What is an equation of the line that passes through the points (2, -7) and (8, -4)?
Answer:
The answer is
[tex]y = \frac{1}{2} x - 8[/tex]Step-by-step explanation:
To find the equation of the line that passes through two points , first find the slope and then use the formula
y - y1 = m(x - x1)
where m is the slope
(x1 , y1) are any of the points
To find the slope of the line using two points we use the formula
[tex]m = \frac{y2 - y1}{x2 - x1} [/tex]
Slope of the line using points
(2, -7) and (8, -4) is
[tex] \frac{ - 4 + 7}{8 - 2} = \frac{3}{6} = \frac{1}{2} [/tex]Now the equation of the line using point (2 , - 7) and slope 1/2 is
[tex] y + 7 = \frac{1}{2} (x - 2)[/tex][tex]y + 7 = \frac{1}{2} x - 1[/tex][tex]y = \frac{1}{2} x - 1 - 7[/tex]We have the final answer as
[tex]y = \frac{1}{2} x - 8[/tex]Hope this helps you
The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are two different procedures for evaluating the oxidation stability of steam turbine oils. An article reported the accompanying observations on x = TOST time hr and y = RBOT time min for 12 oil specimens.TOST 4200 3575 3750 3700 4050 2770 4870 4500 3450 2675 3750 3300RBOT 370 340 375 310 350 200 400 380 285 220 345 280Required:Calculate the value of the sample correlation coefficient. Round your answer to four decimal places. r = _____
Answer:
0.9259
Step-by-step explanation:
Given the following data :
TOST(x) :4200 3575 3750 3700 4050 2770 4870 4500 3450 2675 3750 3300
RBOT(y) : 370 340 375 310 350 200 400 380 285 220 345 280
The correlation Coefficient tells about the strength of the statistical relationship which exists between two variables. The value of correlation Coefficient ranges from - 1 to +1.
The closer the value of correlation Coefficient is to ±1 , the stronger the correlation Coefficient with a negative and positive values signifying a negative and positive relationship respectively. Value of 1 depicts a perfect correlation while 0 means no relationship exists between them. Values close to zero denotes weak relationship.
Using the online Coefficient of correlation calculator ;
The r value of the data above is 0.9259 which signifies a very strong positive relationship between the variables.
In this exercise we have to calculate the value of the coefficient which can be descriptive statistics as:
0.9259
Given the following data :
[tex]TOST(x) :\\4200\\ 3575\\ 3750 \\3700\\ 4050\\ 2770\\ 4870\\ 4500\\ 3450\\ 2675\\ 3750\\ 3300[/tex][tex]RBOT(y) : \\370 \\340 \\375\\ 310\\ 350\\ 200\\ 400\\ 380\\ 285\\ 220\\ 345\\ 280[/tex]
The correlation Coefficient tells about the strength of the statistical relationship which exists between two variables. The value of correlation Coefficient ranges from - 1 to +1.
The closer the value of correlation Coefficient is to ±1 , the stronger the correlation Coefficient with a negative and positive values signifying a negative and positive relationship respectively. Value of 1 depicts a perfect correlation while 0 means no relationship exists between them. Values close to zero denotes weak relationship.
Using the online Coefficient of correlation calculator, the r value of the data above is 0.9259 which signifies a very strong positive relationship between the variables.
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An airline claims that the proportion of no-shows for passengers who booked on its flights is less than 0.06. In a random sample of 380 reservations, 19 were no-shows. A hypothesis test is to be performed to test the airline's claim that the proportion of no-shows on its flights is less than 0.06. Calculate the test statistic associated with this sample. An airline claims that the proportion of no-shows for passengers who booked on its flights is less than 0.06. In a random sample of 380 reservations, 19 were no-shows. A hypothesis test is to be performed to test the airline's claim that the proportion of no-shows on its flights is less than 0.06. Calculate the test statistic associated with this sample.
