Two fair dice are tossed, and the following events are defined: A: {The sum of the numbers showing is odd.} B: {The sum of the numbers showing is 9, 11, or 12.} C. Are events A and B independent? Why?

(a) For an area of 0.5 to the left of the z-score under the standard normal curve, z = 0. This is because the standard normal curve is symmetric, and the area to the left of the mean (which is also the median and mode in this case) is 0.5.
(b) For an area of 0.9826 to the left of the z-score under the standard normal curve, you can look up the corresponding z-score in a standard normal (z) table, or use a calculator or software with an inverse cumulative distribution function. The z-score is approximately z = 2.13.

(c) For an area of 0.1423 to the right of the z-score under the standard normal curve, you first find the area to the left (1 - 0.1423 = 0.8577). Then, look up the corresponding z-score in a standard normal (z) table or use a calculator. The z-score is approximately z = 1.08.
(d) For an area of 0.9394 to the right of the z-score under the standard normal curve, find the area to the left (1 - 0.9394 = 0.0606). Then, look up the corresponding z-score in a standard normal (z) table or use a calculator. The z-score is approximately z = -1.55.

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The minimum number of fish that a family member caught is 1. Based on the given frequency table, the minimum number of fish that a family member caught can be determined by looking at the lowest "Number of fish" value with a non-zero "Number of family members" value.

Here's a step-by-step explanation:

1. Observe the frequency table:

  Number of fish | Number of family members
  -----------------------------------------
  0              | 0
  1              | 3
  2              | 1
  3              | 0
  4              | 4

2. Identify the lowest "Number of fish" value with a non-zero "Number of family members" value. In this case, it's "1 fish" with "3 family members".

Therefore, the minimum number of fish that a family member caught is 1 fish.

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Interpret the Confidence Interval. Based on the information provided, we have the following data: Population 1 (Rural Voters): 85 surveyed, 21 support gun control, Population 2 (City Voters): 85 surveyed, 57 support gun control

Let's assume you've already calculated the confidence interval in Step 1. The confidence interval will show a range within which the true difference in support for gun control between rural and city voters is likely to fall.

To interpret the confidence interval, consider the following example:

Confidence Interval: (X1, X2)

If the entire interval is positive (X1 > 0 and X2 > 0), it indicates that city voters are more likely to support gun control than rural voters, with a certain level of confidence (usually 95% or 99%).

If the entire interval is negative (X1 < 0 and X2 < 0), it indicates that rural voters are more likely to support gun control than city voters, with the same level of confidence.

If the interval contains 0 (X1 < 0 and X2 > 0), it means there is not enough evidence to conclude that there is a significant difference in gun control support between rural and city voters at the chosen confidence level.

Remember to always provide the actual confidence interval values and the chosen confidence level in your interpretation.

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Answer:

10 ft

Step-by-step explanation:

Let x = the distance between the wall and ladder.

So the length of the ladder is x + 2.

Applying Pythagorean theorem:

AC² = AB² + BC² (1)

Substituting the measurements into (1)

(x+2)² = 6² + x²

x² + 4x + 4 = 6² + x²

4x + 4 = 36

x = (36 - 4)/4 = 8

So the length of the ladder is x + 2 = 8 + 2 = 10 ft

The distance from the wall is 8 feet. Using y = x + 2, we get:
y = 10
So the length of the ladder is 10 feet.

To solve this problem, we can use the Pythagorean theorem, which states that for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the ladder forms a right-angled triangle with the wall and the ground. The height at which the ladder touches the wall is 6 ft, and the distance from the wall is represented by x. Since the ladder's length is 2 ft more than its distance from the wall, the ladder's length will be x + 2.

Applying the Pythagorean theorem:

Length of ladder² = (Distance from the wall)² + (Height on wall) ²
(x + 2)² = x² + 6²
   
Expanding and simplifying the equation:

x² + 4x + 4 = x² + 36
4x = 32
x = 8

So, the distance from the wall is 8 ft, and the length of the ladder is x + 2 = 8 + 2 = 10 ft.

