Show that the normal line at any point of the circle x^2+y^2 = a^2 passes through the origin.

We can say with 90% confidence that the increase in proportion of people satisfied with their financial situation is between 1.05% and 6.71%.

To calculate the confidence interval for the increase in proportion of people satisfied with their financial situation, we need to first calculate the proportions for both years:

Proportion in year 1 = 478/2038 = 0.2342
Proportion in year 2 = 537/1967 = 0.2730

The increase in proportion is:
0.2730 - 0.2342 = 0.0388

To calculate the confidence interval, we can use the formula:
CI = (point estimate ± (critical value x standard error))

The point estimate is the increase in proportion we just calculated: 0.0388

The critical value can be found using a z-table for a 90% confidence level. The z-value for a 90% confidence level is 1.645.

The standard error can be calculated using the formula:
sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)]

where p1 and n1 are the proportion and sample size for year 1, and p2 and n2 are the proportion and sample size for year 2.

Plugging in the values, we get:
SE = sqrt[(0.2342(1-0.2342)/2038) + (0.2730(1-0.2730)/1967)] = 0.0174

Now we can plug in all the values to get the confidence interval:
CI = (0.0388 ± (1.645 x 0.0174)) = (0.0105, 0.0671)

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If a scatterplot shows a non-linear relationship between the response and explanatory variable, several options can be considered depending on the purpose of the analysis and the nature of the data.

Here are some possible actions:

Transform the data: One common approach is to transform the data to make the relationship linear. For example, if the relationship appears to be exponential, taking the logarithm of the response variable might make it linear. Similarly, taking the square root, cube root, or inverse of one or both variables might also help.

Use a non-linear model: Another option is to fit a non-linear model that can capture the curvature in the relationship. There are many types of non-linear models, such as quadratic, cubic, exponential, logistic, or spline models. The choice of model depends on the shape of the curve and the underlying theory or hypothesis.

Resample or subset the data: If the non-linear relationship is driven by outliers, influential points, or a subset of the data, it might be helpful to resample or subset the data to remove them. For example, trimming the extreme values, bootstrapping the data, or stratifying the data by a third variable might help.

Explore alternative variables or interactions: If the non-linear relationship is due to an unobserved or omitted variable, it might be useful to explore alternative variables or interactions that could explain the pattern. For example, if the response is sales and the explanatory variable is price, adding a competitor's price or a marketing variable might improve the fit.

Use caution in interpretation: Finally, if the non-linear relationship persists after exploring the above options, it might be necessary to acknowledge the non-linearity and use caution in interpreting the results. Non-linear relationships can be more difficult to interpret and extrapolate, and the statistical inference might be more uncertain or sensitive to assumptions.

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The Shavers Fork watershed is impacted by acid precipitation or other forms of pollution that are affecting the pH levels of the tributary water basin in the springtime.The data of Exercise 17.8 do give good reason to think that the springtime water in the tributary water basin around the Shavers Fork watershed is not neutral. Here's why:

Step 1: State the null hypothesis and the alternative hypothesis.

Null hypothesis: The springtime water in the tributary water basin around the Shavers Fork watershed is neutral (pH 7.0).

Alternative hypothesis: The springtime water in the tributary water basin around the Shavers Fork watershed is not neutral (pH ≠ 7.0).

Step 2: Determine the appropriate test statistic and the level of significance.

In this case, we can use a t-test for a single sample since we are comparing the pH of the springtime water in the tributary water basin to a neutral pH of 7.0. The level of significance is not given in Exercise 17.8, so we will assume it to be 0.05.

Step 3: Calculate the test statistic and the p-value.

Using the data from Exercise 17.8, we find that the sample mean pH is 6.45 and the sample standard deviation is 0.23. The test statistic is calculated as:

t = (6.45 - 7.0) / (0.23 / sqrt(9)) = -9.78

Using a t-table with 8 degrees of freedom (n-1), we find that the p-value is less than 0.001.

Step 4: Make a decision and interpret the results.

