A sports manufacturer produces two products: footballs and baseballs. These products can be produced either during the morning shift or the evening shift. The cost of manufacturing the football and the baseball in the morning shift is $20 each, and the cost of manufacturing the football and the baseball in the evening shift is $25 each. The amounts of labor, leather, inner plastic lining, and demand requirements are given as follows: Resource Football Baseball Labor (hours/unit) 0.75 2 Leather (pounds/unit) 7 15 Inner plastic lining (pounds/unit) 0.5 2 Total demand (units) 1500 1200 Based on the information about the company, we know that the maximum labor hours available in the morning shift and evening shift are 5,000 hours and 2,000 hours, respectively, per month. The maximum amount of leather available for the morning shift is 15,000 pounds per month and 14,000 pounds per month for the evening shift. The maximum amount of inner plastic lining available for the morning shift is 2,000 pounds per month and 1,500 pounds per month for the evening shift.

Answer:

[tex] Range= Max-Min= 10.2-8.7=1.5 inches[/tex]

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 0.50 inches[/tex]

[tex] s^2 = 0.503^2 = 0.25 in^2[/tex]

D. The statistics are representative because they are taken from a random sample

Step-by-step explanation:

For this case we have the following data:

9.9 8.7 10.1 9.2 9.2 9.9 10.1 9.4 9.1 9.3 10.2

The data was colledted from a random sample of people selected in 1988.

We can order the dataset on increasing way and we got:

8.7  9.1  9.2  9.2  9.3  9.4  9.9  9.9 10.1 10.1  10.2

The range is defined as [tex] Range= Max-Min= 10.2-8.7=1.5 inches[/tex]

The mean is defined as:

[tex]\bar X =\frac{\sum_{i=1}^n X_i}{n}= 9.555 inches[/tex]

The standard deviation can be calculated with the following formula:

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 0.50 inches[/tex]

The sample variance would be just the deviation squared:

[tex] s^2 = 0.503^2 = 0.25 in^2[/tex]

And since the data comes from a random sample then is representative fo the population data in 1988. So then the best answer for this case would be:

D. The statistics are representative because they are taken from a random sample

Final answer:

The range, variance, and standard deviation must be calculated from the given data, but their representativeness for the current population is not guaranteed, especially due to changes since 1988 and potential sampling issues.

Explanation:

To calculate the range, variance, and standard deviation for the provided sample data of foot lengths, first, we must find the smallest and largest values to determine the range. In this set, the smallest value is 0.1 inches, and the largest is 10.2 inches, so the range is 10.2 - 0.1 = 10.1 inches.

To find the variance and standard deviation, we need the sum of the squared deviations from the mean, divided by the number of observations minus one for the sample variance, and then the square root of the variance for the standard deviation.

The representativeness of these statistics for the current population depends on various factors, including changes in population demographics and sampling methods. A single small sample, especially with an outlying value such as 0.1, may not be indicative of the entire population's foot sizes today.

Hence, the correct answer is A: 'Since the measurements were made in 1988, they may not necessarily be representative of the population today.'

Answer:

50°, 105°

Step-by-step explanation:

a + 130 = 180 ( sum of angles on a straight line)

a = 180 - 130 = 50°

a + b + 45 = 180 ( sum of angles in a Δ)

30 + b + 45 = 180

75 + b = 180

b = 180 - 75 = 105°

Answer:

A) 3/7

Step-by-step explanation:

We start by calculating the following probabilities:

P(produced by A) = 0.6

P(produced by A and defective) =  P(A ∩ def) = 0.6*0.01 = 0.006

P(produced by A and not defective) = P(A ∩ not def) = 0.6*0.99 = 0.594

P(produced by B and defective) = P(B ∩ def) = 0.4*0.02 = 0.008

P(produced by B and not defective) = P(B ∩ not def) = 0.4*0.98 = 0.392

The probability that it was produced by A given that it is defective is:

P(A|def) = P(A ∩ def) / P(def) = P(A ∩ def) / (P(A ∩ def)+P(B ∩ def)) = 0.006 / (0.006+0.008) = 6/14 = 3/7

Final answer:

The probability that a defective electronic component was produced by Plant A is 3/7. This was found using conditional probability and the information on production percentages and defect rates from both plants.

Explanation:

The question involves applying the concept of conditional probability to determine the probability that a defective electronic component was produced by Plant A. We need to calculate this using Bayes' theorem with the given probabilities for production and defect rates at both plants.