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Answer:

C

Step-by-step explanation:

The distance between the two points can be found using the distance formula, which is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

In this case, the two points are located at (15, -10) and (15, 22), so:

x1 = 15

y1 = -10

x2 = 15

y2 = 22

Substituting these values into the formula, we get:

d = √((15 - 15)^2 + (22 - (-10))^2)

= √(0 + 32^2)

= √1024

= 32

Therefore, the distance between the two points is 32 units.

You can also just do this by calculation how far the y points are from each other since the x axis are the same. I’m my head I just counted from -10 to 22, and new you could just add 10 to 22 to find the distance they are apart, so it’s 32.

Answer: 87.5

Step-by-step explanation:

sum of values/number of values
85+93+91+81/4

350/4

87.5

Answer:

Let's use "a" to represent the length of the shortest side. The remaining two sides are equal in length and double the length of the shortest side, thus we may represent them as "2a" according to the issue.

Because the perimeter of a triangle is equal to the sum of its sides' lengths, we may solve the following equation:

a + 2a + 2a = 36

We may simplify the left side of the equation as follows:

5a = 36

When we divide both sides by 5, we get:

a = 7.2

Now that we know the length of the shortest side, we can calculate the lengths of the other two sides:

2a = 14.4

As a result, the triangle's sides are 7.2 inches, 14.4 inches, and 14.4 inches.

To ensure that these lengths match the problem's requirements, we may check that the two larger sides are equal in length and twice the length of the shortest side:

14.4 = 2(7.2)

14.4 = 14.4

As a result, x = 7.2 inches and 2x = 14.4 inches is our solution.

The length of the shortest side is 7.2 inches, and the equal sides are each 14.4 inches (2x).The three sides of the triangle are 7.2 inches, 14.4 inches, and 14.4 inches.

Let's use x to represent the length of the shortest side. According to the problem, the other two sides are equal in length and double the shortest side, so they must be 2x each.

To find the perimeter of the triangle, we add up the lengths of all three sides:

x + 2x + 2x = 5x

We know that the perimeter is 36 inches, so we can set up an equation:

5x = 36

To solve for x, we divide both sides by 5:

x = 7.2

Now that we know the length of the shortest side, we can find the lengths of the other two sides:

2x = 2(7.2) = 14.4

So the three sides of the triangle are 7.2 inches, 14.4 inches, and 14.4 inches.

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The probability of picking a tile that is the letter N as a fraction 23/50.

We know that the probability of an event is a number between 0 and 1, where, 0 indicates impossibility of the event to happen and 1 indicates certainty of the event to happen.

From the question it is said that a bag of 50 letter tiles in it and 23 of the tiles are the letter N. If you pick a letter tile at random from the​ bag, the probability that it is the letter N is 3 50. Suppose another bag has 400 letter tiles in it and 256 of the tiles are the letter N.

23/50= 0.46 or 46%

And if the bag contain 400 letter tiles in it and 256 of the tiles are the letter N.  then the probability to pick N is 256/400

256/400 =0.64 or 64%

Thus the 1st bag probability is 46% and for 2nd bag is 64%. Hence choose 2nd bag

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The most basic distinction between types of data is that some data are quantitative while other data are qualitative. Quantitative data consists of numerical information that can be measured or counted, allowing for statistical analysis and objective comparisons. This type of data can be further classified into two subcategories: continuous data and discrete data.

Continuous data represent measurements that can take on any value within a specified range, such as height, weight, temperature, or time. These measurements can be represented using fractions or decimals and are typically collected using precise instruments like rulers or thermometers.

Discrete data, on the other hand, consist of distinct, separate values that can be counted or categorized. Examples of discrete data include the number of students in a class, the number of cars in a parking lot, or the number of books sold in a month. Discrete data is often collected through surveys or counting processes.

In contrast, qualitative data are non-numerical and describe attributes, characteristics, or experiences. This type of data is typically obtained through observation, interviews, or open-ended survey questions. Examples of qualitative data include feelings, opinions, beliefs, or descriptions of events.