Since the p-value is less than the level of significance of 0.05, we reject the null hypothesis and conclude that the springtime water in the tributary water basin around the Shavers Fork watershed is not neutral. The data suggest that the pH of the water is significantly lower than a neutral pH of 7.0, indicating that the water is acidic. Therefore, we can infer that the Shavers Fork watershed is impacted by acid precipitation or other forms of pollution that are affecting the pH levels of the tributary water basin in the springtime.

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The order approximately 1.1111 cubic yards of topsoil from the dirt store.

The volume of topsoil required in cubic feet.

The area of the garden is:

The area of the garden is found by multiplying the length and width of the garden. In this case, the garden is 15 feet by 12 feet, so the area is 15 ft x 12 ft = 180 sq ft.

Since we want a 2 inch thick layer of topsoil, we need to convert the thickness to feet:

2 inches = 2/12 feet = 0.1667 feet

The volume of topsoil required in cubic feet is therefore:

180 sq ft × 0.1667 ft = 30 cubic feet

To convert this to cubic yards, we divide by 27 (since there are 27 cubic feet in a cubic yard):

The dirt store sells topsoil by the cubic yard, so we need to convert our answer from cubic feet to cubic yards. Since there are 27 cubic feet in a cubic yard (3 feet x 3 feet x 3 feet)

30 cubic feet ÷ 27 = 1.1111 cubic yards

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The model's rigging length of 326 feet is likely only a fraction of the total rigging on the real Lady Washington. Nonetheless, it is still an astonishing amount of string to work with when creating a model ship!

Assuming that the scale model is an accurate representation of the Lady Washington, we can use the model's rigging length to estimate the total rigging of the actual ship. The model has 326 feet of rigging, and if we know the scale of the model, we can determine the length of the real ship's rigging.
Unfortunately, without knowing the scale of the model or the dimensions of the actual ship, it is impossible to give an exact answer. However, we can make some educated guesses based on typical rigging lengths for ships of a similar size and type.
The Lady Washington is a replica of an 18th-century trading vessel, and based on historical records, we can estimate that a ship of this type and size would have had around 1,000 feet of rigging. This includes the standing rigging (which supports the mast and stays in place all the time), as well as the running rigging (which is used to adjust the sails).

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[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ S = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ n=4 \end{cases}\implies S=180(4-2)\implies S=360[/tex]

so since a quadrilateral will have a total of 360°, minus 240 and 20 that leaves us with only 100° leftover, now to make it in a 3 : 7 ratio, let's simply divide 100 by (3 + 7) and distribute accordingly.

[tex]3~~ : ~~7\implies 3\cdot \frac{100}{3+7}~~ : ~~7\cdot \frac{100}{3+7}\implies 3\cdot 10~~ : ~~7\cdot 10\implies 30^o~~ : ~~70^o[/tex]

The probability of not getting an even number when spinning the spinner once is [tex]\frac{5}{11}[/tex].

You want to know the probability of not getting an even number when spinning a spinner divided into 11 equal sections numbered from 0 to 10.

Step 1: Identify the even numbers in the given range (0 to 10). The even numbers are 0, 2, 4, 6, 8, and 10.

Step 2: Count the number of even numbers. There are 6 even numbers in the given range.

Step 3: Calculate the total number of possible outcomes when spinning the spinner. There are 11 possible outcomes (0 to 10).

Step 4: To find the probability of not getting an even number (P(not even)), subtract the number of even numbers from the total number of outcomes. This will give you the number of odd numbers: 11 - 6 = 5.

Step 5: Now, divide the number of odd numbers by the total number of outcomes to find the probability: P(not even) = 5/11.

So, the probability of not getting an even number when spinning the spinner once is [tex]\frac{5}{11}[/tex].

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Thus, there are approximately 408 students who are taking an online course and prefer basketball to football.

