Answer:
Step-by-step explanation:
Hello!
You have two variables of interest
X₁: failure stress of a NiCrAlZr coating after nine 1-hr cycles.
X₂: failure stress of a NiCrAlZr coating after six 1-hr cycles.
a)
To be able to estimate the difference between the means using a confidence interval, you need that both variables have a normal distribution and to determine whether or not the population variances are equal.
If the population variances are equal, σ₁²=σ₂², you can use a pooled variance t-test
If the population variances are different, σ₁²≠σ₂², you have to use Welch's t-test
Using α: 0.05
The normality test for X₁ shows a p-value of 0.7449 ⇒ You can assume it has a normal distribution.
The normality test for X₂ shows a p-value of 0.9980 ⇒ You can assume it has a normal distribution.
The F-test for variance homogeneity shows a p-value of 0.6968 (H₀:σ₁²=σ₂²) ⇒You can assume both population variances are equal.
b) and c)
You need to test if both population means are the same, the hypotheses are:
H₀: μ₁=μ₂
H₁: μ₁≠μ₂
α: 0.05
[tex]t= \frac{(X[bar]_1-X[bar]_2)-(Mu_1-Mu_2)}{Sa*\sqrt{\frac{1}{n_1} +\frac{1}{n_2} } } ~~t_{n_1+n_2-2}[/tex]
[tex]Sa= \sqrt{\frac{(n_1-1)S^2_1+(n_2-1)S^2_2}{n_1+n_2-2} } = \sqrt{\frac{8*4.28+5*5.62}{9+6-2} }= \sqrt{4.7953}= 2.189= 2.19[/tex]
[tex]t_{H_0}= \frac{(X[bar]_1-X[bar]_2)-(Mu_1-Mu_2)}{Sa*\sqrt{\frac{1}{n_1} +\frac{1}{n_2} } } = \frac{(16.36-11.48)-0}{2.19*\sqrt{\frac{1}{9} +\frac{1}{6} } } = 4.23[/tex]
The distribution of this test is a t with 13 degrees of freedom and the test is two-tailed, so to calculate the p-value you have to do the following:
P(t₁₃≤-4.23)+P(t₁₃≥4.23)= P(t₁₃≤-4.23)+[1-P(t₁₃<4.23)]= 0.000492 + (1-0.999508)= 2*0.000492= 0.000984≅ 0.001
The p-value: 0.001 is less than α: 0.05, the decision is to reject the null hypothesis.
I hope it helps!
HELP! Given the inequality select ALL possible solutions
A gourmet pizza café sells three sizes of pizzas. If you buy all three sizes, it costs $46.24. A medium pizza costs $15.75 and a large pizza costs $17.50. How much does the small pizza cost? What did you need to do to solve this problem?
Answer:
Step-by-step explanation:
You need to subtract the prices of the medium and large from the total
Small = 46.24 - medium - large
17.50 + 15.75 = 33.75
S = 46.24 - 33.75
S = 12.99
As the saying goes, “You can't please everyone.” Studies have shown that in a large
population approximately 4.5% of the population will be displeased, regardless of the
situation. If a random sample of 25 people are selected from such a population, what is the
probability that at least two will be displeased?
A) 0.045
B) 0.311
C) 0.373
D) 0.627
E) 0.689
Answer:
Step-by-step explanation:
The correct answer is (B).
Let X = the number of people that are displeased in a random sample of 25 people selected from a population of which 4.5% will be displeased regardless of the situation. Then X is a binomial random variable with n = 25 and p = 0.045.
P(X ≥ 2) = 1 – P(X ≤ 1) = 1 – binomcdf(n: 25, p: 0.045, x-value: 1) = 0.311.
P(X ≥ 2) = 1 – [P(X = 0) + P(X = 1)] = 1 – 0C25(0.045)0(1 – 0.045)25 – 25C1(0.045)1(1 – 0.045)24 = 0.311.
The probability that at least two people will be displeased in a random sample of 25 people is approximately 0.202.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
Example:
The probability of getting a head in tossing a coin.