Answer:
≈ -0.821
Step-by-step explanation:
Given:
n= 380, samplex= 19, no-shows countp = 0.06, proportion of no-showsThen, the sample proportion is:
p' = x/n = 19/ 380 = 0.05Hypothesis test:
H₀: p = 0.06H₁: p< 0.06Test statistics:
z = (p' - p) /[tex]\sqrt{p(1-p)/n}[/tex] z = (0.05 - 0.06)/[tex]\sqrt{006(1-0.06)/380}[/tex] ≈ -0.821Find the perimeter of the rectangle with the following vertices. (−6, −2), (0, −10), (5, 2), (−1, 10) 23 52 46 40
Answer:
46
Step-by-step explanation:
See attached for reference
The points given:
(−6, −2), (0, −10), (5, 2), (−1, 10)They form a rectangle as seen in the picture.
We can notice that this is a parallelogram, as respective lines have same difference of coordinates.
So calculating only the two of the sides will be sufficient to get its perimeter:
a = √(-1+6)² + (10+2)² = √25+144 = √169= 13b = √(0+6)² + (-10+2)² = √36+64 = √100 = 10So, the perimeter:
P = 2(13+10) = 46
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 5x − 4 x(x2 + 7)2
Answer:
[tex]\frac{5x-4}{x(x^2+7)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+7} + \frac{Dx+E}{(x^2+7)^2}[/tex]
Step-by-step explanation:
Given the expression [tex]\frac{5x-4}{x(x^2+7)^2}[/tex], we are to re-write the expression in form of a partial fraction.
Before we write in form of a partial fraction, we need to note the expression at the denominator. Since the expression in parenthesis is a quadratic equation, the equivalent numerator must be a linear expression.
Also the quadratic equation is a repeated form since it is squared. This means that we are to repeat the quadratic equation twice when writing as a partial fraction.
[tex]\frac{5x-4}{x(x^2+7)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+7} + \frac{Dx+E}{(x^2+7)^2}[/tex]
From the above partial fraction, it can be seen that x² + 7 in parenthesis was repeated twice and their equivalent expressions at the numerator are both linear i.e Bx+E and Dx+ E where A, B, C, D and E are the unknown constant.
Sharvay spends $15 to buy 17 pieces of candy. M&M’s cost $0.75 and candy bars cost $1. How many M&M’s and candy bars did Sharvay buy?
Answer:
8 M&Ms and 9 Candy Bars
Step-by-step explanation:
$15 dollars could buy 15 candy bars, and there are 17 pieces of candy total.
Prioritizing the number of bars:
0.75 * 2 = 1.50
1.50 * 2 = 3
At least $3 were spend on M&Ms, meaning 4 M&Ms and 12 candy bars, which is only 16 candy pieces...
8 M&Ms and 9 candy bars is equivalent to 17 total candy pieces.
Suppose that a box contains 6 cameras and that 3 of them are defective. A sample of 2 cameras is selected at random. Define the random variable X as the number of defective cameras in the sample. Write the binomial probability distribution for X . Round to two decimal places.
Answer:
X ~ Binom (n = 6, p = 0.50)
Step-by-step explanation:
We are given that a box contains 6 cameras and that 3 of them are defective.
A sample of 2 cameras is selected at random.
Let X = Number of defective cameras in the sample.
The above situation can be represented through binomial distribution;
[tex]P(X=r)= \binom{n}{r}\times p^{r} \times (1-p)^{n-r} ;x = 0,1,2,3,......[/tex]
where, n = number of trials (samples) taken = 2 cameras
x = number of success
p = probabilitiy of success which in our question is probability that
cameras are defective, i.e. p = [tex]\frac{3}{6}[/tex] = 0.50
So, X ~ Binom (n = 2, p = 0.50)
Now, the binomial probability distribution for X is given by;
[tex]P(X=r)= \binom{6}{r}\times 0.5^{r} \times (1-0.5)^{6-r} ;r = 0,1,2[/tex]
Here, the number of success can be 0, 1, or 2 defective cameras.