To solve this problem, we need to use the formula for the intersection of two independent events:

P(A and B) = P(A) * P(B)

where P(A) is the probability of event A occurring, P(B) is the probability of event B occurring, and P(A and B) is the probability of both events A and B occurring simultaneously.

In this case, let A be the event of taking an online class, and let B be the event of preferring basketball to football. We are asked to find the number of students who are in the intersection of these two events, or P(A and B).

We are given that there are 1200 students taking an online class, out of a total of 5000 students. Therefore, the probability of taking an online class is:

P(A) = 1200/5000 = 0.24

We are also given that 1700 students prefer basketball to football. Since this event is independent of taking an online class, the probability of preferring basketball to football is simply:

P(B) = 1700/5000 = 0.34

Now we can use the formula to find the probability of both events occurring simultaneously:

P(A and B) = P(A) * P(B) = 0.24 * 0.34 = 0.0816

Finally, we can convert this probability to a number of students by multiplying by the total number of students:

0.0816 * 5000 = 408

Therefore, there are approximately 408 students who are taking an online course and prefer basketball to football.

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Hence, the margin of error for this 95% confidence interval of a population proportion is 5.4%.

The margin of error for a confidence interval is the distance between the sample statistic (in this case, the sample proportion) and the confidence interval limits. To find the margin of error, we can use the formula:

Margin of error = (upper limit - lower limit) / 2

In this case, the lower limit of the 95% confidence interval is 64.5% and the upper limit is 75.3%. So the margin of error is:

Margin of error = (75.3% - 64.5%) / 2

Margin of error = 10.8% / 2

Margin of error = 5.4%

Therefore, the margin of error for this 95% confidence interval of a population proportion is 5.4%.

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The value of x if the angles are congruent angles is 4

From the question, we have the following parameters that can be used in our computation:

(5x+2)° (4x+6)°

Assuming the angles are congruent angles

Then we have

(5x+2)° = (4x+6)°

Remove the bracket and the degree sign

So, we have

5x + 2 = 4x + 6

When the like terms are evaluated, we have

x = 4

This means that the value of x is 4

Note that the condition is that the angles (5x+2)° and (4x+6)° are congruent angles

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Answer:

-14°C

Step-by-step explanation:

The temperature at 12 noon = 10°C (given)

The temperature decreases by 2°C in 1 hour (given)

Thus, the temperature decreases by 1°C in 1/2 hour

Temperature 10°C above zero - Temperature 8°C below zero = 10 - (- 8) = 10 + 8 = 18°C

The temperature decreases by 18°C in 1/2 × 18 = 9 hours

Thus, from 10°C above zero to 8°C below zero it takes 9 hours

Total time = 12 noon + 9 hours

= 21 hours = 9 pm

Thus, at 9 pm temperature would be 8°C below zero.

(ii) The temperature at 12 noon = 10°C

The temperature decreases by 2°C every hour

The temperature decrease in 12 hours = - 2°C × 12 = - 24°C

At midnight, the temperature will be = 10°C + (-24°C) = -14 °C

Therefore, the temperature at mid night will be 14°C below 0.

This equation tells us that the distance Alanna ran (ddd) is equal to her average speed (vvv) multiplied by the time she spent running (ttt - 0.25).

To find the equation that relates ddd, vvv, and ttt, we need to use the formula for average speed, which is:

Average speed = distance ÷ time

In this case, Alanna ran ddd kilometers at an average speed of vvv kilometers per hour, so we can write:

vvv = ddd ÷ t1

where t1 is the time it took Alanna to run ddd kilometers.

After running, Alanna walked to cool down for 0.25 hours, so the total time for the trip was ttt = t1 + 0.25. We can substitute this into our equation to get:

vvv = ddd ÷ (ttt - 0.25)

Finally, we can rearrange this equation to solve for ddd:

ddd = vvv × (ttt - 0.25)

So the equation that relates ddd, vvv, and ttt is:

ddd = vvv × (ttt - 0.25)

This equation tells us that the distance Alanna ran (ddd) is equal to her average speed (vvv) multiplied by the time she spent running (ttt - 0.25).