P(H) = 1/2
We have,
This problem can be solved using the binomial distribution since we have a fixed number of trials (selecting 25 people) and each trial has two possible outcomes (displeased or not displeased).
Let p be the probability of an individual being displeased, which is given as 0.045 (or 4.5% as a decimal).
Then, the probability of an individual not being displeased is:
1 - p = 0.955.
Let X be the number of displeased people in a random sample of 25.
We want to find the probability that at least two people are displeased, which can be expressed as:
P(X ≥ 2) = 1 - P(X < 2)
To calculate P(X < 2), we can use the binomial distribution formula:
[tex]P(X = k) = (^n C_k) \times p^k \times (1 - p)^{n-k}[/tex]
where n is the sample size (25), k is the number of displeased people, and (n choose k) is the binomial coefficient which represents the number of ways to choose k items from a set of n items.
For k = 0, we have:
[tex]P(X = 0) = (^{25}C_ 0) \times 0.045^0 \times 0.955^{25}[/tex]
≈ 0.378
For k = 1, we have:
[tex]P(X = 1) = (^{25}C_1) \times 0.045^1 \times 0.955^{24}[/tex]
≈ 0.42
Therefore,
P(X < 2) = P(X = 0) + P(X = 1) ≈ 0.798.
Finally, we can calculate,
P(X ≥ 2) = 1 - P(X < 2)
= 1 - 0.798
= 0.202.
Thus,
The probability that at least two people will be displeased in a random sample of 25 people is approximately 0.202.
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Consider the diagram below.
3.5 in
2 in.
For the above circular shape, chord FE is a perpendicular bisector of chord BC. Which of the following
represents the diameter of the circle?
Answer: C. 8.1
I just got that one wrong, but there is the right answer
The diameter of the circle is C. 8.125 inches.
What is Circle?Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.
Given is a circle.
Given, chord FE is a perpendicular bisector of chord BC.
Here FE is the diameter of the circle.
We have a chord theorem which states that products of the lengths of the segments of line formed by two intersecting chords on each chord are equal.
Let X be the intersecting point of the chords.
Here, using the theorem,
BX . CX = FX . EX
3.5 × 3.5 = 2 × EX
EX = 6.125
Diameter = FE = FX + EX = 2 + 6.125 = 8.125
Hence the diameter is 8.125 inches.
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The complete question is given below.
Olivia is carpeting her living room. It is 6-by-7.5 feet. If she wants to buy 10 percent extra for waste, how m any square feet of carpet should she buy? A. 45 square feet B. 49.5 square feet C. 55 square feet D. 59.5 square feet
Answer:
49.5 square feet.
Step-by-step explanation:
6 by 7.5 means 6 x 7.5.
6 x 7.5 = 45
She wants to by 10% extra.
45 is 100% so multiply 45 by 1.1.
45 x 1.1 = 49.5
Olivia's room area is 45 square feet. Considering a 10% extra for waste, she should buy 49.5 square feet of carpet.
Explanation:To calculate how much carpet Olivia should buy including waste, we first need to figure out the area of her room. The area of a rectangle is found by multiplying the length by the width, so in this case, we multiply 6 feet by 7.5 feet, which equals 45 square feet. Then we factor in the 10 percent extra for waste - which is 4.5 square feet (10% of 45). Adding these together, we get a total of 49.5 square feet. Therefore, Olivia should buy 49.5 square feet of carpet. The correct answer is B. 49.5 square feet.
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Circle O is shown. Secant K L intersects tangent K J at point K outside of the circle. Tangent K J intersects circle O at point M. Secant K L intersects the circle at point N. The measure of arc M N is 62 degrees. The measure of arc M L is 138 degrees.
Shondra wants to find the measure of angle JKL. Her work is started. What is the measure of angle JKL?
m∠JKL = One-half (138 – 62)
m∠JKL =
°
Answer: 38
138 - 62 = 76
1/2 > 0.5
so 76 x 0.5 = 38
Step-by-step explanation:
Answer:
38
Step-by-step explanation:
The heights of a group of boys and girls at a local middle school are shown on the dot plots below
Boys' Heights
40 41 44 46 48 60 62 64
C
Inches
M
Girls' Heights
40 41 44 46 48 50 52 54 56 58
40 is the height of the boy and the inch is C
What is 2.888888 as a fraction?