To answer the question: The probability of a component being defective from either Plant A or Plant B is calculated as follows:

P(Defective) = P(Defective | A)P(A) + P(Defective | B)P(B)
= (0.01)(0.60) + (0.02)(0.40)
= 0.006 + 0.008
= 0.014

Next, the probability that the component was produced at Plant A given that it is defective is:

P(A | Defective) = P(Defective | A)P(A) / P(Defective)
= (0.01)(0.60) / 0.014
= 0.006 / 0.014
= 3/7

Therefore, the correct answer is A. 3/7.

Answer:

(a) 0.023835

(b) 0.6281

Step-by-step explanation:

(a) The chance someone from this population will test positive is given by the percentage of people who have the virus multiplied by the change of testing positive (1 - false-negative rate) added to the percentage of people who do not have the virus multiplied by the change of testing positive (false-positive rate)

[tex]P(+) = 0.015*(1-0.002)+(1-0.015)*0.009\\P(+) = 0.023835[/tex]

(b) The probability that someone actually has the virus given that they have tested positive is determined as the probability of having the virus and testing positive divided by the probability of testing positive:

[tex]P(V|+) = \frac{ 0.015*(1-0.002)}{0.023835}\\P(V|+) = 0.6281[/tex]

Answer:

F is assigned the value of 15

[tex]74AF_{16} = 29871_{10}[/tex]

Step-by-step explanation:

Hexadecimal number system is base 16 and it contain the following numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

A has a value of 10

B has a value of 11

C has a value of 12

D has a value of 13

E has a value of 14

F has a value of 15

By completing the expanded notation:

[tex](7*4096) + (4*256) + (A * 16) + (F *1)\\= (7*4096) + (4*256) + (10 * 16) + (15 *1)\\= 28672 + 1024 + 160 + 15\\= 29871[/tex]

In hexadecimal notation, the letter 'F' corresponds to the decimal value 15. Therefore, when converting from hexadecimal to decimal, you would assign the value 15 to 'F'.

In hexadecimal notation, the letters A through F correspond to the decimal values 10 through 15, respectively. When converting from hexadecimal to decimal, you would replace the hexadecimal digit 'F' with its decimal equivalent. Therefore, in the hexadecimal system, the letter 'F' signifies the decimal number 15. To calculate the value of 74AF16 in decimal, you replace 'F' with 15 and compute the expression (7*4096) + (4*256) + (10*16) + (15*1).

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Answer:

The answer is a. 50%

Explanation:

Beverages (which include energy drinks, fruit drinks, sweetened coffee and tea, soft drinks, energy drinks, alcoholic beverages, etc.) are the major source of added sugars that are being consumed by the population of the United States: it accounts for almost half (around 50%) of added sugars consumed.

Beverages account for about 75% of the added sugars consumed in the U.S.

Beverages account for about 75% of the added sugars consumed in the U.S. This includes soda, fruit drinks, sports drinks, and energy drinks. These beverages are often high in sugar and can contribute to weight gain and other health issues.

It's important to be mindful of our consumption of sugary beverages and choose healthier alternatives like water, unsweetened tea, or low-sugar options.

Therefore, The correct answer is d. 75%.

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Nickel a: 4.874

Nickel b: 4.7

The difference will be:

Higher- lower

4.874 - 4.7

0.174

So the difference is 0.174g

The difference in mass between the two nickels is 0.174 grams.

Given that:

Weight of first nickel, n = 4.7 g

Weight of second nickel, N = 4.7 g

To find the difference in mass between the two nickels, subtract the weight of one nickel from the weight of the other nickel:

Difference in mass = Weight of the second nickel - Weight of the first nickel

Let's calculate the difference:

Weight of the second nickel = 4.874 g

Weight of the first nickel = 4.7 g

Difference in mass = 4.874 g - 4.7 g

Difference in mass = 0.174 g

Hence, the difference in mass between the two nickels is 0.174 grams.