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The answer is: ∫c xy^2 ds = 0To evaluate the line integral of c xy2 ds, we need to first parameterize the given curve c, which is the right half of the circle x2 y2 = 4 oriented counterclockwise.

Let's use the parameterization x = 2cos(t) and y = 2sin(t) for t in [0, pi], which traces out the right half of the circle as t varies from 0 to pi.

Next, we need to find ds, which represents an infinitesimal length element along the curve c. We can use the formula ds = sqrt(dx/dt)^2 + (dy/dt)^2 dt, which simplifies to ds = 2dt.

Substituting our parameterization and ds into the line integral, we get:

∫c xy^2 ds = ∫0^π (2cos(t) * 2sin(t)^2)(2dt)

Simplifying, we get:

∫c xy^2 ds = 8 ∫0^π sin^2(t)cos(t) dt

Using the identity sin^2(t) = (1/2)(1-cos(2t)), we can rewrite the integral as:

∫c xy^2 ds = 8 ∫0^π (1/2)(1-cos(2t))cos(t) dt

Expanding and simplifying, we get:

∫c xy^2 ds = 4 ∫0^π cos(t) dt - 4 ∫0^π cos(2t)cos(t) dt

Evaluating the first integral gives us:

4sin(π) - 4sin(0) = 0

For the second integral, we can use the identity cos(2t)cos(t) = (1/2)cos(3t) + (1/2)cos(t) and simplify:

∫0^π cos(2t)cos(t) dt = (1/2) ∫0^π cos(3t) dt + (1/2) ∫0^π cos(t) dt

The first integral evaluates to 0, and the second integral evaluates to 2sin(π) - 2sin(0) = 0.

Therefore, the final answer is:

∫c xy^2 ds = 0

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The calculated value of the unit rate of the turkey is $7.2 per pound

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

Unit rate = Cost/Pounds of turkey

Substitute the known values in the above equation, so, we have the following representation

Unit rate = 9/(1 1/4)

Evaluate

Unit rate = 7.2

Hence, the unit rate is 7.2

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Based on the 2012 Gallup survey that interviewed a random sample of 474,195 U.S. adults, it can be reasonably concluded that there is a slightly higher prevalence of obesity among unemployed individuals who are actively seeking work (28.6%) compared to those who are employed (24.9%).

However, it is important to note that this survey only provides a snapshot of a specific time period and may not be representative of the entire U.S. population. Additionally, other factors such as age, gender, and socio-economic status may also influence obesity rates and were not accounted for in this survey.

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To compute the 99% confidence interval for the population mean, you need to determine the appropriate sample size to ensure that the sample mean does not deviate from the population mean by more than 1.3. The key terms involved in this process are the confidence interval, sample size, population mean, and sample mean.

The confidence interval represents the range within which the population parameter (in this case, the population mean) is likely to fall, given a certain level of confidence. A 99% confidence interval means that you are 99% confident that the true population mean falls within the specified range.

To calculate the required sample size, you will need to use the formula for the margin of error (E), which is E = (Zα/2 * σ) / √n, where Zα/2 is the critical value associated with the desired level of confidence (99%), σ is the population standard deviation, and n is the sample size.

Since you want the sample mean to not deviate from the population mean by more than 1.3, you will need to set E = 1.3 and solve for n. After finding the critical value for a 99% confidence interval (which is approximately 2.576) and assuming you know the population standard deviation, you can plug these values into the formula and solve for n.

By doing this, you will be able to determine the appropriate sample size to ensure that the 99% confidence interval for the population mean is within 1.3 units of the sample mean.

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The gradient of the river is approximately 0.3947%, calculated by dividing the change in elevation of 1500 feet by the horizontal distance of 72 miles (converted to 380,160 feet).

The gradient of a river is the change in elevation divided by the horizontal distance.