Answer:
26/9
Step-by-step explanation:
8/9=.88888888888889 (on calc)
2+8/9
18/9+8/9
=26/9
0.273 repeating decimal into a fraction
Decimal:0.273 or 0.273273273
Fraction:273/999=91/333
Answer:
0.273 as a fraction is 273/1000
0.273273273... as a repeating decimal, (273/999) as repeating decimal fraction.
g 6. Provide an example of (a) a geometric series that diverges. (b) a geometric series PN n=0 an, that starts at n = 0 and converges. Find its sum. (c) a geometric series PN n=1 an, that starts at n = 1 and converges. Find its sum. (d) Explain how the sums for a geometric series that starts at n = 0 differs from the same series that starts at n = 1.
Answer:
Check step-by-step-explanation.
Step-by-step explanation:
A given criteria for geometric series of the form [tex]\sum_{n=0}^{\infty} r^n[/tex] is that [tex]|r|<1[/tex]. Other wise, the series diverges. When it converges, we know that
[tex] \sum_{n=0}^\infty r^n = \frac{1}{1-r}[/tex].
So,
a)[tex]\sum_{n=0}^\infty (\frac{3}{2})^n[/tex] diverges since [tex]\frac{3}{2}>1[/tex]
b)[tex]\sum_{n=0}^\infty (\frac{1}{2})^n [/tex]converges since [tex]\frac{1}{2}<1[/tex], and
[tex]\sum_{n=0}^\infty (\frac{1}{2})^n= \frac{1}{1-\frac{1}{2}} = \frac{2}{2-1} = 2[/tex]
c)We can use the series in b) but starting at n=1 instead of n=0. Since they differ only on one term, we know it also converges and
[tex]\sum_{n=1}^{\infty}(\frac{1}{2})^n = \sum_{n=0}^{\infty}(\frac{1}{2})^n-(\frac{1}{2})^0 = 2-1 = 1[/tex].
d)Based on point c, we can easily generalize that if we consider the following difference
[tex]\sum_{n=1}^\infty r^n-\sum_{n=0}^\infty r^n = r^0 = 1[/tex]
So, they differ only by 1 if the series converges.
Final answer:
A divergent geometric series has a common ratio (r) greater than 1. A convergent geometric series has a common ratio (r) between -1 and 1. The sums of geometric series that start at n = 0 and n = 1 are different because the first term is included or omitted.
Explanation:
The questions can be answered as -
(a) A geometric series that diverges is an example where the common ratio (r) is greater than 1. An example of a divergent geometric series is: 2 + 4 + 8 + 16 + ...
(b) A geometric series that converges is an example where the common ratio (r) is between -1 and 1. An example of a convergent geometric series starting at n = 0 is: 1 - 1/2 + 1/4 - 1/8 + ... To find its sum, we can use the formula for the sum of a geometric series: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Plugging in the values, the sum of this series is 2/3.
(c) A geometric series that starts at n = 1 and converges can have a different sum since the first term is omitted from the calculation. An example of such a series is: 1/2 + 1/4 + 1/8 + 1/16 + ... To find its sum, we use the same formula as in part (b) but with a different first term. In this case, the sum of the series is 1/2.
(d) The sums for a geometric series that starts at n = 0 and n = 1 are different because the first term is included in the sum for n = 0 but omitted in the sum for n = 1.
A national consumer magazine reported the following correlations.
The correlation between car weight and car reliability is -0.30.
The correlation between car weight and annual maintenance cost is 0.20.
Which of the following statements are true?
I. Heavier cars tend to be less reliable.
II. Heavier cars tend to cost more to maintain.
III. Car weight is related more strongly to reliability than to maintenance cost.
a. III only
b. I, II, and III
c. I and II
d. I only
Answer:
Correct option: (b).
