Answer:
5/9
Step-by-step explanation:
Answer:
A = 0.55 or 5/9
Step-by-step explanation:
Area = HL
A = (5/6)(2/3)
A = 0.55 or 5/9
For women aged 18-24. systolic blood pressures (in mm Hg) are normally distributed with a mean of 114 8 and a standard deviation of 13.1. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122.
The probability that their mean systolic blood pressure is between 119 and 122 is 0.0807.
The given distribution is normal.
So, the formula for the standardized random variable, z can be used.
Here,Mean of the given distribution, μ = 114.8
Standard deviation of the given distribution, σ = 13.1
Number of women aged 18-24 randomly selected, n = 23
Let X be the mean systolic blood pressure of 23 randomly selected women aged 18-24.
P(X is between 119 and 122) = P((X-μ)/σ is between (119-μ)/(σ/√n) and (122-μ)/(σ/√n))
= P((X-μ)/σ is between (119-114.8)/(13.1/√23) and (122-114.8)/(13.1/√23))
= P((X-μ)/σ is between 1.35 and 2.45)
Using standard normal distribution table,P(1.35 < z < 2.45)= P(z < 2.45) - P(z < 1.35)≈ 0.9922 - 0.9115= 0.0807
Thus, the probability that the mean systolic blood pressure of the randomly selected 23 women aged 18-24 is between 119 and 122 is approximately equal to 0.0807.
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Maya buys ice cream and onions at the store. She pays a total of $21.29. She pays a total of $3.80 for the ice cream. She buys 3 bags of onions that each cost the same amount. Write and solve an equation which can be used to determine xx, how much each bag of onions costs.
Using the Excel file Weddings, apply the Excel Regression tool using the wedding cost as the dependent variable and attendance as the independent variable. Interpret all key regression results, hypothesis tests, and confidence intervals in the output. Analyze the residuals to determine if the assumptions underlying the regression analysis are valid. Use the standard residuals to determine if any possible outliers exist. If a couple is planning a wedding for 175 guests, how much should they budget?
Using the Excel Regression tool on the Weddings dataset with wedding cost as the dependent variable and attendance as the independent variable, the key regression results, hypothesis tests, and confidence intervals should be interpreted.
After applying the Excel Regression tool with wedding cost as the dependent variable and attendance as the independent variable, the key regression results should be examined. These results typically include coefficients, standard errors, t-values, p-values, and confidence intervals. The coefficient for the independent variable (attendance) represents the estimated change in the wedding cost for each unit increase in attendance. Hypothesis tests and confidence intervals help assess the statistical significance and precision of the estimated coefficients. The t-value indicates the strength of evidence against the null hypothesis of no relationship between the variables, and the p-value indicates the probability of observing the estimated coefficient under the null hypothesis. Lower p-values suggest stronger evidence against the null hypothesis. To validate the assumptions of the regression analysis, the residuals (the differences between the observed and predicted values) should be analyzed. Residual plots can be examined to check for patterns or deviations from assumptions, such as linearity, independence, and constant variance. Outliers can be identified by examining the standardized residuals. Values with high absolute values indicate potential outliers. To estimate the budget for a wedding with 175 guests, the regression model can be used to predict the corresponding cost. The attendance value of 175 can be plugged into the regression equation, and the predicted wedding cost can be obtained.
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a)Estimate a model relating annual salary to firm sales and market value. Make the
model of constant elasticity variety for both independent variables. Write the results
out in equation form (s.e. under parameter estimates). summary (lm(formula= salary ∼ sales + mktval, data = ceosal2)) Call: lm(formula = salary sales + mktval, data = ceosal2) Residuals: Coefficients: segnitr. coues: v Residual standard error: 535.9 on 174 degrees of freedom Multiple R-squared: 0.1777, Adjusted R-squared: 0.1682 F-statistic: 18.8 on 2 and 174 DF, p-value: 4.065e−08 > lm(formula = lsalary ∼ lsales + lmktval, data = ceosal2) Call: lm(formula = lsalary ∼ lsales + lmktval, data = ceosal2) Coefficients: (Intercept) 4.6209
Lsales 0.1621
Lmktval 0.1067
b) A friend of yours is about to start as a CEO at a firm. She is thinking of asking for
$500.000 as annual salaries. The firm sales last year was $5.000.000 and the market
value of the firm is $20 million. According to your model from part (a) would she be
asking too much? What are the expected salaries according to the model?