Given that the change of elevation is 1500ft and the length of the river is 72 miles, we first need to convert the units to a consistent system. Let's convert the length from miles to feet, since the change in elevation is given in feet

72 miles = 72 x 5280 feet

72 miles = 380,160 feet

Now we can calculate the gradient using the formula

gradient = change in elevation / horizontal distance

gradient = 1500 ft / 380,160 ft

Simplifying, we get

gradient = 0.003947

Therefore, the gradient of the river is approximately 0.003947, which can be expressed as a percentage by multiplying by 100

gradient = 0.003947 * 100

gradient = 0.3947%

So, the gradient of the river is approximately 0.3947%.

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In British Columbia, Canada, the number y of purple martin nesting pairs x years since 2000 can be modeled by the equation y = 0.63x² +51.8x + 144 there were about 1000 nesting pairs around the time of year 2011.5.

To find when there were about 1000 nesting pairs, we need to solve the given equation for x.

y = 0.63x² +51.8x + 144

We substitute y = 1000 and solve for x:

1000 = 0.63x² +51.8x + 144

0.63x² + 51.8x - 856 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± sqrt(b² - 4ac)) / 2a

where a = 0.63, b = 51.8, and c = -856.

x = (-51.8 ± sqrt(51.8² - 4(0.63)(-856))) / 2(0.63)

x ≈ -86.3 or x ≈ 11.5

Since we are interested in the time after 2000, we discard the negative solution and get:

x ≈ 11.5

This means that there were about 1000 nesting pairs approximately 11.5 years after 2000. To find the actual year, we add 11.5 to 2000:

2000 + 11.5 = 2011.5

Therefore, there were about 1000 nesting pairs around the time of  year 2011.5.

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Answer:

Yes, it is TRUE

Hope my answer helps you ✌️

If x = 2, y = 3 and z = 4 then the value of ;

1. 3yz = 36

2. xy = 6

Substitution means replacing a particular variable with another term, either a constant or another variable. For example, if x = 2 and y = 1

3x²+y² will be calculated by replacing 2 for x and 1 for y in the expression.

Similarly;

if x = 2 , y = 3 and z = 4

then;

3yz = 3 × 3 × 4

= 9× 4

= 36

and xy = 2 × 3

= 6

therefore the value of 3yz and xy are 36 and 6; respectively.

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Answer:

22/30

33/45

7.33333...

%73.33..

Step-by-step explanation:

to find it in a different form as a fraction you can simply multiply it by any number greater than 1 and just make sure you multiply the numerator and the denominator by the same number

to get a decimal simply divide it

to get a percentage you divide the numbers and multiply that by 10.

Q1 = median(2,4,4,5,7,8) = (4+5)/2 = 4.5

The third quartile (Q3) is the median of the 7th through 12th values:

Q3 = median(8,9,12,15,17,28) = (12+15)/2 = 13.5

Thus, the first quartile is 4.5 and the third quartile is 13.5.

The mean of the given data is:

mean = (2+4+4+5+7+8+8+9+12+15+17+28)/12 = 10.5

Thus, the mean is 10.5.

To find the quartiles and median, we first need to order the data:

2, 4, 4, 5, 7, 8, 8, 9, 12, 15, 17, 28

The median is the middle value of the ordered data. Since we have 12 data points, the median is the average of the 6th and 7th values:

median = (7+8)/2 = 7.5

To find the quartiles, we need to divide the ordered data into four equal parts. Since we have 12 data points, the first quartile (Q1) is the median of the 1st through 6th values:

Q1 = median(2,4,4,5,7,8) = (4+5)/2 = 4.5

The third quartile (Q3) is the median of the 7th through 12th values:

Q3 = median(8,9,12,15,17,28) = (12+15)/2 = 13.5

Thus, the first quartile is 4.5 and the third quartile is 13.5.

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The total number of different choices for the representative is 114

The number of choices for the representative to the university committee is the sum of the number of CS faculty members and the number of CS majors who are not faculty members.

Since no one can be both a faculty member and a student.

The number of choices for a faculty member is simply the number of members of the CS faculty, which is 9.