Step-by-step explanation:
The correlation coefficient is a statistical degree that computes the strength of the linear relationship between the relative movements of the two variables (i.e. dependent and independent). It ranges from -1 to +1.
Negative correlation is a relationship amid two variables in which one variable rises as the other falls, and vice versa. A positive correlation occurs when one variable declines as the other variable declines, or one variable escalates while the other escalates.
In statistics, a perfect positive correlation is represented by +1 and -1 indicates a perfect negative correlation.
The closer the correlation value is to 1 the stronger the relationship between the two variables.
Let,
X = car weight
Y = car reliability
Z = annual maintenance cost
It is provided that:
Corr. (X, Y) = -0.30
Corr. (X, Z) = 0.20
The correlation coefficient of car weight and car reliability is negative. This implies that there is a negative relation between the two variables, i.e. as the weight of the car increases the reliability decreases and vice-versa.
And the correlation coefficient of car weight and annual maintenance cost is positive. That is, the two variables move in the same direction, i.e. as the weight of the car increases the annual maintenance cost also increases.
The correlation coefficient value of (X, Y) is closer to 1 than that of (X, Z).
This implies that the relationship between car weight and car reliability is much more stronger that the relationship between car weight and annual maintenance cost.
Thus, all the provided statement are correct.
Hence, the correct option is (b).
Final answer:
The correlations indicate that heavier cars tend to be less reliable and cost more to maintain. (option b) I, II, and III.
Explanation:
The given correlations between car weight and car reliability (-0.30) and car weight and annual maintenance cost (0.20) can be used to determine the relationship between these variables.
I. Heavier cars tend to be less reliable.
II. Heavier cars tend to cost more to maintain.III. Car weight is related more strongly to reliability than to maintenance cost.Based on the given correlations, the following statements are true:
I. Heavier cars tend to be less reliable.II. Heavier cars tend to cost more to maintain.III. Car weight is related more strongly to reliability than to maintenance cost.Therefore, the correct answer is (option b) I, II, and III.
A school has one computer for every 17 students. If the school has 714 students, how many computers does it have?
Answer:
42 computers
Step-by-step explanation:
If there is 1 computer per 17 students, then we simply divide the amount of students by 17 to get the amount of computers.
Answer:
42
Step-by-step explanation:
714/17= 42
[tex]\sqrt{91 - 40\sqrt{3} }[/tex]
Answer:
[tex]5\sqrt{3} - 4[/tex]
Step-by-step explanation:
[tex]\sqrt{(4-5\sqrt{3} )^{2} }[/tex]
[tex]5\sqrt{3} - 4[/tex]
Sheryl takes a summer job selling hats for a local soccer team. She realizes that there is a relationship between the number of games the team wins each season and the number of hats vendors like her tend to sell. She collects data from the past several seasons. The scatter plot shows her data and the line of best fit.
Using technology, she finds that the equation of the line of best fit is y = 3.75x + 13.75.
Based on the equation for the line of best fit, about how many hats can Sheryl predict she will sell if the team wins 9 games this season?
A.
48 hats
B.
42 hats
C.
9 hats
D.
10 hats
Answer:
47
Step-by-step explanation:
x is games won
y is hats sold
[tex]y=3.75x+13.75\\y=3.75(9) + 13.75\\y=47.5[/tex]
Since you can't sell .5 of a hat- you need to round this answer.
Answer:
48
Step-by-step explanation:
suppose that no one demanded a hotel room at $150. At this price how much profit would a hotel owner earn
Answer:
Depending on how many people buy hotel rooms, then there is no answer
Step-by-step explanation:
(The question is not specific enough)
a consumer magazine counts the number of tissues per box in a random sample of 15 boxes of No- Rasp facial tissues. The sample standard deviation of the number of tissues per box is 97. Assume that the population is normally distributed. What is the 95% confidence interval for the population variance of the number of tissues per box?
Answer:
95% confidence interval for the population variance of the number of tissues per box is [5043.11 , 23401.31].
Step-by-step explanation:
We are given that a consumer magazine counts the number of tissues per box in a random sample of 15 boxes of No- Rasp facial tissues. The sample standard deviation of the number of tissues per box is 97.