a) The estimated model relating annual salary to firm sales and market value, in equation form, is: Salary = 4.6209 + 0.1621 * log(sales) + 0.1067 * log(mktval), where log denotes the natural logarithm.
b) Calculating this expression will give us the expected salary according to the model. If the expected salary is higher than $500,000, then your friend would be asking too much.
a) The estimated model relating annual salary to firm sales and market value, in equation form, is:
Salary = 4.6209 + 0.1621 * log(sales) + 0.1067 * log(mktval)
where log denotes the natural logarithm.
b) To determine if your friend would be asking too much for an annual salary of $500,000, we need to plug the values of firm sales and market value into the model and calculate the expected salary.
Using the given values:
- Firm sales (sales) = $5,000,000
- Market value (mktval) = $20,000,000
We first need to take the logarithm of the sales and market value:
log(sales) = log(5,000,000)
log(mktval) = log(20,000,000)
Then, we can substitute these values into the equation:
Expected Salary = 4.6209 + 0.1621 * log(5,000,000) + 0.1067 * log(20,000,000)
Calculating this expression will give us the expected salary according to the model. If the expected salary is higher than $500,000, then your friend would be asking too much.
Note: Make sure to use the natural logarithm (ln) in the calculations.
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Which answer choice below lists only personal financial assets?
А
car, savings, mortgage loan
B
personal savings, college fund, 401K retirement fund
C
credit cards, home equity loan, retirement fund
D
boat, personal loans, savings
Answer:
c
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
Can you give me the answer to this
Answer: I think A
Step-by-step explanation:
How many different simple random samples of size 5 can be obtained from a population whose size is 35? The number of simple random samples which can be obtained is a (Type a whole number.)
The number of different simple random samples of size 5 that can be obtained from a population of size 35 is 324,632.
To calculate the number of different simple random samples of size 5 that can be obtained from a population of size 35, we can use the combination formula. The formula for combination is given by:
C(n, r) = n! / (r! * (n-r)!)
where n is the population size and r is the sample size.
In this case, we have n = 35 (population size) and r = 5 (sample size). Plugging in these values into the formula:
C(35, 5) = 35! / (5! * (35-5)!)
Simplifying this expression:
C(35, 5) = 35! / (5! * 30!)
Calculating the factorial values:
C(35, 5) = 35 * 34 * 33 * 32 * 31 / (5 * 4 * 3 * 2 * 1)
C(35, 5) = 324,632
Therefore, the number of different simple random samples of size 5 that can be obtained from a population of size 35 is 324,632.
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Let RP* be a complexity class defined as follows. A language L is in RP* if and only if there is a polynomial time Turing machine M and a polynomial p such that (i) if we L, then Prųe{0,1 }p(w) [M accepts (w,t)]21-(1/2)lwl and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] = 1. Thus M comes "exponentially close" to deciding L with certainty. Prove that RP* = RP.
Main answer: RP* = RP.
Supporting answer:
RP* is defined as the set of languages L for which there exists a polynomial-time Turing machine M that, on input w, halts with probability at least 1/2 if w is not in L, and halts with probability at least 1 - 2-l|w| if w is in L. This definition is almost identical to the definition of RP, except that the probability of error is reduced from 1/2 to 2-l|w|. However, RP* and RP are equivalent.
To see that RP* is a subset of RP, note that if a language L is in RP*, then there exists a polynomial-time Turing machine M and a polynomial p such that (i) if we L, then Prųe{0,1 }p(w) [M accepts (w,t)]21-(1/2)lwl and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] = 1. This means that M can be used as a randomized algorithm for L with error probability at most 1/2. Therefore, L is in RP.
To see that RP is a subset of RP*, note that if a language L is in RP, then there exists a polynomial-time Turing machine M and a constant c such that (i) if we L, then Prųe{0,1}c|w|[M accepts (w,t)] >= 1/2 and (ii) if w & L, then Prge{0,1}P(w) [M rejects (w,t)] < 1/2. By setting p(n) = c * 2n, we obtain a Turing machine M' that satisfies the conditions of RP*. Therefore, L is in RP*.