The number of choices for a CS major who is not a faculty member can be calculated by subtracting the number of CS faculty members from the total number of CS majors: 114 - 9 = 105.

Therefore, the total number of different choices for the representative is:

9 + 105 = 114

So there are 114 different choices for the representative to the university committee.

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The probability that the return of the portfolio will be less than -43.5% in any given year is 0.0139, or approximately 1.39%.

To solve this problem, we need to standardize the value of -43.5% using the given mean and standard deviation.

z = (x - mu) / sigma

where z is the z-score, x is the value we want to find the probability for (-43.5%), mu is the expected return (13.2%), and sigma is the standard deviation (18.9%).

Substituting the given values:

z = (-0.435 - 0.132) / 0.189

z = -2.22

We can use a standard normal distribution table or calculator to find the probability that a standard normal random variable is less than -2.22.

P(Z < -2.22) = 0.0139

Therefore, the probability that the return of the portfolio will be less than -43.5% in any given year is 0.0139, or approximately 1.39%.

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Normal probability distribution is applied to a continuous random variable. The correct option is D.

The normal probability distribution, also known as the Gaussian distribution, is a probability distribution that is commonly used in statistics and probability theory. It is a continuous probability distribution that is often used to model the behavior of a wide range of variables, such as physical measurements like height, weight, and temperature.

The normal distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ). It is a bell-shaped curve that is symmetrical around the mean, with the highest point of the curve being located at the mean. The standard deviation determines the width of the curve, and 68% of the data falls within one standard deviation of the mean, while 95% falls within two standard deviations.

The normal distribution is widely used in statistical inference and hypothesis testing, as many test statistics are approximately normally distributed under certain conditions. It is also used in modeling various phenomena, including financial markets, population growth, and natural phenomena like earthquakes and weather patterns.

Overall, the normal probability distribution is a powerful tool for modeling and analyzing a wide range of continuous random variables in a variety of fields.

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For the data generated by the questionnaire item that asked students to indicate their class rank in college, the appropriate measure of location would be the mode.

The mode is the value that occurs most frequently in a dataset and represents the most common response. In this case, the mode would indicate the most common class rank among the students surveyed. It is important to note that the use of the mode as a measure of location is most appropriate when dealing with nominal or ordinal data, such as class rank, where there is no inherent numerical relationship between the categories.

Other measures of location, such as the mean or median, are more appropriate for interval or ratio data where there is a meaningful numerical relationship between the values.

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The final cost if paid within 15 days is $51.91 ($53.52 - $1.61).

The electrician purchased 30 double-pole, double-branch cutouts listed at $6.40 per box of 5, with a 25% discount, and 8 surface panels listed at $4.75 each, with a 35% discount. If the bill is paid within 15 days, an additional 3% discount is applied.

First, we need to calculate the cost of cutouts and surface panels separately. For the cutouts, 30 cutouts are equivalent to 6 boxes (30/5). With the 25% discount, the price per box becomes $4.80 ($6.40 * 0.75). So, the total cost for cutouts is $28.80 (6 * $4.80).

For the surface panels, with the 35% discount, the price per panel becomes $3.09 ($4.75 * 0.65). The total cost for 8 panels is $24.72 (8 * $3.09).

Now, we add the costs together: $28.80 (cutouts) + $24.72 (panels) = $53.52. Lastly, we apply the 3% discount for paying the bill within 15 days, which is $1.61 ($53.52 * 0.03).

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The probability that a basketball player with a 71% free throw shooting accuracy does not make his next free throw shot is 29%.

The probability that a basketball player with a 71% free throw shooting accuracy does not make his next free throw shot.
To calculate the probability of missing a free throw shot, we need to subtract the shooting accuracy percentage from 100%.

In this case, the probability of making a free throw is 71%, which means the probability of missing the free throw is 29%.

Therefore, the probability that the basketball player does not make his next free throw shot is 29%.
It is important to note that free throw shooting accuracy can vary depending on the player's physical and mental condition, as well as external factors such as the audience's noise, the game's pressure, and the distance from the basket.