Firstly, the pivotal quantity for 95% confidence interval for the population variance is given by;
P.Q. = [tex]\frac{(n-1)s^{2} }{\sigma^{2} }[/tex] ~ [tex]\chi^{2}__n_-_1[/tex]
where, [tex]s^{2}[/tex] = sample variance = [tex]97^{2}[/tex] = 9409
n = sample of boxes = 15
[tex]\sigma^{2}[/tex] = population variance
Here for constructing 95% confidence interval we have used chi-square test statistics.
So, 95% confidence interval for the population variance, [tex]\sigma^{2}[/tex] is ;
P(5.629 < [tex]\chi^{2}__1_4[/tex] < 26.12) = 0.95 {As the critical value of chi-square at 14
degree of freedom are 5.629 & 26.12}
P(5.629 < [tex]\frac{(n-1)s^{2} }{\sigma^{2} }[/tex] < 26.12) = 0.95
P( [tex]\frac{5.629 }{(n-1)s^{2} }[/tex] < [tex]\frac{1}{\sigma^{2} }[/tex] < [tex]\frac{26.12 }{(n-1)s^{2} }[/tex] ) = 0.95
P( [tex]\frac{(n-1)s^{2} }{26.12 }[/tex] < [tex]\sigma^{2}[/tex] < [tex]\frac{(n-1)s^{2} }{5.629 }[/tex] ) = 0.95
95% confidence interval for [tex]\sigma^{2}[/tex] = [ [tex]\frac{(n-1)s^{2} }{26.12 }[/tex] , [tex]\frac{(n-1)s^{2} }{5.629 }[/tex] ]
= [ [tex]\frac{14 \times 9409 }{26.12 }[/tex] , [tex]\frac{14 \times 9409 }{5.629 }[/tex] ]
= [5043.11 , 23401.31]
Therefore, 95% confidence interval for the population variance of the number of tissues per box is [5043.11 , 23401.31].
you have $15,000 to invest for 5 years at 5.5% annual interest rate that is compounded continuously. how much money will you have at the end of 5 years?
Answer:
$19,747.96
Step-by-step explanation:
You are going to want to use the continuous compound interest formula, which is shown below:
[tex]A = Pe^{rt}[/tex]
A = total
P = principal amount
r = interest rate (decimal)
t = time (years)
First, lets change 5.5% into a decimal:
5.5% -> [tex]\frac{5.5}{100}[/tex] -> 0.055
Next, plug in the values into the equation:
[tex]A=15,000e^{0.055(5)}[/tex]
[tex]A=19,747.96[/tex]
After 5 years, you will have $19,747.96
Diane has $10,000 in savings account that earns interest annually at the rate 5%. How much money In interest will she earn in 1 year?
Answer:
$500
Step-by-step explanation:
The amount of interest in one year is the product of the interest rate and the account balance:
I = Prt = $10,000×0.05×1 = $500 . . . . . . interest earned in 1 year
In a large population of college-educated adults, the average IQ is 112 with standard deviation 25. Suppose 300 adults from this population are randomly selected for a market research campaign. The distribution of the sample means for IQ is
a. approximately Normal, mean 112, standard deviation 25.
b. approximately Normal, mean 112, standard deviation 1.443.
c. approximately Normal, mean 112, standard deviation 0.083.
d. approximately Normal, mean equal to the observed value of the sample mean, standard deviation 25.
Answer:
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
[tex]\mu_{\bar X}= 112[/tex]
[tex]\sigma_{\bar X}=\frac{25}{\sqrt[300}}= 1.443[/tex]
And the best option for this case would be:
b. approximately Normal, mean 112, standard deviation 1.443.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the IQ of a population, and for this case we know the following info:
Where [tex]\mu=65.5[/tex] and [tex]\sigma=2.6[/tex]
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
[tex]\mu_{\bar X}= 112[/tex]
[tex]\sigma_{\bar X}=\frac{25}{\sqrt[300}}= 1.443[/tex]
And the best option for this case would be:
b. approximately Normal, mean 112, standard deviation 1.443.