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Answer:

f(t)= 11 in - 1.3 in/h *t

Step-by-step explanation:

defining the length of the candle as L , then since the candle burns at a constant rate , then

-dL/dt = 1.3 in/h = a

therefore

-∫dL = a∫dt

-L(t)=a*t + C , C=constant

at t=0 , the length of the candle is L₀= 11 in ,thus

-L₀=a*0 + C →  C= -L₀

replacing the value of C

-L(t)=a*t - L₀

L(t) = L₀ - a*t = 11 in - 1.3 in/h *t

then

f(t)= 11 in - 1.3 in/h *t

Answer:

240 cm²

Step-by-step explanation:

We are required to determine the surface area of the figure;

To get the area we add the are of all the surfaces;

Area of triangle;

Area = 0.5 × b × h

There are two triangles;

Therefore;

Area of the two triangles;

Area = 0.5 × 6 × 8 × 2

        = 48 cm²

Area of the rectangles;

Area of a rectangle = Length × width

Area of the first rectangle;

 = 6 cm × 8 cm

 = 48 cm²

Area of the second rectangle

  = 8 cm × 8 cm

  = 64 cm²

Area of the third rectangle

 = 10 cm × 8 cm

 = 80 cm²

The total surface area will be;

 Area = 48 cm² + 48 cm² + 64 cm² + 80 cm²

          = 240 cm²

Answer:

Elements of AxBb and BxA have been listed in the attached file

Step-by-step explanation:

The concept applied is that of binary operation and generally using the rule of combining more than one operations sign in either communitative or associative property as shown in the attachment.

Answer:

   [tex]\large\boxed{\large\boxed{slope=-2/5}}[/tex]

Explanation:

The problem is: given the points (19,−2) and (−11,10) find the slope of the line that joins them.

The slope of a line is the change in the y-coordinate over the change of the x-coordinate:

Thus:

         [tex]slope=[10-(-2)]/[-11-19]\\\\slope=12/(-30)\\\\slope=-12/30[/tex]

Simplify, dividing both numerator and denominator by 6:

            [tex]slope=-2/5\leftarrow answer[/tex]

Answer:

We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.

And if the distribution is not bell shaped or symmetric then we can't use it.

Step-by-step explanation:

Previous concepts

The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). "Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ)".

Solution to the problem

We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.

And if the distribution is not bell shaped or symmetric then we can't use it.

The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False. The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.

The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False.

The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.

It states that approximately 68% of the data is within one standard deviation of the mean, 95% is within two standard deviations, and more than 99% is within three standard deviations.

Therefore, the Empirical Rule is not applied when the data is not bell-shaped or symmetric.

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Answer:

a) 0.259

b) 0.297

c) 0.497

Step-by-step explanation:

To solve this problem, it is important to know the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

[tex]\mu = 11, \sigma = 3[/tex]

a. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 10.8 and 11.2 ​minutes?

Here we have that [tex]n = 25, s = \frac{3}{\sqrt{25}} = 0.6[/tex]

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11.2 - 11}{0.6}[/tex]

[tex]Z = 0.33[/tex]

[tex]Z = 0.33[/tex] has a pvalue of 0.6293.

X = 10.8

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.8 - 11}{0.6}[/tex]

[tex]Z = -0.33[/tex]

[tex]Z = -0.33[/tex] has a pvalue of 0.3707.

0.6293 - 0.3707 = 0.2586

0.259 probability, rounded to three decimal places.

b. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 10.5 and 11 ​minutes?

Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So

X = 11

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11 - 11}{0.6}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5.

X = 10.5

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.5 - 11}{0.6}[/tex]

[tex]Z = -0.83[/tex]

[tex]Z = -0.83[/tex] has a pvalue of 0.2033.

0.5 - 0.2033 = 0.2967

0.297, rounded to three decimal places.

c. If you select a random sample of 100 ​sessions, what is the probability that the sample mean is between 10.8 and 11.2 ​minutes?

Here we have that [tex]n = 100, s = \frac{3}{\sqrt{100}} = 0.3[/tex]

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11.2 - 11}{0.3}[/tex]

[tex]Z = 0.67[/tex]

[tex]Z = 0.67[/tex] has a pvalue of 0.7486.

X = 10.8

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.8 - 11}{0.3}[/tex]

[tex]Z = -0.67[/tex]

[tex]Z = -0.67[/tex] has a pvalue of 0.2514.

0.7486 - 0.2514 = 0.4972

0.497, rounded to three decimal places.

To find the probability that the sample mean is between eight minutes and 8.5 minutes, calculate the z-scores for both values and find the area under the standard normal distribution curve between these z-scores.

To find the probability that the sample mean is between eight minutes and 8.5 minutes, we need to calculate the z-scores for both values and then find the area under the standard normal distribution curve between these two z-scores.

The formula to calculate the z-score is: z = (x - mu) / (sigma / sqrt(n))

where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.