Thus, it is crucial for basketball players to train and practice regularly to improve their shooting skills and increase their chances of making free throw shots.
To answer this, we need to consider the complement of the success probability.

Since the player has a 71% chance of making the free throw, it means there is a 29% chance that he will not make it (100% - 71% = 29%).

The probability can also be expressed as a decimal, which is 0.29 (29/100 = 0.29).

Therefore, the probability that your favorite basketball player does not make his next free throw shot is 29% or 0.29.
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Answer:

C

Step-by-step explanation:

given the data in ascending order

1 , 3 , 5 , 6 , 7 , 9 , 100

              ↑ middle value

then the second quartile Q₂ ( the median ) is the middle value of the set

thus Q₂ = 6

the first quartile Q₁ is the middle value of the data to the left of the median

1 , 3 , 5

    ↑

Q₁ = 3

the third quartile Q₃ is the middle value of the data to the right of the median

7 , 9 , 100

     ↑

Q₃ = 9

the first , second and third quartiles are 3 , 6 and 9

The first, second, and third quartiles are 3, 6, and 9. The correct answer is option C) 3, 6, and 9.

The first quartile (Q1) is the value that divides the data set into quarters, with 25% of the data falling below this value. To find Q1, we need to locate the median of the first half of the data set. The first half of the data set consists of (1, 3, 5). The median of this set is 3, so Q1 is 3.

The second quartile (Q2) is the median of the entire data set, which is 6.

The third quartile (Q3) is the value that divides the data set into quarters, with 75% of the data falling below this value. To find Q3, we need to locate the median of the second half of the data set. The second half of the data set consists of (7, 9, 100). The median of this set is 9, so Q3 is 9.

Therefore, the first, second, and third quartiles of the given data set (1, 3, 5, 6, 7, 9, 100) are 3, 6, and 9 respectively. The correct answer is option C) 3, 6, and 9.

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Town B population had a greater average rate of change. Town B will eventually have a greater population.

To determine which town had a greater average rate of change from year 4 to year 6, we need to find the average rate of change of each town over that time period.

For Town A, we can find the population in year 4 by plugging in x = 4 into the function P(x):

P(4) = 6[tex](4)^{2}[/tex] + 9(4) + 300 = 372

Similarly, the population in year 6 is:

P(6) = 6[tex](6)^{2}[/tex] + 9(6) + 300 = 498

So the average rate of change of Town A from year 4 to year 6 is:

(498 - 372) / (6 - 4) = 63

For Town B, we can use the formula for compound interest to find the population in year 4 and year 6:

Population in year 4 = 500[tex](1+0.07)^{4}[/tex] = 669.66

Population in year 6 = 500[tex](1+0.007)^{6}[/tex] = 802.86

So the average rate of change of Town B from year 4 to year 6 is:

(802.86 - 669.66) / (6 - 4) = 66.6

Therefore, Town B had a greater average rate of change from year 4 to year 6.

To determine which town will eventually have a greater population, we can compare the population functions for each town. For Town A, the population function is:

P(x) = 6[tex]x^{2}[/tex] + 9x + 300

For Town B, the population function is given by the formula for compound interest:

P(x) = 500[tex](1+0.07)^{x}[/tex]

To compare the growth of these functions, we can take the limit as x approaches infinity:

lim P(x) = lim (6[tex]x^{2}[/tex]+ 9x + 300) = ∞

x→∞

lim P(x) = lim [500[tex](1+0.07)^{x}[/tex] ] = ∞

x→∞

Both functions approach infinity as x approaches infinity, so neither town will eventually have a greater population. However, the population growth rate of Town B (7% per year) is constant and faster than the population growth rate of Town A (which is quadratic and slows down as x increases). Therefore, over a long enough time period, Town B's population growth rate will eventually surpass Town A's population growth rate, even though neither town will eventually have a greater population.

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