Makayla wants to make 200 mL of a 18% saline solution but only has access to 8% and 24% saline mixtures.
Which of the following system of equations correctly describes this situation if x represents the amount of the 8% solution used, and y represents the amount of the 24% solution used?
Answer:
x + y = 200
0.08x + 0.24y = 0.18(200)
Step-by-step explanation:
x + y = 200
0.08x + 0.24y = 0.18(200)
The equations which describes the amount of 8% solution used and the amount of 24% solution used are: x+ y=200 and x+3y=450.
What is equation?An equation is a relationship between two or more variables. They are mostly present in equal to form and are equated to find the value of variables present in them.
How to form equation?let the amount of 8% solution used be x and the amount of 24% solution used be y.
According to question the amount of total solution will be 200ml, So, the equation will be:
x +y=200
Now we have been said that the solution will be 18% saline and 8% saline mixture and 24% saline mixtures are used, So the next equation will be:
0.08x+ 0.24y=0.18*200
8x/100+24/100=18/100*200
8x+24y=3600
8(x+3y)=3600
x+3y=450
Hence the equations which shows the amount of solutions will be x+y=200 and x+3y=450.
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Activity 12-4: A large monohybrid crossa corn ear with purple and yellow kernels The total number of purple and yellow kernels on 8 different corn ears were counted: Purple kernels 3593 Yellow kernels 1102 What is the ratio of purple kernels to yellow kernels
Complete Question
The complete question is shown on the first uploaded image
Answer:
The correct option is C
Step-by-step explanation:
From the question we are told that
The number of purple kernel is [tex]n_k = 3593[/tex]
The number of yellow kernel is [tex]n_y = 1102[/tex]
Generally the ration of the purple to the yellow kernels is mathematically evaluated as
[tex]r = \frac{n_k}{n_y}[/tex]
substituting values
[tex]r = \frac{3593}{1102}[/tex]
[tex]r = 3.3[/tex]
[tex]r \approx 3[/tex]
Therefore the ratio is
[tex]1 \ Yellow : 3 \ Purple[/tex]
Find the cost of fencing a square plot area 11025 sq m. at the rate of rupees 85 per sq m.
Answer:
Rate per meter=85 Rs. hence , the cost of fencing the square plot at rate of 85 rs. per meter is 937125 Rs.
A man saves 4% of his monthly
income of $19,540, the percentage
Savings is increased in the ratio
3:2 Calculate the savings from
the monthly
income.
Answer:
Although the question is not clear, It most likely looks like you were asking for the calculation of the savings for the month after increase.
savings for the month after increase = $1172.4
Step-by-step explanation:
First, let us calculate how much was saved before the increase in savings:
monthly income = $19,540
Percentage saved = 4% of monthly income
= 4/100 × 19,540 = 0.04 × 19,540 = $781.6
Next, we are given the ratio of increase in savings as 3:2
Let the new savings amount be x
3 : 2 = x : 781.6
[tex]\frac{3}{2} = \frac{x}{781.6} \\781.6\ \times 3\ =2x\\2344.8 = 2x\\x =\frac{2344.8}{2} \\x = \$1172.4[/tex]
therefore savings for the month after increase = $1172.4
Just incase you were looking for the savings before the increase, the answer is $781.6 (as calculated above)
The expression −50x+100 represents the balance, in dollars, of a bank account after x months. What is the rate of change, in dollars per month, of the bank account balance?
Answer:
-50
Step-by-step explanation:
Basically get two slopes -50(1)+100 will get you 1,50 (1 is x and 50 is y since its the answer)
-50(0)+100 (0,100) Y₂-Y₁/X₂-X₁ 50-100/1-0
Rate of change per month = -$50
Let R be a system consisting of rational expressions. Which operations are closed for R?