Using the given information, we can calculate the z-scores as follows:

z1 = (8 - 11) / (3 / sqrt(25))

z2 = (8.5 - 11) / (3 / sqrt(25))

Next, we use a standard normal distribution table or a calculator to find the area between these two z-scores, which represents the probability that the sample mean is between eight minutes and 8.5 minutes.

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Answer:

dilation

Step-by-step explanation:

When the pulmonary valve does not work properly, it can interfere with blood flow from the heart to the lungs, as well as force the heart to work harder to carry the blood that is needed to the rest of the body. Some children have heart conditions present at the time of birth and may require repair or replacement of the pulmonary valve, this option has a lower risk of infection, preserves the strength and functioning of the valve, and eliminates the need to take medication.

Final answer:

The root operation for treating pulmonary valve stenosis in newborns using balloon valvuloplasty is the valvuloplasty itself, which is a non-surgical procedure to widen the stenosed heart valve.

Explanation:

The root operation for treating newborn pulmonary valve stenosis with percutaneous balloon pulmonary valvuloplasty is the procedure known as valvuloplasty.

This procedure involves the insertion of a specialized catheter with a balloon at its tip into a blood vessel, usually via the leg, and navigating it to the valve.

The balloon is then inflated to widen the stenosed valve and allow better blood flow. Subsequently, the balloon is deflated and removed, completing the valvuloplasty.

The percentage of cans expected to be rejected based on given mean and standard deviation are calculated using the Z score and standard normal distribution table. By adjusting the mean and standard deviation, the rejection rates will change accordingly.

This question is about calculating the expected rejection rate of cans based on different conditions using statistical concepts like mean and standard deviation.

(a) The Z score for 7.9 is (7.9 - 8.05) / 0.05 = -3. We use the standard normal distribution table to find the probability of a can having weight less than 7.9 ounces. That's almost 0.1% (0.001), so about 0.1% of cans are expected to be rejected.

(b) After adjusting the average weight to 8.1 oz, the Z score for 7.9 becomes (7.9 - 8.1) / 0.05 = -4. Again, find the probability in the standard normal distribution table, it is almost 0, so the rejection rate will drastically decrease.

(c) When the standard deviation is reduced to 0.03 but mean remains 8.05, the Z score becomes (7.9 - 8.05) / 0.03 = -5. The rejection rate will be extremely close to 0 as per standard normal distribution table reference.

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Answer:

D. 0.821

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The combinations formula is important to solve this question:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

Desired outcomes

The order is not important. For example, Elisa, Laura and Roze is the same outcome as Roze, Elisa and Laura. This is why we use the combinations formula.

At least 3 girls.

3 girls

3 girls from a set of 5 and 2 boys from a set of 3. So

[tex]C_{5,3}*C_{3,2} = 30[/tex]

4 girls

4 girls from a set of 5 and 1 boy from a set of 3. So

[tex]C_{5,4}*C_{3,1} = 15[/tex]

5 girls

5 girls from a set of 5

[tex]C_{5,5} = 1[/tex]

[tex]D = 30+15+1 = 46[/tex]

Total outcomes

5 from a set of 8. So

[tex]T = C_{8,5} = 56[/tex]

Probability

[tex]P = \frac{D}{T} = \frac{46}{56} = 0.821[/tex]

So the correct answer is:

D. 0.821

Answer:

I got answer choice D

Hope this helps :)

Step-by-step explanation:

Answer:

[tex]P(X=5)=(30C5)(0.2)^5 (1-0.2)^{30-5}=0.172[/tex]

Step-by-step explanation:

Assuming this complete question :"Assume that a procedure yields a binomial distribution with a trial repeated n times. Using the binomial probability formula, what is the probability of x successes given the probability p of success on a single trial? Round your answer to three decimal places.

n=30, x= 5, p=1/5"

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we know that:

[tex]X \sim Binom(n=30, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

And for this case if we find the probability for x=5 we got:

[tex]P(X=5)=(30C5)(0.2)^5 (1-0.2)^{30-5}=0.172[/tex]

Answer:

1) 2x+7

2) -3x+11

3) 0.75x-2

4) -2x+0

5) -1.5x+2

6) -4x+16

Step-by-step explanation:

1) y = mx + c

m = 2 when x=1 , y=9

9 = 2(1)+c

c = 7

y = 2x + 7

2) m = -3

When x=4, y= -1

-1 = -3(4) + c

c = -1+12 = 11

y = -3x + 11

3) m = 0.75

When x= -4, y= -5

-5 = 0.75(-4) + c

-5 = -3 + c

c = -2

y = 0.75x - 2

4) m = (y2-y1)/(x2-x1)

m = (2-(-6))/(-1-3) = 8/-4 = -2

y = -2x + c

When x= -1, y= 2

2 = -2(-1) + c

2 = 2 + c

c = 0

y = -2x + 0

5) m = (-10-(-4))/(8-4)

m = (-10+4)/4 = -6/4 = -1.5

y = -1.5x + c

When x= 4, y= -4

-4 = -1.5(4) + c

-4 = -6 + c

c = 2

y = -1.5x + 2

6) m = (-4-4)/(5-3) = -8/2 = -4

When x= 3, y= 4

4 = -4(3) + c

4 = -12 + c

c = 16

y = -4x + 16

Answer:

1) 2x+7

2) -3x+11

3) 0.75x-2

4) -2x+0

5) -1.5x+2

6) -4x+16

Step-by-step explanation:

Answer:

68.26% of the data would be between 16.4 and 21.6.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 19, \sigma = 2.6[/tex]

What percentage of the data would be between 16.4 and 21.6?

This is the pvalue of Z when X = 21.6 subtracted by the pvalue of Z when X = 16.4. So

X = 21.6

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{21.6 - 19}{2.6}[/tex]

[tex]Z = 1[/tex]

[tex]Z = 1[/tex] has a pvalue of 0.8413.

X = 16.4

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{16.4 - 19}{2.6}[/tex]

[tex]Z = -1[/tex]

[tex]Z = -1[/tex] has a pvalue of 0.1587

So 0.8413 - 0.1587 = 0.6826 = 68.26% of the data would be between 16.4 and 21.6.

Answer: Percentage = 0.6826 X 100 = 68.26%

Step-by-step explanation: Please find the attached document for the step by step explanation

Step-by-step explanation:

Let's say S is the number of small boxes and L is the number of large boxes.

S + L = 26

7S + 13L = 254

Solve the system of equations using substitution.

7S + 13(26 − S) = 254

7S + 338 − 13S = 254

84 − 6S = 0

S = 14

L = 26 − S

L = 12

The company shipped 14 small boxes and 12 large boxes.

Answer:14 small boxes and 12 large boxes were shipped.

Step-by-step explanation:

Let x represent the number of small boxes of paper that were shipped.

Let y represent the number of large boxes of paper that were shipped.

A total of 26 boxes of paper were shipped. This means that

x + y = 26

The volume of each small box is 7 cubic feet and the volume of each large box is 13 cubic feet. The total number of boxes shipped have a combined volume of 254 cubic feet. This means that

7x + 13y = 254 - - - - - - - - - - - - 1

Substituting x = 26 - y into equation 1, it becomes

7(26 - y) + 13y = 254

182 - 7y + 13y = 254

- 7y + 13y = 254 - 182

6y = 72

y = 72/6 = 12

x = 26 - y = 26 - 12

x = 14

Answer:

This statistic is misleading because Howard surveys only his department, and not membes of all the departments that the company has.

Step-by-step explanation:

This is a common statistics practice, when we want to study something from a population, we find a sample of this population.

However, the sample has to be representative

For example:

I want to estimate the proportion of New York state residents who are Buffalo Bills fans. So i ask, lets say, 1000 randomly selected Buffalo residents wheter they are Buffalo Bills fans, and expand this to the entire population of New York State residents. This is not representative of all New York State residents, just Buffalo residents.

In this problem, we have that:

Howard wants to know the proportion of employees of a company who use the company's healthcare. He asks only his department. However, a company as multiple departments, which leads to the statistics found in Howard's survey being misleading.

Final answer:

Howard's statistic that 48% of people in his company use the healthcare may be misleading due to potential issues like non-representative sampling, biased survey questions, response bias, and lack of current context.

Explanation:

The question about Howard reporting that 48% of the people in his company use the company's healthcare is misleading may have several underlying reasons. Firstly, the sample size and how the sample was obtained are critical factors that determine the reliability and representation of the poll. A random sample of 600 people is generally a good size, but if the sample is not representative of the entire company's demographics, the statistic could be misleading.

Another potential issue could be the phrasing of the survey question. Questions that are biased or leading can influence the way people respond and thus skew the results. In addition, respondents might have reasons to provide socially desirable answers rather than true ones, or they might misremember their actual usage of healthcare. This is an example of response bias, which can lead to inaccurate data.

Finally, it's vital to consider the context, such as whether the data is recent and if there have been any significant changes in the company's healthcare policy or overall employee demographics since the survey. All these factors can contribute to why a statistic might be considered misleading, as it may not accurately reflect the true situation.