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It is commonly recognized that a poor environment (for example, poverty, low educational level of parents) and the presence of stressors (for example divorce of parents, abuse) are associated with lower IQs in children. Recent research indicates that a child's IQ is related to the number of such risk factors in the children's background, regardless of which specific factors are present. Below are the numbers of risk factors and the IQs of 12 children.
Child
Risk factors (x)
IQ (y)
A
3
112
B
6
82
C
0
105
D
5
102
E
1
115
F
1
101
G
5
94
H
4
89
I
3
98
J
3
109
K
4
91
L
2
107
A. Draw a scatterplot of the data. What does the plot tell you? Determine the regression equation. Be sure to include an interpretation of each parameter in the regression equation.
B. What IQ do you predict for James, who has 5 risk factors, versus Elizabeth, who has only 2?
A) The scatterplot of the data is as follows:What can we interpret from the scatterplot of the given data?We can observe that the higher the number of risk factors, the lower the IQ of the children. This fact is supported by the trend line that shows a negative correlation. This is what the plot is telling us.A regression equation is given by the formula: $y=\beta_0+\beta_1x$Where, $y$ is the IQ of the child, $x$ is the number of risk factors, $\beta_0$ is the y-intercept and $\beta_1$ is the slope of the line.Both $\beta_0$ and $\beta_1$ can be calculated using the following formulae: $$\beta_1=r_{xy}\frac{s_y}{s_x}$$$$\beta_0=\bar{y}-\beta_1\bar{x}$$Where, $r_{xy}$ is the correlation coefficient between $x$ and $y$, $s_x$ and $s_y$ are the standard deviations of $x$ and $y$ respectively, and $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively.From the scatterplot, we can observe that $r_{xy}$ is negative. Therefore, there exists a negative correlation between $x$ and $y$.Let us calculate the values of $\beta_1$, $\beta_0$ using the above formulae.$$r_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}}$$$$r_{xy}=\frac{(3)(112)+(6)(82)+(0)(105)+(5)(102)+(1)(115)+(1)(101)+(5)(94)+(4)(89)+(3)(98)+(3)(109)+(4)(91)+(2)(107)}{\sqrt{(3^2+6^2+0^2+5^2+1^2+1^2+5^2+4^2+3^2+3^2+4^2+2^2)(112^2+82^2+105^2+102^2+115^2+101^2+94^2+89^2+98^2+109^2+91^2+107^2)}}$$$$r_{xy}=-0.835$$Now, let's calculate the standard deviations of $x$ and $y$:$$s_x=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}$$$$s_y=\sqrt{\frac{\sum(y-\bar{y})^2}{n-1}}$$$$s_x=\sqrt{\frac{(3-2)^2+(6-2)^2+(0-2)^2+(5-2)^2+(1-2)^2+(1-2)^2+(5-2)^2+(4-2)^2+(3-2)^2+(3-2)^2+(4-2)^2+(2-2)^2}{12-1}}$$$$s_x=\sqrt{\frac{44}{11}}=2$$$$s_y=\sqrt{\frac{\sum(y-\bar{y})^2}{n-1}}$$$$s_y=\sqrt{\frac{(112-99.17)^2+(82-99.17)^2+(105-99.17)^2+(102-99.17)^2+(115-99.17)^2+(101-99.17)^2+(94-99.17)^2+(89-99.17)^2+(98-99.17)^2+(109-99.17)^2+(91-99.17)^2+(107-99.17)^2}{12-1}}$$$$s_y=\sqrt{\frac{5626.1}{11}}=9.025$$Finally, let's calculate $\beta_1$ and $\beta_0$:$$\beta_1=r_{xy}\frac{s_y}{s_x}$$$$\beta_1=-0.835\frac{9.025}{2}=-3.762$$$$\beta_0=\bar{y}-\beta_1\bar{x}$$$$\beta_0=99.17-(-3.762)(3.08)=112.91$$Therefore, the regression equation is $y=112.91-3.762x$.Here, $\beta_0=112.91$ represents the expected IQ of a child with zero risk factors, while $\beta_1=-3.762$ represents the decrease in IQ per risk factor.B) James has 5 risk factors. Substituting $x=5$ in the regression equation, we get:$$y=112.91-3.762x$$$$y=112.91-3.762(5)$$$$y=93.13$$Therefore, the IQ predicted for James is $93.13$.Similarly, Elizabeth has 2 risk factors. Substituting $x=2$ in the regression equation, we get:$$y=112.91-3.762x$$$$y=112.91-3.762(2)$$$$y=105.39$$Therefore, the IQ predicted for Elizabeth is $105.39$.