Answer:
D
Step-by-step explanation:A set is said to be closed under an operation when the application of the operation between any two elements of the set leads to an element that belongs to the same set. If a set is closed under an operation, it is said to have the closure property of that operation. When we combine two rational expressions by adding, subtracting, multiplying, or dividing, we get a rational expression. This pattern indicates that rational expressions are closed for all four operations.
A potato chip company makes potato chips in two flavors, Regular and Salt & Vinegar. Riley is a production manager for the company who is trying to ensure that each bag contains about the same number of chips, regardless of flavor. He collects two random samples of 10 bags of chips of each flavor and counts the number of chips in each bag. Assume that the population variances of the number of chips per bag for both flavors are equal and that the number of chips per bag for both flavors are normally distributed. Let the Regular chips be the first sample, and let the Salt & Vinegar chips be the second sample. Riley conducts a two-mean hypothesis test at the 0.05 level of significance, to test if there is evidence that both flavors have the same number of chips in each bag. (a) H0:μ1=μ2; Ha:μ1≠μ2, which is a two-tailed test. (b) t≈1.44 , p-value is approximately 0.167 (c) Which of the following are appropriate conclusions for this hypothesis test?
A. There is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.B. There is sufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.C. Reject H0.D. Fail to reject H0.
Answer:
A. There is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag.
D. Fail to reject H0.
Step-by-step explanation:
From the summary of the given test statistics.
The null and the alternative hypothesis are:
[tex]H_0:\mu_1=\mu_2 \\ \\ Ha:\mu_1 \neq \mu_2[/tex]
This test is also a two tailed test.
Similarly, the t value for the test statistics = 1.44
The p- value - 0.167
The level of significance ∝ = 0.05
The objective we are meant to achieve here is to determine which of the following from the given options are appropriate conclusions for this hypothesis test.
From what we have above:
Decision Rule: We fail to reject the null hypothesis since the p-value is greater than the level of significance at 0.05
CONCLUSION: Therefore, we can conclude that there is insufficient evidence at the 0.05 level of significance to conclude that Regular and Salt & Vinegar chips have different amounts of chips per bag as we fail to reject H0.
Given below are descriptions of two lines. Line 1: Goes through (-2,10) and (1,1) Line 2: Goes through (-2,8) and (2,-4)
Answer:
Option (2)
Step-by-step explanation:
1). If two lines have the same slope, lines are defined as parallel.
m₁ = m₂
2). If the multiplication of the slopes of two lines is (-1), lines will be perpendicular.
m₁ × m₂ = (-1)
Line 1 : It passes through two points (-2, 10) and (1, 1).
Slope of the line 1 = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
= [tex]\frac{1+2}{10-1}[/tex]
= [tex]\frac{3}{9}[/tex]
m₁ = [tex]\frac{1}{3}[/tex]
Line 2 : It passes through two points (-2, 8) and (2, -4).
Slope of the line 2 = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
= [tex]\frac{8+4}{-2-2}[/tex]
= [tex]-\frac{12}{4}[/tex]
m₂ = -3
Since, m₁ × m₂ = [tex]\frac{1}{3}\times (-3)[/tex]
= (-1)
Therefore, given lines are perpendicular to each other.
Option (2) is the correct option.
A box contain 12 balls in which 4 are white 3 are blue and 5 are red.3 balls are drawn at random from the box.find the chance that all three are selected
Answer:
3/11
Step-by-step explanation:
In the above question, we have the following information
Total number of balls = 12
White balls = 4
Blue balls = 3
Red balls = 5
We are to find the chance of probability that if we select 3 balls, all the three are selected.
