same across all four species
Answer:
[tex] p_v= 0.01[/tex]
Since the significance level is 0.05 we see that [tex]pv<\alpha[/tex] so we have enough evidence to reject the null hypothesis. And the best conclusion for this case would be:
b. at least some, but not all, of the gestation periods across all four species are the same
Because is only to identify if AT LEAST one mean is different, NOT to conclude that the all the means are different.
Step-by-step explanation:
Previous concepts
Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".
The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"
Solution to the problem
The hypothesis for this case are:
Null hypothesis: [tex]\mu_{A}=\mu_{B}=\mu_{C}= \mu_D[/tex]
Alternative hypothesis: Not all the means are equal [tex]\mu_{i}\neq \mu_{j}, i,j=A,B,C,D[/tex]
In order to find the mean square between treatments (MSTR), we need to find first the sum of squares and the degrees of freedom.
If we assume that we have [tex]p=4[/tex] groups and on each group from [tex]j=1,\dots,p[/tex] we have [tex]n_j[/tex] individuals on each group we can define the following formulas of variation:
[tex]SS_{total}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x)^2 [/tex]
[tex]SS_{between}=SS_{model}=\sum_{j=1}^p n_j (\bar x_{j}-\bar x)^2 [/tex]
[tex]SS_{within}=SS_{error}=\sum_{j=1}^p \sum_{i=1}^{n_j} (x_{ij}-\bar x_j)^2 [/tex]
And we have this property
[tex]SST=SS_{between}+SS_{within}[/tex]
And in order to test this hypothesis we need to ue an F statistic and for this case the p value calculated is
[tex] p_v= 0.01[/tex]
Since the significance level is 0.05 we see that [tex]pv<\alpha[/tex] so we have enough evidence to reject the null hypothesis. And the best conclusion for this case would be:
b. at least some, but not all, of the gestation periods across all four species are the same
Because is only to identify if AT LEAST one mean is different NOT to conclude that the all the means are different.
The correct option is (d)not all of the average gestation periods are the same across all four species.
Test Statistic:In a test of hypothesis, the test statistic is a function of the sample data used to decide whether or not to reject the null hypothesis.
Given that,
[tex]\alpha=0.05\\p-value=0.009[/tex]
[tex]H_0:[/tex]The average gestation period is the same for each of the four spices.
Since the p-value is less than the value of the [tex]\alpha[/tex].
So, we reject the [tex]H_0[/tex]
Not all of the average gestation periods are the same across all four species.
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Suppose that prior to conducting a coin-flipping experiment, we suspect that the coin is fair. How many times would we have to flip the coin in order to obtain a 99% confidence interval of width of at most .18 for the probability of flipping a head? (note that the z-score was rounded to three decimal places in the calculation)
To determine the sample size needed for a 99% confidence interval with a maximum width of .18 for a coin flip experiment, we must rely upon statistical principles such as the law of large numbers and the relationship between sample size and confidence interval width. Although we cannot give a specific number without more details, it is generally true that a larger sample size (i.e., more coin flips) results in a narrower confidence interval.
Explanation:Your question pertains to finding the necessary sample size for a particular width of a confidence interval in a coin-flip experiment. A confidence interval represents the range where we are certain that the parameter, in this case, the probability of getting a head, lies to a certain degree, say 99%. Narrowing down this interval or making it smaller would require a larger number of coin flips or trials. The probable outcome of our experiment does indeed align with the theoretical probability when a large number of trials is conducted, which is also known as the law of large numbers.
The calculation of the size of a confidence interval involves standard deviation, confidence level (z-score) and sample size. By adjusting these parameters, we can alter the size of a confidence interval. For example, a 90% confidence interval would be smaller compared to a 95% interval due to the decreased degree of certainty. However, without exact numbers we cannot directly calculate how many flips are required for a 99% confidence interval of a certain width.
As a general rule of thumb, when conducting confidence interval tests, it would be safe to suggest that a larger sample size (which for a coin flip experiment would mean more coin flips) will result in a narrower confidence interval. Therefore, to achieve a 99% confidence interval of width at most .18, one would need net a large number of trials, or flips.
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Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation.
Centerville is located at (8,0) in the xy-plane, Springfield is at (0,7), and Shelbyville is at (0,- 7). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed.
The optimal point where the cable splits into two branches for Springfield and Shelbyville is at point (4,0) on the x-axis. This is computed through calculus principles for optimization and the distance formula. The minimum total cable length connecting all three towns is about 11.66 units.
Explanation:The subject of this problem is optimization in mathematics, specifically in coordinate geometry and calculus. The problem can be solved using the distance formula in the xy-plane as well as the principles of differential calculus.
Let's denote the point (x,0) where cable splits as P, Centerville as C, Springfield as S and Shelbyville as Sh. By using the distance formula, we can determine the lengths of the branch cables, CS and CSh.
CS = sqrt[(8-x)²+7²] CSh = sqrt[(8-x)²+(-7)²]
The total length of the cable is the sum of these two distances. That gives us: Cable Length = sqrt[(8-x)²+7²] + sqrt[(8-x)²+(-7)²].
To find the minimum cable length, we differentiate the above function and equate it to zero to find the critical points. Using differential calculus, we can see that minimum cable length reduces to x = 4. Therefore, the point where cable splits is (4, 0).
Substitute x = 4 into the cable length equation to get the minimum total cable length, which is roughly 11.66 units.
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The location that minimizes the amount of cable is [tex](\frac{7}{\sqrt 3},0)[/tex], total length being 20.12 units.
To minimize the amount of cable needed in the Y-shaped configuration, we need to find the optimal point (x, 0) on the x-axis where the cable splits. This point should minimize the total length of cable from Centerville to Springfield and Shelbyville.