Please help and thanks
a right angle so 90 degrees
Evaluate for f(8), show all your work: f(x)=x2+10
Answer:
[tex]f(8)=74[/tex]
Step-by-step explanation:
[tex]f(8)=8^2+10=64+10=74[/tex]
Maya cuts sandwiches into halves. She has 15 sandwiches. How many halves does she make
Answer:
30
Step-by-step explanation:
15 x 2
I need help quick please
i’ll tell you on discord, add me there my user is ri#0511
Answer: 0.1562 I believe
ASAP PLEASE
Type in your response.
What is the area of kite ABCD?
sq yd
Answer:
51.5?
Step-by-step explanation:
I think it's just length times width divided by 2
The area of the kite ABCD is 51.56 square yards.
What is the area of the kite?A kite's area is the amount of space enclosed or surrounded by a kite in a two-dimensional plane.
A kite's area equals half the product of its diagonal lengths. The formula for calculating a kite's area is:
Area = 1/2 × d₁ × d₂
Where d₁ and d₂ are long and short diagonals of a kite.
Here, AC = 12.5 yards and BD = 8.25 yards
Substitute the values of d₁ = 12.5 yards and d₂ in the formula of the kite's area:
The area of the kite = 1/2 × 12.5 × 8.25
The area of the kite = 51.5625
The area of the kite ≈ 51.56 square yards
Thus, the area of the kite ABCD is 51.56 square yards.
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if an analyst wants to estimate the mean for an entire population (mu), the estimate would be more accurate if the analyst computed:
If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the sample mean.
If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the sample mean.
What is a population?In statistics, a population is a complete set of events that a statistician desires to investigate. A population is a collection of individuals, items, or data points that have a specific attribute of interest to the analyst.
What is an estimate? An estimate is an approximation of an unknown quantity that is dependent on imperfect or incomplete information. In statistics, an estimate is a projection of a population parameter dependent on data collected from a sample. An estimate is a numerical value generated from a statistical formula that is intended to provide an approximate value for an unknown population parameter.
What is an analyst?An analyst is an individual who examines a company or business's financial and business data to evaluate their health and determine their future development.
What does it mean to estimate the mean for an entire population?In statistics, estimating the mean for an entire population entails using a sample of data to calculate an approximate value of the mean for the entire population. The mean is the numerical value that provides information about the data set's central tendency. The analyst should choose a representative sample of the population to ensure the estimate is accurate.
What is the best way to estimate the mean of the entire population?
The estimate would be more accurate if the analyst computed the sample mean. The sample mean is an estimate of the population mean, denoted as μ. Sample mean is the average of the sampled data, and it is computed as follows;$$\overline{x}=\frac{\sum_{i=1}^{n}x_i}{n}$$
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If an analyst wants to estimate the mean for an entire population (μ), the estimate would be more accurate if the analyst computed the mean of a larger random sample.
A random sample is a method of selecting a subset from the entire population. It should be such that every member of the population has an equal chance of being chosen.
The sample should be sufficiently large and representative of the population as a whole to make reasonable inferences regarding the population. The mean value computed from a sample is used as an estimate of the true population mean. The sample mean is a random variable that can fluctuate from one sample to the next, depending on the sample that is taken. It has a standard deviation, called the standard error of the mean, that can be calculated using the following formula:standard error of the mean = standard deviation of the population / square root of the sample size.The larger the sample size, the smaller the standard error of the mean, and hence the more accurate the estimate of the population mean will be.
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A filing cabinet is 2 yards long by 2 feet wide by 48 inches high. What is the volume of the filing cabinet in cubic inches? Enter your answer in the box.
Answer:
82944 in^3
Step-by-step explanation:
Volume = length x breadth x height
Before volume can be calculated, feet and yards needs to be converted to inches
1 yard = 36 inches
2 x 36 = 72 inches
1 foot = 12 inches
2 x 12 = 24 inches
Volume = 72 x 48 x 24 = 82944 in^3
Consider the first three terms of the sequence below.