Hence,
Probability ( all the three balls are selected) = P(White ball) × P(Blue ball) × P( Red ball)
Probability ( all the three balls are selected) = 4/12 × 3/11 × 5/10
= 60/1320
= 1/22
The number of ways by which we can selected all the three balls is a total of 6 ways:
WBR = White, Blue, Red
WRB = White, Red, Blue
RBW = Red, Blue, White
RWB = Red, White, Blue
BRW = Blue, Red, White
BWR = Blue, White, Red
Therefore, the chance that all three are selected :
1/22 × 6 ways = 6/22 = 3/11
A research center poll showed that % of people believe that it is morally wrong to not report all income on tax returns. What is the probability that someone does not have this belief? 78 The probability that someone does not believe that it is morally wrong to not report all income on tax returns is . (Type an integer or a decimal.)
Question:
A research center poll showed that 78% of people believe that it is morally wrong to not report all income on tax returns. What is the probability that someone does not have this belief?
The probability that someone does not believe that it is morally wrong to not report all income on tax returns is . (Type an integer or a decimal.)
Answer:
[tex]q = 0.22[/tex]
Step-by-step explanation:
Given
Let p represent the given proportion
p = 78%
Required
Determine the probability that someone holds a contrary belief
Start by converting the given proportion to decimal
[tex]p = 78\%[/tex]
[tex]p = \frac{78}{100}[/tex]
[tex]p = 0.78[/tex]
In probability, the sum of opposite probability is equal to 1
Represent the probability that someone holds a contrary belief with q
So;
[tex]p + q = 1[/tex]
Make q the subject of formula
[tex]q = 1 - p[/tex]
Substitute 0.78 for p
[tex]q = 1 - 0.78[/tex]
[tex]q = 0.22[/tex]
Hence, the probability that someone does not believe is 0.22
a westward moving motorcycle slows down from 24.0 m/a to 12.0 m/s in 3.0 seconds. what is the magnitude and direction of the acceleration
Answer:
0
Step-by-step explanation:
How to graph the line y=4/3x
Answer:
make a table of values
Step-by-step explanation:
then plot using those values
The required graph has been attached which represents the line y = 4/3x
What is a graph?A graph can be defined as a pictorial representation or a diagram that represents data or values.
We have been given the equation of a line below as:
y = 4/3x
Rewrite in slope-intercept form.
y = (4/3)x
Use the slope-intercept form to discover the slope and y-intercept.
Here the slope is 4/3 and y-intercept = (0, 0)
Any line can be graphed using two points. Select two x values, and plug them into the equation to find the corresponding y values.
When substitute the value of x = 0, then the value of y = 0, and When substitute the value of x = 3, then the value of y = -4,
Hence, the graph represents the line y = 4/3x
Therefore, the required graph of the line y=4/3x will be shown in the as attached file.
Learn more about the graphs here:
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A sprinkler system is being installed in a newly renovated building on campus. The average activation time is supposed to be at most 20 seconds. A series of 12 fire alarm/sprinkler system tests results in an average activation time of 21.5 seconds. Do these data indicate that the design specifications have not been met? The hypotheses to be tested are H0: m = 20 versus Ha: m > 20, where m = the true average activation time of the sprinkler system. Assume that activation times for this system are Normally distributed with s = 3 seconds.
(a) What is the value of the observed test statistic?
(b) What is the value of the P-value?
(c) Are the data statistically significant at the 5% significance level? Explain briefly.
(d) What does the decision you made mean with respect to the question "Do these data indicate that the design specifications have not been met?"
(e) If the true average activation time of the sprinkler system is, in fact, equal to 20 seconds, what type of error would you have made?
Answer:
A) t = 1.73
B) p-value = 0.0558
C) Data is not statistically significant because the p-value of 0.0558 is more than the significance value of 0.05
D) The decision means that the design specifications are not met.