The probability that sales will be less than 4,300 oranges is 21.19%

The z score is used to determine by how many standard deviations the raw score is above or below the mean. The z score is given by:

[tex]z=\frac{x-\mu}{\sigma} \\\\where\ x=raw\ score, \mu=mean, \sigma=standard\ deviation\\\\Given\ that:\\\mu=4700,\sigma=500\\\\For\ x=4300:\\\\z=\frac{4300-4700}{500} =-0.8[/tex]

From the normal distribution table:

P(x < 4300) = P(z < -0.8) = 0.2119 = 21.19%

The probability that sales will be less than 4,300 oranges is 21.19%

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The probability that sales will be less than 4,300 oranges is approximately 21.23%.

To find the probability that sales will be less than 4,300 oranges, we need to standardize the value using the z-score formula.

The z-score formula is given by z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 4,300, μ = 4,700, and σ = 500.

Substituting these values into the formula, we get z = (4,300 - 4,700) / 500 = -0.8. We can then use a z-table or a calculator to find the probability associated with a z-score of -0.8, which is approximately 0.2123 or 21.23%.

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Answer:

[tex]V=226.08\ ft^3[/tex]

Step-by-step explanation:

we know that

The volume of a cylinder is equal to

[tex]V=\pi r^{2} h[/tex]

where

r is the radius of the base of the cylinder

h is the height of the cylinder

we have

[tex]r=6\ ft\\h=2\ ft\\\pi=3.14[/tex]

substitute the given values in the formula

[tex]V=(3.14)(6)^{2}(2)\\ V=226.08\ ft^3[/tex]

Answer: the system of linear equations to represent this scenario are

8x + 4y = 40

3x + 10y = 32

Step-by-step explanation:

Let x represent the cost of one hamburger.

Let y represent the cost of one hotdog.

You spend $40 for 8 hamburgers and 4 hotdogs at a ballgame. This means that

8x + 4y = 40 - - - - - - - - - - - - 1

The next game you spend $32 for 3 hamburgers and 10 hotdogs. This means that

3x + 10y = 32- - - - - - - - - - - - 2

Multiplying equation 1 by 3 and equation 2 by 8, it becomes

24x + 12y = 120

24x + 80y = 256

- 68y = - 136

y = - 136 /- 68

y = 2

Substituting y = 2 into equation 2, it becomes

3x + 10y = 32

3x + 10 × 2 = 32

3x = 32 - 20 = 12

x = 12/3 = 4

Answer:

a) [tex]E(A)=\frac{1+6}{2}=3.5 ppm[/tex]

b) [tex] P(2.875 <X < 3.5) = F(3.5) -F(2.875) = \frac{3.5-1}{5}- \frac{2.875-1}{5}= \frac{1}{8}= 0.125[/tex]

c) [tex] P(X<4.125) = F(4.125) = \frac{4.125-1}{5}= 0.625[/tex]

Step-by-step explanation:

If we work with the limits defined from 5 to 10 then part b and c from this question not makes sense. If we work with the limits 1 and 6 all the parts for the question makes sense because if we work with 5 and 10 the only thing that we can find is the expected value [tex] E(A) = \frac{5+10}{2}= 7.5[/tex]

Assuming the following correct question : "A publication released the results of a study of the evolution of a certain mineral in the​ Earth's crust. Researchers estimate that the trace amount of this mineral x in reservoirs follows a uniform distribution ranging between 1 and 6 parts per million"

Solution to the problem

Let A the random variable that represent " amount of the mineral x ". And we know that the distribution of A is given by:

[tex]A\sim Uniform(1 ,6)[/tex]

Part a

For this uniform distribution the expected value is given by [tex]E(X) =\frac{a+b}{2}[/tex] where X is the random variable, and a,b represent the limits for the distribution. If we apply this for our case we got:

[tex]E(A)=\frac{1+6}{2}=3.5 ppm[/tex]

Part b

For this case we can use the cumulative distribution function for the uniform distribution given by:

[tex] F(X=x)= \frac{x-a}{b-a} = \frac{x}{6-1} =\frac{x-1}{5} , 1 \leq X \leq 6[/tex]

And we want this probability:[tex] P(2.875 <X < 3.5) = F(3.5) -F(2.875) = \frac{3.5-1}{5}- \frac{2.875-1}{5}= \frac{1}{8}= 0.125[/tex]Part c

For this case we want this probability:

[tex] P(X<4.125) = F(4.125) = \frac{4.125-1}{5}= 0.625[/tex]

Answer: it is verified that:

* y1 and y2 are solutions to the differential equation,

* c1 + c2t^(1/2) is not a solution.