In summary, the primary distinction between types of data lies in their nature: quantitative data is numerical and allows for objective measurement, while qualitative data is descriptive and explores subjective aspects. Understanding the difference between these two types of data is essential for conducting accurate and meaningful research.

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To build a box with a square base, no top, and a volume of 100 m³ that uses the minimum amount of material, we need to minimize the surface area while maintaining the volume.

Let x represent the side length of the square base, and h represent the height of the box.

Volume (V) = x²h = 100 m³
Surface Area (SA) = x² + 4xh

First, solve the volume equation for h:

h = 100/x²

Now, substitute this expression for h into the surface area equation:

SA = x² + 4x(100/x²) = x² + 400/x

To minimize the surface area, we'll find the derivative of the surface area equation with respect to x and set it equal to zero:

d(SA)/dx = 2x - 400/x²

Setting the derivative equal to zero and solving for x:

2x - 400/x² = 0
2x³ - 400 = 0
x³ = 200
x = (200)^(1/3) ≈ 5.85 m

Now, plug the value of x back into the equation for h:

h = 100/(5.85²) ≈ 2.93 m

So, the dimensions of the box that minimize the amount of material used are approximately 5.85 m × 5.85 m × 2.93 m.

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Answer:

1/

Step-by-step explanation:

A standard dice has 6 equally likely outcomes, which are the numbers 1, 2, 3, 4, 5, and 6. Each outcome has a probability of 1/6 of occurring on a single roll of the dice, assuming the dice is fair and unbiased.

Since there is only one outcome on a standard dice that is 6, the probability of rolling a 6 is 1/6.

Therefore, the probability of a dice that is 6 as a fraction is 1/6.

To rework problem 27 in section 6.3 with the given data, we need to use the same approach as in the textbook. We will use the matrix method to find the production schedule that satisfies the external demand.

Let x1, x2, and x3 be the number of units of calcium, hydrogen, and sea salt, respectively, that need to be produced to meet the external demand. Then the production constraints can be written as:

0.3x1 + 0.8x2 + 0.3x3 >= 57 (for calcium)
0.2x1 + 0.2x2 + 0x3 >= 32 (for hydrogen)
0.5x1 + 0.4x2 + 0.6x3 >= 13 (for sea salt)

We can write these constraints in matrix form as:

| 0.3  0.8  0.3 |   | x1 |   | 57 |
| 0.2  0.2  0   | * | x2 | >= | 32 |
| 0.5  0.4  0.6 |   | x3 |   | 13 |

Solving this system of inequalities, we get:

x1 = 150
x2 = 40
x3 = 25

Therefore, the production schedule that satisfies the external demand for 57 units of calcium, 32 units of hydrogen, and 13 units of sea salt is to produce 150 units of calcium, 40 units of hydrogen, and 25 units of sea salt.
To find the production schedule that satisfies the external demand for 57 units of calcium, 32 units of hydrogen, and 13 units of sea salt, we can set up a system of linear equations.

Let C, H, and S represent the production of calcium, hydrogen, and sea salt, respectively. We have:

1. 0.3C + 0.2H + 0.5S = 57 (calcium requirements)
2. 0.8C + 0.2H + 0.4S = 32 (hydrogen requirements)
3. 0.3C + 0H + 0.6S = 13 (sea salt requirements)

Solving this system of linear equations will give us the production schedule for calcium, hydrogen, and sea salt. Using a matrix solver or similar tool, we find the solutions:

C = 40
H = 20
S = 30

So, to satisfy the external demand, the production schedule should include 40 units of calcium, 20 units of hydrogen, and 30 units of sea salt.

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To use the comparison test or limit comparison test, we need to find a known series that either diverges or converges and is similar to the given series.

Using the comparison test, we can compare the given series to the series 1/n^2, which is a p-series with p=2 and converges.

To see if our series is smaller or larger than 1/n^2, we can simplify the given series by canceling out k^4 from the numerator and denominator:

k^4 / (16k^6 5) = 1 / (16k^2 5)

Now, we can compare 1 / (16k^2 5) to 1/n^2:

1 / (16k^2 5) < 1/n^2 for all k > 1

Therefore, since our series is smaller than a convergent series, it must also converge.