E) Type II error
Step-by-step explanation:
The hypotheses are:
H₀: μ = 20
H₁: μ > 20
A) Formula for the test statistic is;
t = (x' - μ)/(s/√n)
Now, we are given;
x' = 21.5
μ = 20
s = 3
n = 12
Thus;
t = (21.5 - 20)/(3/√12)
t = 1.73
B) we have our t-value as 1.73
Now, Degree of freedom(DF) = n - 1
So,DF = 12 - 1 = 11
Using significance level of α = 0.05, t-value = 1.73 and DF = 11, one tailed hypothesis, from online P-value calculator attached, we have;
p-value = 0.0558
C) Data is not statistically significant because the p-value of 0.0558 is more than the significance value of 0.05
D) We will not reject the null hypothesis. The decision means that the design specifications are not met.
E) If the true average activation time of the sprinkler system is, in fact, equal to 20 seconds, then the null hypothesis is false.
Since we did not reject the null hypothesis even though it is false, the error that was committed was therefore a type II error.
A professor graded the final exams and found that the mean score was 70 points. Which of the following can you conclude?
A- All of the above.
B- The median score was 70 points.
C- 50% of the students scored below 70 points.
D- This would be a normal distribution.
Answer: C) 50% of the students scored below 70%
Step-by-step explanation:
Mean is the average. To find the mean (aka average) you add up all of the scores and divide by the number of tests.
B) The mean can be 70 without any test scoring 70% so B is not true.
A) Since B is not true, then A is not a valid option.
D) We don't know any of the other data so don't know if it is skewed left, skewed right, or normal. Therefore, option D is not true.
C) If the average is 70%, then half received grades above that score and half received grades below that score. So, option C is TRUE!
find the sum 7+7(2)+7(2^2)+...+7(2^9)
Answer:
7161
Step-by-step explanation:
7 + 7(2) + 7(2)² + ... + 7(2)⁹
= ∑₁¹⁰ 7(2)ⁿ⁻¹
= 7 (1 − 2¹⁰) / (1 − 2)
= 7161
heeeeeeeeelpppppppppppp
Answer:
1). x = 2.67 units
2). x = 4.80 units
3). x = 6.00 units
Step-by-step explanation:
1). By applying Pythagoras theorem,
Hypotenuse² = [Leg(1)]² + [leg(2)]²
12² = x² + b² [Let the base of both the triangles = b units]
144 = x² + b² ------(1)
Similarly, 13² = (x + 3)² + b²
169 = x² + 6x + 9 + b²
169 - 9 - 6x = x² + b²
160 - 6x = x² + b² ------(2)
From equation (1) and (2)
144 = 160 - 6x
6x = 160 - 144
x = [tex]\frac{16}{6}[/tex]
x = 2.67 units
2). By applying Pythagoras theorem,
10² = x² + h² [Let the height of the triangle = h]
100 = x² + h² ------(1)
13² = (2x)² + h²
169 = 4x² + h² -----(2)
By substituting equation (1) from equation (2),
169 - 100 = (4x² + h²) - (x² + h²)
69 = 3x²
x² = 23
x = √23
x = 4.795
x ≈ 4.80 units
3). By applying Pythagoras theorem,
9² = x² + h² [Let the height of the triangle = h units]
81 = x² + h² ------(1)
7² = (x - 4)² + h²
49 = x² + 16 - 8x + h²
49 - 16 = x² + h² - 8x
33 + 8x = x² + h² -------(2)
From equation (1) and (2)
81 = 33 + 8x
8x = 48
x = 6.00 units
Which part of an I-statement involves a description of your needs or feelings?
Answer:
the answer is c
Step-by-step explanation:
I need to know what’s 500+0+12+44+55+500+0+12+44+55+ 500+0+12+44+55+500+0+12+44+55+
Answer:
Step-by-step explanation:
2,444
Answer:
2,444.
Step-by-step explanation:
500+0+12+44+55+500+0+12+44+55+ 500+0+12+44+55+500+0+12+44+55 = 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 + 500 + 12 + 44 + 55 = 4(500 + 12 + 44 + 55) = 4(512 + 99) = 4(611) = 2,444.
Hope this helps!