Step-by-step explanation:

Given the differential equation

yy'' + (y')² = 0

To verify that y1 solutions to the DE, differentiate y1 twice and substitute the values of y1'' for y'', y1' for y', and y1 for y into the DE. If it is equal to 0, then it is a solution. Do this for y2 as well.

Now,

y1 = 1

y1' = 0

y'' = 0

So,

y1y1'' + (y1')² = (1)(0) + (0)² = 0

Hence, y1 is a solution.

y2 = t^(1/2)

y2' = (1/2)t^(-1/2)

y2'' = (-1/4)t^(-3/2)

So,

y2y2'' + (y2')² = t^(1/2)×(-1/4)t^(-3/2) + [(1/2)t^(-1/2)]² = (-1/4)t^(-1) + (1/4)t^(-1) = 0

Hence, y2 is a solution.

Now, for some nonzero constants, c1 and c2, suppose c1 + c2t^(1/2) is a solution, then y = c1 + c2t^(1/2) satisfies the differential equation.

Let us differentiate this twice, and verify if it satisfies the differential equation.

y = c1 + c2t^(1/2)

y' = (1/2)c2t^(-1/2)

y'' = (-1/4)c2t(-3/2)

yy'' + (y')² = [c1 + c2t^(1/2)][(-1/4)c2t(-3/2)] + [(1/2)c2t^(-1/2)]²

= (-1/4)c1c2t(-3/2) + (-1/4)(c2)²t(-3/2) + (1/4)(c2)²t^(-1)

= (-1/4)c1c2t(-3/2)

≠ 0

This clearly doesn't satisfy the differential equation, hence, it is not a solution.

Final answer:

The provided solutions y₁(t) = 1 and y₂(t) = [tex]t^{(1/2)[/tex] satisfy the given differential equation, verified through substitution and simplification. However, a linear combination of the form [tex]c_1 + c_2t^{(1/2)[/tex] is not a solution as it does not satisfy the original equation when its derivatives are substituted.

Explanation:

We are given the second-order differential equation yy'' + (y')² = 0, where y = y(t) is a function of t, and we are asked to verify solutions and understand properties of certain types of solutions to this equation.

To verify that y₁(t) = 1 is a solution, we calculate the derivatives: y₁' = 0 and y₁'' = 0. Substituting these into the differential equation yields (1)(0)+(0)² = 0, which holds true, confirming that y₁(t) = 1 is indeed a solution.

Next, to verify y₂(t) = [tex]t^{(1/2)[/tex], we find y₂' = [tex](1/2)t^{(-1/2)[/tex] and [tex]y_2'' = -(1/4)t^{(-3/2)[/tex]. Substituting these values gives [tex](t^{(1/2)})(-(1/4)t^{(-3/2)}) + ((1/2)t^{(-1/2)})^2 = 0[/tex], which simplifies to 0, showing that y₂(t) is also a solution.

For the linear combination of solutions, we consider [tex]y(t) = c_1 + c_2t^{(1/2)[/tex]. Derivatives are [tex]y' = c_2(1/2)t^{(-1/2)[/tex] and [tex]y'' = -c_2(1/4)t^{(-3/2)[/tex]. Substituting into the given differential equation does not yield zero, thus [tex]c_1 + c_2t^{(1/2)[/tex] is not a solution.

Answer:

Susan should pick 6 cherries from box, so the probability of picking the 2 rotten cherries is 1/20

Step-by-step explanation:

assuming that each cherry is equally probable to be chosen , since each cherry is independent from the others and sampling is done without replacement , the random variable X= number of cherries that are rotten from the picked ones follows a hyper geometrical distribution , where

P(X=k)= C(M,k) * C(N-M, n-k) / C(N,n)

where

N= population size = 25

n= number of picks

M = total number of rotten cherries =2

k = number of rotten cherries picked =2

C( ) = combination

then

1/20=C(2,2)*C(25-2,n-2)/C(25,n) = 1 * (23!/(n-2)!*(25-n)! / (25!/(n!*(25-n)!