14000, 12600, 11340
Complete a recursively-defined function to describe this sequence.
f(1) = _____
f(n) = f(n - 1) · ___, for n ≥ 2
The next term in the sequence is ____.
Answer:
f(1) = 14000
f(n) = f(n - 1) · 0.9 , for n ≥ 2
The next term in the sequence is 10206 .
Step-by-step explanation:
First we try to find the type of sequence it is.
Is there a common difference?
12600 - 14000 = -1400
11340 - 12600 = -1260
There is no common difference, so it is not an arithmetic sequence.
12600/14000 = 0.9
11340/12600 = 0.9
There is a common ratio, so this is a geometric sequence.
Each term is 0.9 times the previous term.
The next term is:
11340 * 0.9 = 10206
Answer:
f(1) = 14000
f(n) = f(n - 1) · 0.9 , for n ≥ 2
The next term in the sequence is 10206 .
Question is in picture
Answer:
27.5
Step-by-step explanation:
pythagorean theorem. a^2+b^2=c^2.
so 12^2+b^2=30^2
meaning 144+b^2=900
900-144=756
square root 756
b=27.5
a pair of dice is rolled, and the number that appears uppermost on each die is observed. refer to this experiment and find the probability of the given event. (enter your answer as a fraction.)
If a pair of dice is rolled, and the number that appears uppermost on each die is observed, the probability of the sum of the numbers being either 7 or 11 is 2/9.
To find the probability of the sum of the numbers rolled on a pair of dice being either 7 or 11, we need to determine the number of favorable outcomes and the total number of possible outcomes.
There are six possible outcomes for each die, ranging from 1 to 6. Since we are rolling two dice, the total number of possible outcomes is 6 multiplied by 6, which is 36.
To calculate the number of favorable outcomes, we need to determine the combinations that result in a sum of either 7 or 11.
For the sum of 7, there are six possible combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
For the sum of 11, there are two possible combinations: (5, 6) and (6, 5).
Therefore, the number of favorable outcomes is 6 + 2 = 8.
The probability of the sum of the numbers being either 7 or 11 is given by the ratio of favorable outcomes to the total number of outcomes:
P(sum is 7 or 11) = favorable outcomes / total outcomes = 8/36 = 2/9.
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Complete question is:
A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. (Enter your answer as a fraction.)
The sum of the numbers is either 7 or 11.
For a standard normal distribution, find:
P(-2.46 < z < 2.82)
Given: For a standard normal distribution, we need to find the probability between P(-2.46 < z < 2.82). It is found that P(-2.46 < z < 2.82) = 0.9905.
Explanation: Given, P(-2.46 < z < 2.82)
The standard normal distribution has a mean of μ=0 and a standard deviation of σ=1. It is called the standard normal distribution, because it is the normal distribution where z-scores correspond to the number of standard deviations above or below the mean.
A z-score tells us how many standard deviations a value is from the mean.
A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.
To find the probability of P(-2.46 < z < 2.82), we need to find the area under the standard normal distribution curve between -2.46 and 2.82.
To find this probability, we can use a standard normal distribution table or a calculator that has a normal distribution function.
Using a standard normal distribution table, we can find the area to the left of z=2.82 and the area to the left of z=-2.46 and then subtract the two values to find the area between these z-scores.
The area to the left of z=2.82 is 0.9974, and the area to the left of z=-2.46 is 0.0069.
Therefore, the area between these z-scores is:
P(-2.46 < z < 2.82) = 0.9974 - 0.0069
= 0.9905
Therefore, P(-2.46 < z < 2.82) = 0.9905.
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What is the value of c^2-d^2 if c+d=7 and c-d=-2?
Answer:
- 14
Step-by-step explanation:
c² - d² ← is a difference of squares and factors as
= (c + d)(c - d) ← substitute values for factors
= 7 × - 2
= - 14
tony runs 3 miles in 25 minutes. at the same rate, how many miles would he run in 20 minutes?
Answer:
2.4
Step-by-step explanation:
3 / 28 = 0.12
(we do this to see how many miles he ran in a minute)
0.12 x 20 = 2.4
(then we multiply how many miles he can run in 20 minutes)
Use the GCF and distributive property to write an equivalent expression.
Answer:
2
6
9
Step-by-step explanation:
Because if you pay attention to our teacher he told us how to respond it!