In a study of academic procrastination, researchers reported that for a random sample of 41 undergraduate students preparing for a psychology exam, the mean time spent studying was 11.9 hours with a standard deviation of 4.5 hours. Compute a 95% confidence interval for μ, the mean time spent studying for the exam among all students taking this course.
Answer:
The 95% confidence interval is [tex]10.5 < \mu <13.3[/tex]
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 41[/tex]
The sample mean is [tex]\= x = 11.9 \ hr[/tex]
The standard deviation is [tex]\sigma = 4.5[/tex]
For a 95% confidence interval the confidence level is 95%
Given that the confidence level is 95% then the level of significance can be mathematically represented as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical values of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table
The values is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{ \sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{ 4.5 }{ \sqrt{41} }[/tex]
[tex]E = 1.377[/tex]
The 95% confidence interval is mathematically represented as
[tex]\= x - E < \mu < \= x - E[/tex]
substituting values
[tex]11.9 - 1.377 < \mu <11.9 + 1.377[/tex]
[tex]10.5 < \mu <13.3[/tex]
If S is a compact subset of R and T is a closed subset of S, then T is compact. Prove this using the definition of compactness.
Answer:
It has been proved that T is compact
Step-by-step explanation:
To prove this using the definition of compactness, let's assume that T is
not compact. Now, if that be the case, an open cover of T will exist. Let's call this open cover "A". Now, this open cover will have no finite subcover.
Now, from the question, since T is closed, it’s complement R\T will be open.
Therefore, if we add the set R\T to the collection of sets A, then we'll have an open cover of R and also of S.
Due to the fact that S is compact, this
cover will have a finite sub - cover which we will call B.
Finally, either B itself or B\{R\T} would be a finite sub - cover of A. This is a contradiction.
Thus, it proves that T has to be compact if S is to be a compact subset of R and T is to be a closed subset of S.
Can somebody explain how these would be done? The selected answer is incorrect, and I was told "Nice try...express the product by first multiplying the coefficients...then adding your "like term" angles...for instance, cos (2pi/5) + cos (-pi/2) = cos (2pi/5 + -pi/2)...then use the calculator in RADIAN mode to evaluate." Doing those steps, I got the correct constant but a coefficient that was completely off. For the second one, I was told "Good effort...express the quotient by first dividing the coefficients...then subtract your "like term" angles...for instance, cos (2pi/5) - cos (-pi/2) = cos (pi/6 - pi/3)...Finally, use the calculator (in radian MODE) to evaluate."
Answer:
Solution ( Second Attachment ) : - 2.017 + 0.656i
Solution ( First Attachment ) : 16.140 - 5.244i
Step-by-step explanation:
Second Attachment : The quotient of the two expressions would be the following,
[tex]6\left[\cos \left(\frac{2\pi }{5}\right)+i\sin \left(\frac{2\pi \:}{5}\right)\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
So if we want to determine this expression in standard complex form, we can first convert it into trigonometric form, then apply trivial identities. Either that, or we can straight away apply the following identities and substitute,
( 1 ) cos(x) = sin(π / 2 - x)
( 2 ) sin(x) = cos(π / 2 - x)
If cos(x) = sin(π / 2 - x), then cos(2π / 5) = sin(π / 2 - 2π / 5) = sin(π / 10). Respectively sin(2π / 5) = cos(π / 2 - 2π / 5) = cos(π / 10). Let's simplify sin(π / 10) and cos(π / 10) with two more identities,
( 1 ) [tex]\cos \left(\frac{x}{2}\right)=\sqrt{\frac{1+\cos \left(x\right)}{2}}[/tex]
( 2 ) [tex]\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos \left(x\right)}{2}}[/tex]
These two identities makes sin(π / 10) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and cos(π / 10) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex].