1/20 = n!/(n-2)!  * 1/(24*25)

24*25/20 = n*(n-1)

n²-n-30 =0

n= (1 +√(1+4*1*30))/2 = 12/2= 6

n=6

then Susan should pick 6 cherries from box, so the probability of picking the 2 rotten cherries is 1/20

To find the number of cherries Susan should pick, use the combination formula and probability calculation. After setting up an equation with probability equal to 1/20, solve for 'x' using trial and error methods.

To answer the question, we need to use the combination formula. This formula in the field of statistics is used to find the number of possible combinations that can be obtained by taking 'r' elements from a set of 'n' elements.

The formula is: C(n, r) = n! / [(n - r)! * r!]

Given 25 cherries, 2 of which are rotten, Susan wants to choose some cherries such that the probability of getting exactly 2 rotten cherries is 1/20. Let's assume she needs to pick 'x' cherries.

Now we can write the probability equation: Probability = [C(2, 2) * C(23, x - 2)] / C(25, x) = 1/20

Unfortunately, we can't explicitly solve this equation because it would require checking different values of 'x'. However, it can be solved manually or through trial and error using software or a calculator. After checking different values, you can find the 'x' that satisfies the equation.

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Complete Question

The complete question is shown on the first uploaded image

Answer:

a) 0.88

b) 0.02

c) 0.01

d) 0.99

Step-by-step explanation:

Step one: State the given parameters

            [tex]P(D_{1} ) = 0.12[/tex]                                   [tex]P(D_{2} ) = 0.07[/tex]

           [tex]P(D_{3} ) = 0.05[/tex]                                    [tex]P (D_{1} U D_{2} ) = 0.13[/tex]

          [tex]P(D_{1}n D_{2}n D_{3}) = 0.01[/tex]                        [tex]P(D_{1} U D_{3}) = 0.14[/tex]

Step 2 : Obtain the probability that a unit does not have a type 1 defect

         [tex]P(\frac{}{D_{1} })[/tex] =  [tex]1 -P(D_{1} )[/tex]

                    = [tex]1 - 0.12[/tex]

                    = 0.88  

Step 3 : Obtain the probability that a unit has both type 2 and 3 defect?

          The probability of the unit having both type 2 and type 3 defect is denoted as [tex]P(D_{2} n D_{3} )[/tex]

   This is calculated as

                    [tex]P(D_{2}n D_{3}) =P(D_{2} ) + P(D_{3}) - P(D_{2} U D_{3})\\\\ = 0.07 + 0,05 - 0.13[/tex]

                    =   0.02

Therefore P(D_{2} n D_{3} ) = 0.02

Step 4 : Obtain the probability that the unit has both a type 2 and type 3 ,but not a type 1 defect

                  Let [tex]P(\frac{}{D_{1}} n D_{2} n D_{3} )[/tex] denote the  probability that the unit has both a type 2 and type 3 ,but not a type 1 defect.

This can be calculated as follows :

                      [tex]P(\frac{}{D_{1}} n D_{2} n D_{3} ) = P(D_{2} n D_{3}) - P(D_{1} n D_{2}nD_{3})[/tex]

                                               =   0.02 - 0.01

                                               =  0.01

Step 4 : Obtain the probability that a unit has at most two defects

               P(at most 2 defects)  = 1 - P(all three defects)

                                                  = [tex]1- P(D_{1} n D_{2}nD_{3})[/tex]

                                                  =  1 - 0.01

                                                  = 0.99

Since we're given the femour's length, we'll have to use the first function.

If we substitute [tex]f=49[/tex] in the expression we have

[tex]g(f)=2.56f+47.24 \implies g(49)=2.56\cdot 49+47.24=125.44+47.24=172.68[/tex]

The height of the woman with a femur length of 49 cm is 172.68 cm.

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

G(f) = 2.56f + 47.24

f = 49 cm

G(49) = 2.56 x 49 + 47.24

G(49) = 125.44 + 47.24

G(49) = 172.68  cm

Thus,

The height of the woman is 172.68 cm.

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Answer: k = 200/3

Step-by-step explanation:

If a variable, a varies directly with a variable, c, it means that as a increases, c increases and as a decreases, c decreases.

Also, If a variable, a varies inversely with a variable, b, it means that as a increases, b decreases and as a decreases, c increases.

The variable c varies directly with a and inversely with b. We would introduce a constant of variation, k. Therefore

a = kc/b

If c = 3/20 when a = 2 and b =5, then

2 = (k × 3/20)/5 = 3k/100

Cross multiplying, it becomes

3k = 100 × 2 = 200

k = 200/3

Answer: k=3/8 c=3/40

Step-by-step explanation:

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