Write an expression that gives the requested sum.
The sum of the first 16 terms of the geometric sequence with first term 9 and common ratio 2
Answer:
The sum of the first 16 terms of the geometric sequence
[tex]S_{16} = \frac{9(2^{16}-1) }{2-1}[/tex]
S₁₆ = 5,89,815
Step-by-step explanation:
Explanation:-
Geometric series:-
The geometric sequence has its sequence Formation
a , a r, ar² , ar³,...…..a rⁿ be the n t h sequence
Given first term a=9 and common ratio 'r' = 2
The sum of the first 16 terms of the geometric sequence
[tex]S_{n} = \frac{a(r^{n}-1) }{r-1} if r>1[/tex]
Given first term a=9 , 'r' = 2 and n=16
[tex]S_{16} = \frac{9(2^{16}-1) }{2-1}[/tex]
[tex]S_{16} = \frac{9(2^{16}-1) }{1}= 9(65,536-1)=5,89,815[/tex]
Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such than ∠ C A B = 46.5 ° . Find the distance across the lake from A to B.
Answer:
The distance across the lake from A to B = 690.7 ft
Step-by-step explanation:
Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that
∠CAB=46.5∘. He also measures CA as 312 ft and CB as 527 ft. Find the distance between A and B.
Given
A = 46.5°
a = 527 ft
b = 312 ft
To find; c = ?
Using the sine rule
[a/sin A] = [b/sin B] = [c/sin C]
We first obtain angle B, that is, ∠ABC
[a/sin A] = [b/sin B]
[527/sin 46.5°] = [312/sin B]
sin B = 0.4294
B = 25.43°
Note that: The sum of angles in a triangle = 180°
A + B + C = 180°
46.5° + 25.43° + C = 108.07°
C = 108.07°
We then solve for c now,
[b/sin B] = [c/sin C]
[312/sin 25.43°] = [c/sin 108.07°]
c = 690.745 ft
Hope this Helps!!!
To find the distance across the lake from A to B, use trigonometry and the length of side AC and angle CAB to find the length of side AB.
Explanation:To find the distance across the lake from point A to point B, we can use trigonometry.
Given that ∠CAB = 46.5°, we can find the distance across the lake by finding the length of side AB in triangle CAB.
Since we know the length of side AC (53 m) and the angle CAB (46.5°), we can use the sine function to find side AB:
AB = AC * sin(CAB) = 53 m * sin(46.5°) = 39.6 m
Therefore, the distance across the lake from point A to point B is approximately 39.6 meters.
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A sphere has a diameter of 8 cm. Which statements about the sphere are true?
Answer:
I believe the correct statements are 1), 2), and 3).
The statements provided in the question are all true. The sphere has a radius of 4 cm (A), the diameter's length is twice the length of the radius (B), and the volume of the sphere is 256/3 π cm³ (C).
A) The sphere has a diameter of 8 cm, which is the distance between two points on the surface of the sphere passing through its center. By definition, the diameter is twice the length of the radius. Thus, to find the radius of the sphere, we divide the diameter by 2:
Radius = Diameter / 2
Radius = 8 cm / 2
Radius = 4 cm
B) This statement is true. As mentioned earlier, the diameter of the sphere (8 cm) is indeed twice the length of its radius (4 cm).
C) The volume of a sphere can be calculated using the formula:
Volume = (4/3) * π * Radius³
Now, let's calculate the volume using the given radius:
Volume = (4/3) * π * (4 cm)³
Volume = (4/3) * π * 64 cm³
Volume = 256/3 * π cm³
Hence the correct option is (a), (b) and (c).
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Complete Question:
A sphere has a diameter of 8 cm.
A) The sphere has a radius of 4 cm.
B) The diameter’s length is twice the length of the radius.
C) The volume of the sphere is 256/3 π cm³
Manny, Rachelle, and Peg race each other in carts. There are no other racers, and exactly one racer wins each race (no ties allowed!). The following graph shows an incomplete probability model for who will win any given race.