Please help me with this question mark as A branlist please ASAP ASAP please ASAP
Answer:
x=18
y=27
explanation: 4 times 3 is 12
The null hypothesis is that 30% people are unemployed in Karachi city. In a sample of 100 people, 35 are unemployed. Test the hypothesis with the alternative hypothesis is not equal to 30%. What is the p-value?
After testing hypothesis with alternative-hypothesis is not equal to 30%, the p-value is 0.278.
To test the hypothesis that proportion of unemployed people in Karachi city is not equal to 30%, we perform a two-tailed test using binomial distribution.
The null-hypothesis (H₀) is that the proportion of unemployed people is 30% (p = 0.30), and the alternative-hypothesis (H₁) is that the proportion is not equal to 30% (p ≠ 0.30),
We have a sample of 100 people, and 35 of them are unemployed. we will calculate "test-statistic" "z-score" and use it to find p-value,
The "test-statistic" formula for "two-tailed" test is :
z = (p' - p₀)/√((p₀ × (1 - p₀))/n),
where p' = sample proportion, p₀ = hypothesized-proportion under the null-hypothesis, and n = sample-size,
In this case, p' = 35/100 = 0.35, p₀ = 0.30, and n = 100,
Calculating the test statistic:
z = (0.35 - 0.30)/√((0.30 × (1 - 0.30)) / 100)
= 0.05/√((0.30 * 0.70) / 100)
≈ 0.05/√(0.21 / 100)
≈ 0.05/√(0.0021)
≈ 0.05/0.0458258
≈ 1.0905
To find the p-value, we calculate probability of obtaining "test-statistic" as extreme as 1.0905 in a two-tailed test.
We know that the cumulative-probability (area) to left of 1.0905 is approximately 0.861.
Since this is a two-tailed test, we double this probability to get "p-value":
So, p-value = 2 × (1 - 0.861),
= 2 × 0.139,
= 0.278
Therefore, the p-value for hypothesis test is 0.278.
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Determine an expression for dy/d x = y' if [1+y]²-x+y=4 10.
The integration method you must use here is
Logarithmic q_23 = 1 Implicit q_23 = 2 Product rule q_23 = 3
The simplified expression for y' = 1/q_24y + q_25
The required expression is: y' = (1/2)(1-y)/(1+y)
Given [1 + y]² - x + y = 4 10. We need to determine an expression for dy/dx = y'.
Simplification of the given expression:
[1 + y]² + y - 4 10 = x
Differentiating w.r.t x by using the chain rule, we get:
(2[1 + y])*(dy/dx) + dy/dx + 1 = 0
(dy/dx)[2(1 + y) + 1] = - 1 - [1 + y]²
(dy/dx) = [- 1 - (1 + y)²]/[2(1 + y)]
The given expression is [1+y]²-x+y=4 10. We need to determine an expression for dy/d x = y'.
Differentiating the given equation with respect to x, we get:
2(1+y).dy/dx - 1 + dy/dx = 0
dy/dx(2+2y) = 1 - y(2+dy/dx)
dy/dx(2+2y) = (1-y)(2+dy/dx)
dy/dx = (1-y)/(2+2y)
dy/dx = (1/2)(1-y)/(1+y)
Hence, the required expression is: y' = (1/2)(1-y)/(1+y)
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Philip is recording what kind of shoes people are wearing at the mall. Out of the 12 people he has seen, 6 are wearing high heels. Considering this data, how many of the next 10 people Philip sees would you expect to be wearing high heels?
Answer: 5 people
Step-by-step explanation:
The easiest way to do this is to set up a proportion. 6/12=x/10. To solve, cross-multiply and divide. Multiply 6 and 10 (6*10=60) and divide by 12 (60/12=5).
You can also find that 6/12 is equal to 1/2 by dividing each side by 6. Then multiply 10 by 1/2 (or divide 10 by 2) to get the final answer of 5 people.
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Let a € (-1,1). Evaluate TT cos 20 de. 1-2a cos 0 + a2 d0
The value of the given integral is 2π [sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [sin(22e)/22 + sin(18e)/18] + C]
To evaluate the given expression:
∫∫ cos(20e) (1 - 2a cos(e) + a²) de dθ
We need to integrate with respect to both e and θ.