Therefore cos(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], and sin(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex]. Substitute,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[\cos \left(\frac{-\pi }{2}\right)+i\sin \left(\frac{-\pi \:}{2}\right)\right][/tex]
Remember that cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting those values,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex]
And now simplify this expression to receive our answer,
[tex]6\left[ \left\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}+i\left\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}\right][/tex] ÷ [tex]2\sqrt{2}\left[0-i\right][/tex] = [tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}+\frac{3\sqrt{3-\sqrt{5}}}{4}i[/tex],
[tex]-\frac{3\sqrt{5+\sqrt{5}}}{4}[/tex] = [tex]-2.01749\dots[/tex] and [tex]\:\frac{3\sqrt{3-\sqrt{5}}}{4}[/tex] = [tex]0.65552\dots[/tex]
= [tex]-2.01749+0.65552i[/tex]
As you can see our solution is option c. - 2.01749 was rounded to - 2.017, and 0.65552 was rounded to 0.656.
________________________________________
First Attachment : We know from the previous problem that cos(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{3-\sqrt{5}}}{4}[/tex], sin(2π / 5) = [tex]\frac{\sqrt{2}\sqrt{5+\sqrt{5}}}{4}[/tex], cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting we receive a simplified expression,
[tex]6\sqrt{5+\sqrt{5}}-6i\sqrt{3-\sqrt{5}}[/tex]
We know that [tex]6\sqrt{5+\sqrt{5}} = 16.13996\dots[/tex] and [tex]-\:6\sqrt{3-\sqrt{5}} = -5.24419\dots[/tex] . Therefore,
Solution : [tex]16.13996 - 5.24419i[/tex]
Which rounds to about option b.
According to a government study among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $1,999. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $574. (Round your z-score computation to 2 decimal places and final answers to 2 decimal places.) What percent of the adults spend more than $2,550 per year on reading and entertainment?
Answer:
The probability is [tex]P(X > x ) = 0.19215[/tex]
Step-by-step explanation:
From the question we are told that
Th The population mean [tex]\mu = \$ 1,999[/tex]
The standard deviation is [tex]\sigma = \$ 574[/tex]
The values considered is [tex]x = \$ 2,500[/tex]
Given that the distribution of the amounts spent follows the normal distribution then the percent of the adults spend more than $2,550 per year on reading and entertainment is mathematically represented as
[tex]P(X > x ) = P(\frac{ X - \mu}{\sigma } > \frac{ x - \mu}{\sigma } )[/tex]
Generally
[tex]X - \mu}{\sigma } = Z (The \ standardized \ value \ of \ X )[/tex]
So
[tex]P(X > x ) = P(Z > \frac{ x - \mu}{\sigma } )[/tex]
substituting values
[tex]P(X > 2500 ) = P(Z > \frac{ 2500 - 1999}{574 } )[/tex]
[tex]P(X > 2500 ) = P(Z >0.87 )[/tex]
From the normal distribution table the value of [tex]P(Z >0.87 )[/tex] is
[tex]P(Z >0.87 ) = 0.19215[/tex]
Thus
[tex]P(X > x ) = 0.19215[/tex]
A football team starts on the 10 yard line moving toward the 50 yard line so they can score on the other side of the field. In three plays they gain 14 yards, lose 12 yards, and gain 4 more yards. What yard line do they start their fourth play?
Answer:
16 yard line
Step-by-step explanation:
The football team is starting on the 10 yard line. In the first play, they move up to the 24 yard line. Then in the second play, they go back to the 12 yard line since they lost 12 yards. Then in the third play, they gain 4 yards so you add 4 to 12. They end up at the 16 yard line after the third play. This means that they're going to start their fourth play at the 16 yard line.
Answer:
16 yards.
Step-by-step explanation:
They start at 10 yards. They are moving towards the 50 yard line, so gaining yards will add to the 10 yards instead of subtract from the 10 yards.
In the first play, they gain 14 yards. 10 + 14 = 24 yards.
In the second play, they lose 12 yards. 24 - 12 = 12 yards.
In the third play, they gain 4 yards. 12 + 4 = 16 yards, which is where they start their fourth play.
Hope this helps!