Answer:
Step-by-step explanation:
0.4
Answer: 0.4
P (Manny wins) + P (Rachelle wins) + P (Peg wins) = 1
0.35 + P (Rachelle wins) + 0.25 = 1
P (Rachelle wins) = 1 - 0.35 - 0.35
P (Rachelle wins) = 0.4
The probability that Rachelle will win any given race is 0.4.
g Using calculus and the SDT (then FDT if necessary), find all global and local maximum and minimums given the function f(x) = x 3 + x 2 − x + 1 where x ∈ [−2, 1 2 ]. Clearly identify critical values and show the SDT then the FDT if the SDT didn’t provide an answer and then interpret the solution.
Answer:
[tex]S_{1 } (x,y) = (0.333, 0.815)[/tex] (Absolute minimum) and [tex]S_{2} (x,y) = (-1, 2)[/tex] (Absolute maximum)
Step-by-step explanation:
The critical points are determined with the help of the First Derivative Test:
[tex]f'(x) = 3\cdot x ^{2} +2\cdot x -1[/tex]
[tex]3\cdot x^{2} + 2\cdot x - 1 = 0[/tex]
The critical points are:
[tex]x_{1} \approx 0.333[/tex] and [tex]x_{2} \approx -1[/tex]
The Second Derivative Test offers a criterion to decide whether critical point is an absolute maximum and whether is an absolute minimum:
[tex]f''(x) = 6\cdot x +2[/tex]
[tex]f''(x_{1}) = 3.998[/tex] (Absolute minimum)
[tex]f''(x_{2}) = -4[/tex] (Absolute maximum)
The critical points are:
[tex]S_{1 } (x,y) = (0.333, 0.815)[/tex] (Absolute minimum) and [tex]S_{2} (x,y) = (-1, 2)[/tex] (Absolute maximum)
The global maximum is at x = -1 with f(-1) = 2, and the global minimum is at x = -2 with f(-2) = -1.
To find the global and local maxima and minima of the function f(x) = [tex]x^3 + x^2[/tex] − x + 1 over the interval [-2, 1/2], follow these steps:
Find critical points: First, we need to find the derivative of the function f(x):
f'(x) = [tex]3x^2[/tex]+ 2x - 1
Solve f'(x) = 0 to find critical points within the interval:
[tex]3x^2[/tex]+ 2x - 1 = 0
Using the quadratic formula x = (-b ± √(b²-4ac)) / 2a, where a = 3, b = 2, and c = -1:
x = [-2 ± √(4 + 12)] / 6
x = [-2 ± 4] / 6
This gives two solutions:
x = 1/3 and x = -1
Both are within the interval [-2, 1/2], so we consider these as critical points.
Second Derivative Test (SDT): Find the second derivative to apply the SDT:
f''(x) = 6x + 2
Evaluate f''(x) at the critical points:
f''(-1) = 6(-1) + 2 = -4 (which is negative, indicating a local maximum)
f''(1/3) = 6(1/3) + 2 = 4 (which is positive, indicating a local minimum)
Evaluate the function at the endpoints: Compute f(x) at the bounds of the interval:
f(-2) = (-2)³ + (-2)² - (-2) + 1 = -8 + 4 + 2 + 1 = -1
f(1/2) = (1/2)³ + (1/2)² - (1/2) + 1 = 1/8 + 1/4 - 1/2 + 1 = 7/8
Additionally, compute the function at the critical points:
f(-1) = (-1)³ + (-1)² - (-1) + 1 = -1 + 1 + 1 + 1 = 2
f(1/3) = (1/3)³ + (1/3)² - (1/3) + 1 = 1/27 + 1/9 - 1/3 + 1 ≈ 0.963
Interpret the results:
The maximum and minimum values for f(x) over [-2, 1/2] are:
Local maximum at x = -1 with f(-1) = 2
Local minimum at x = 1/3 with f(1/3) ≈ 0.963
Global maximum at x = -1 with f(-1) = 2
Global minimum at x = -2 with f(-2) = -1
Jaylon spent $21.60 of the $80 he got for his birthday. What percent of his money did he spend? Oh
Answer:
27%
Step-by-step explanation:
Jaylon spent 27% of his birthday money.