First, let's integrate with respect to e:
∫ cos(20e) (1 - 2a cos(e) + a²) de
Using the power reduction formula for cosine, we can rewrite the integrand:
= ∫ cos(20e) (1 - a² + a² cos²(e) - 2a cos(e)) de
= ∫ cos(20e) (1 - a² - a² sin²(e)) de
Now, we can integrate term by term:
= ∫ cos(20e) de - a² ∫ cos(20e) sin²(e) de - a² ∫ cos(20e) de
The first and third integrals are straightforward to evaluate:
= ∫ cos(20e) de - a² ∫ cos(20e) de - a² ∫ cos(20e) sin²(e) de
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) sin²(e) de
Now, let's evaluate the remaining integral with respect to e:
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) sin²(e) de
Using the trigonometric identity sin²(e) = (1 - cos(2e))/2, we can simplify the integrand:
= sin(20e) - a² sin(20e) - a² ∫ cos(20e) ((1 - cos(2e))/2) de
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) - cos(20e) cos(2e) de
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) de + (a²/2) ∫ cos(20e) cos(2e) de
Integrating the first term gives:
= sin(20e) - a² sin(20e) - (a²/2) ∫ cos(20e) de + (a²/2) ∫ cos(20e) cos(2e) de
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ cos(20e) cos(2e) de
Next, let's evaluate the remaining integral with respect to e:
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ cos(20e) cos(2e) de
Using the trigonometric identity cos(a) cos(b) = (1/2) [cos(a + b) + cos(a - b)], we can simplify the integrand:
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) ∫ (1/2) [cos(20e + 2e) + cos(20e - 2e)] de
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [∫ cos(22e) de + ∫ cos(18e) de]
= sin(20e) - a² sin(20e) - (a²/2) sin(20e) + (a²/2) (1/2) [sin(22e)/22 + sin(18e)/18] + C
where C is the constant of integration.
Now, we have the integral with respect to e evaluated. To find the final result, we need to integrate with respect to θ. However, we don't have any dependence on θ in the original expression. Therefore, we can simply multiply the result obtained above by 2π, as we're integrating over the entire range of θ.
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Complete question:
"The life time, in tens of hours, of a certain delicate electrical component is modeled by the random variable X with probability density function:
w= c(9-x) 0 <= x <=9
0 otherwise
(a) show that ca 2 (b) Find the mean life time of a component.
(a) The integral of w(x) over its entire domain is zero, it cannot be equal to 1. This means that the given function does not satisfy the normalization property and hence cannot be a probability density function.
To show that the given probability density function is a valid one, we need to verify that it satisfies the two properties of a probability density function:
1. Non-negativity:
For 0 <= x <= 9, c(9-x) is always non-negative, since c is a positive constant and (9-x) is also non-negative in this range. For any other value of x, w(x) is zero. Hence, w(x) is non-negative for all x.
2. Normalization:
[tex]a[0,9] w(x) dx = a[0,9] c(9-x) dx[/tex]
= [tex]c a[0,9] (9-x) dx[/tex]
= [tex]c [(9x - (x^2)/2)] [from 0 to 9][/tex]
= [tex]c [(81/2) - (81/2)][/tex]
= [tex]c (0)[/tex]
=[tex]0[/tex]
(b) The given probability density function does not have a valid normalization constant and hence does not represent a valid probability distribution.
To find the mean life time of a component, we need to calculate the expected value of X using the formula:
[tex]E(X) = a[a,b] x (w(x) dx)[/tex]
where a and b are the lower and upper bounds of the domain respectively.
In this case, we have:
a = 0 and b = 9
w(x) = c(9-x)
Hence,
[tex]E(X) = a[0,9] x*c(9-x) dx[/tex]
= [tex]c a[0,9] (9x - x^2) dx[/tex]
= [tex]c [(81 x^2/2) - (x^3/3)] [from 0 to 9][/tex]
=[tex]c [(6561/2) - (729/3)][/tex]
= [tex]c (2958/3)[/tex]
To find the value of c, we can use the normalization property:
[tex]a[0,9] w(x) dx = 1[/tex]
[tex]a[0,9] c(9-x) dx = 1[/tex]
[tex]c a[0,9] (9-x) dx = 1[/tex]
[tex]c [(81/2) - (81/2)] = 1[/tex]
[tex]c * 0 = 1[/tex]
This is not possible, since c cannot be infinite.
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