Step-by-Step Solution:
1. Identify the distances involved:
- The distance from Centerville (8,0) to (x,0) on the x-axis.
- The distance from (x,0) to Springfield (0,7).
- The distance from (x,0) to Shelbyville (0,-7).
2. Define the distances mathematically:
- The distance from Centerville to (x,0) is:
[tex]\[ L_1 = |8 - x| \] - The distance from \((x,0)\) to Springfield \((0,7)\) is: \[ L_2 = \sqrt{x^2 + 7^2} = \sqrt{x^2 + 49} \] - The distance from \((x,0)\) to Shelbyville \((0,-7)\) is: \[ L_3 = \sqrt{x^2 + (-7)^2} = \sqrt{x^2 + 49} \][/tex]
3. Total length of the cable L:
[tex]\[ L = L_1 + L_2 + L_3 = |8 - x| + \sqrt{x^2 + 49} + \sqrt{x^2 + 49} \] \[ L = |8 - x| + 2\sqrt{x^2 + 49} \][/tex]
4. Optimize the total length L:
To find the minimum, we need to consider the derivative of L with respect to x. Since L involves absolute value, we'll consider two cases: x ≤ 8) and x > 8.
Case 1: x ≤ 8
[tex]\[ L = (8 - x) + 2\sqrt{x^2 + 49} \] \[ \frac{dL}{dx} = -1 + 2 \cdot \frac{x}{\sqrt{x^2 + 49}} \][/tex]
Set the derivative equal to zero to find critical points:
[tex]\[ -1 + 2 \cdot \frac{x}{\sqrt{x^2 + 49}} = 0 \] \[ 2 \cdot \frac{x}{\sqrt{x^2 + 49}} = 1 \] \[ \frac{x}{\sqrt{x^2 + 49}} = \frac{1}{2} \] \[ x = \frac{\sqrt{x^2 + 49}}{2} \][/tex]
Square both sides to solve for x:
[tex]\[ x^2 = \frac{x^2 + 49}{4} \] \[ 4x^2 = x^2 + 49 \] \[ 3x^2 = 49 \] \[ x^2 = \frac{49}{3} \] \[ x = \frac{7}{\sqrt{3}} = \frac{7\sqrt{3}}{3} \][/tex]
Case 2: (x > 8)
This case would lead to a contradiction because the optimal point must lie on the interval [tex]\(0 \leq x \leq 8\)[/tex] for the Y-configuration to be practical.
Conclusion:
The point (x, 0) that minimizes the total length is:
[tex]\[x = \frac{7\sqrt{3}}{3}\][/tex]
[tex]1. \(d_1 = |x - 8| = \left| \frac{7}{\sqrt{3}} - 8 \right|\)\\2. \(d_2 = \sqrt{x^2 + 49} = \sqrt{\left(\frac{7}{\sqrt{3}}\right)^2 + 49}\)\\3. \(d_3 = \sqrt{x^2 + 49} = \sqrt{\left(\frac{7}{\sqrt{3}}\right)^2 + 49}\)[/tex]
[tex]\[ \text{Total length} = \left| \frac{7}{\sqrt{3}} - 8 \right| + 2\sqrt{\left(\frac{7}{\sqrt{3}}\right)^2 + 49} \]\[ \text{Total length} = \left| \frac{7}{\sqrt{3}} - 8 \right| + 2\sqrt{\frac{49}{3} + 49} \]\[ \text{Total length} = \left| \frac{7}{\sqrt{3}} - 8 \right| + 2\times \sqrt{16.33+49}\]\[ \text{Total length} = \left| 4.04 - 8 \right| + 2\times 8.08\][/tex]
[tex]\text{Total length}=3.96+16.16=20.12[/tex]
Complete the statements to apply the triangle inequality rule to the given triangle. QS + QR > QR + RS > RS + QS >
Answer:
QS + QR > RS
QR + RS > QS
RS + QS > QR
Step-by-step explanation:
JUST TOOK THE TEST
The complete Triangle Inequality is
QS + QR > RSQR + RS > QSRS + QS > QRWhat is Triangle Inequality?The triangle inequality theorem defines the relationship between a triangle's three sides. The total of the lengths of the two sides of any triangle is always greater than the length of the third side, according to this theorem. In other words, the shortest distance between two unique points is always a straight line, according to this theorem.
According to Triangle Inequality " the sum of two side of the triangle is greater than the third side of Triangle."
Then, applying Triangle Inequality
QS + QR > RS
QR + RS > QS
RS + QS > QR
Learn more about Triangle Inequality here:
https://brainly.com/question/30298845
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Which method would determine the volume of the prism with dimensions 2 times 2 and one-fourth times 4 shown below?
A prism has a length of 2 and one-fourth, height of 4, and width of 2 inches.
There are 8 one-quarter cubes and 16 unit cubes so (8 times one-fourth) + 16 will give the volume.
There are 8 one-quarter cubes and 8 unit cubes so (8 times one-fourth) + 8 will give the volume.
There are 8 one-quarter cubes and 16 unit cubes so (8 times 4) + 16 will give the volume.
There are 8 one-quarter cubes and 8 unit cubes so (8 times 4) + 8 will give the volume.
Answer:
THE CORRECT ANSWERS (B.)Step-by-step explanation:
Answer:
The Answer is B.
Step-by-step explanation:
I took the test on Edg 2021 and got it right the first time.
Cual conjunto de pares ordenados representa los vértices del triángulo?
Los tres puntos son (-3,0), (3,0), (0,4).
Por lo tanto, la respuesta es A, espero que esto ayude, que tenga un buen día.
8x-y+3x what is the answer for that
Answer:
11x-y
Step-by-step explanation:
You add 8x+3x and get 11x then you put the equation together
11x-y
HOPE THIS HELPS!!!