Answer:
27%
Step-by-step explanation:
To find the percent spent, take the amount spent and put it over the total
21.60 /80
.27
Multiply by 100% to put in percent form
27%
rewrite the expression in its simplest form
Answer:
[tex]2\sqrt[3]{x^{2}y^{2} }[/tex]
Step-by-step explanation:
10. Use the trigonometric ratio tan 0 = 3/4 to write the other five trigonometric ratios for 0
I hope this helps you
Find the value of x in the triangle shown below.
Answer:
x=8
Step-by-step explanation:
We can use the Pythagorean theorem to solve
a^2+b^2 = c^2 where a and b are the legs and c is the hypotenuse
6^2 + x^2 = 10^2
36 + x^2 = 100
Subtract 36 from each side
36-36 +x^2 = 100-36
x^2 = 64
Take the square root of each side
sqrt(x^2) = sqrt(64)
x = 8
Answer:
x = 8
Step-by-step explanation:
According to Pythagorean Theorem
[tex]AB {}^{2} + BC {}^{2} = AC {}^{2} [/tex]
here
[tex]AB = x \\ BC = 6 \\ AC = 10[/tex]
Now,
[tex] {x}^{2} + {6}^{2} = 10 {}^{2} \\ {x}^{2} = {10}^{2} - {6}^{2} \\ x {}^{2} = 100 - 36 \\ {x}^{2} = 64 \\ x = \sqrt{64} \\ x = 8[/tex]
AB = x = 8
Is -8 less or greater than -10
Answer: -8 is greater than -10
Step-by-step explanation: When in the negatives, the smaller the number, the greater it is
Over the past decade, the mean number of hacking attacks experienced by members of the Information Systems Security Association is 510 per year with a standard deviation of 14.28 attacks. The distribution of number of attacks per year is normally distributed. Suppose nothing in this environment changes.
1. What is the likelihood that this group will suffer an average of more than 600 attacks in the next 10 years?
Answer:
[tex]P(X>600)=P(\frac{X-\mu}{\sigma}>\frac{600-\mu}{\sigma})=P(Z>\frac{600-510}{14.28})=P(z>6.302)[/tex]
And we can find this probability using the complement rule and the normal standard distribution and we got:
[tex]P(z>6.302)=1-P(z<6.302)=1-0.99999 \approx 0[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the number of attacks of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(510,14.28)[/tex]
Where [tex]\mu=510[/tex] and [tex]\sigma=14.28[/tex]
We are interested on this probability
[tex]P(X>600)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X>600)=P(\frac{X-\mu}{\sigma}>\frac{600-\mu}{\sigma})=P(Z>\frac{600-510}{14.28})=P(z>6.302)[/tex]
And we can find this probability using the complement rule and the normal standard distribution and we got:
[tex]P(z>6.302)=1-P(z<6.302)=1-0.99999 \approx 0[/tex]
Given the mean and standard deviation for the number of hacking attacks, the z-score for 600 attacks is about 6.30.
The question asks for the likelihood that the group will suffer an average of more than 600 hacking attacks per year over the next 10 years, given that over the past decade the mean number of hacking attacks experienced by members is 510 per year with a standard deviation of 14.28 attacks. Considering the normal distribution of these hacking attacks, we can calculate this probability by finding the z-score corresponding to 600 attacks and then using the standard normal distribution table to find the probability of exceeding this value.
To calculate the z-score for 600 attacks, we use the formula:
Z = (X - μ) / σ
Where X is the value in question (600 attacks), μ (mu) is the mean (510 attacks), and σ (sigma) is the standard deviation (14.28 attacks). Substituting the given values:
Z = (600 - 510) / 14.28 ≈ 6.30
Looking up a z-score of 6.30 in the standard normal distribution table, we find that the area to the left of this z-score is almost 1, meaning the probability of experiencing more than 600 attacks is extremely small, approaching 0. Thus, it is incredibly unlikely that this group will suffer an average of more than 600 attacks in the next 10 years if nothing changes in their environment.