Alternatively, we can use the limit comparison test by finding the limit of the ratio of the given series to 1/n^2 as n approaches infinity:

lim (n→∞) [k^4/(16k^6 5) / (1/n^2)] = lim (n→∞) (n^2 / (16k^6 5))

This limit is zero for all k > 1, which means the given series and the series 1/n^2 have the same behavior and both converge.

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Answer:

71 is about 72, and 33% is about 1/3, so 33% of 71 is approximately 1/3 × 72 = 24.

.33 × 71 = 23.43, so the approximation is reasonable

The difference between the highest and lowest yield values.

Range = highest_ yield - lowest_ yield

Based on the information provided, I understand that you have a table with yield data for ten consecutive production lots. To compute the sample statistics, follow these steps:

1. Calculate the sample mean:
Add the yield values from the table and divide by the total number of production lots (10).

Mean = (yield_ 1 + yield_ 2 + ... + yield_ 10) / 10

2. Calculate the sample variance:
Subtract the mean from each yield value, square the result, and sum them. Divide the sum by the number of production lots minus 1 (9).

Variance = [(yield_1 - mean)^2 + (yield_2 - mean)^2 + ... + (yield_10 - mean)^2] / 9

3. Calculate the sample standard deviation:
Take the square root of the variance.

Standard deviation = √(variance)

4. Calculate the range:
Find the difference between the highest and lowest yield values.

Range = highest_yield - lowest_yield

Once you have calculated these sample statistics, you can better understand the yield performance of the manufacturing process and analyze the production data. Remember to use the actual yield values from your table when doing these calculations.

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In order to derive the sampling distribution, it is necessary to have a solid understanding of the rules of probability and the laws of expected value and variance.

The sampling distribution refers to the distribution of a statistic, such as the mean or standard deviation, calculated from multiple samples taken from the same population. In order to calculate the probability of obtaining a certain value for the statistic, one must understand the rules of probability, such as the addition and multiplication rules. Additionally, the laws of expected value and variance provide a framework for understanding how the sampling distribution behaves, including its central tendency and variability.

Without knowledge of these concepts, it would be difficult to accurately derive and interpret the sampling distribution. Therefore, a solid understanding of the rules of probability and the laws of expected value and variance is essential for working with sampling distributions.

The answer is False.

To derive the sampling distribution, understanding the rules of probability, the laws of expected value, and variance is essential. The rules of probability help in determining the likelihood of various outcomes within a sample. The laws of expected value provide the average of all possible outcomes, weighted by their probability, while variance measures the dispersion of data points in a distribution.

Sampling refers to the process of selecting a subset of individuals from a larger population. The sampling distribution is the probability distribution of a sample statistic, such as the mean or variance, based on repeated random sampling from the same population.

To create an accurate sampling distribution, it is important to comprehend and apply the rules of probability to identify the likelihood of different sample outcomes. The laws of expected value and variance play a crucial role in summarizing the central tendency and variability of the sampling distribution, respectively.

In summary, it is false to assume that one does not need to know the rules of probability, expected value, and variance when deriving the sampling distribution. These concepts are fundamental to understanding and constructing a valid sampling distribution that reflects the properties of the larger population.

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The answers to all parts are shown below.

1. 5a + 3 -4a² + 2a³

writing into standard form of polynomial

2a³- 4a² + 5a +3.

2. 7xy + x³ - y³ -5x²y²

writing into standard form of polynomial

= x³ - y³ - 5x²y² + 7xy

3. The degree of the polynomials are

3x+1 = 1 degree

5x² - 2 degree

10- 0 degree

3x³ -2x² + 10= 3 degree

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Answer:

40(1.25-t)

Step-by-step explanation:

There are 3 components to consider; time, speed and distance

Time and Speed are given.

The distance has to be calculated.

Speed to work = 55 miles per hour

Time to work = 1.25-T

Speed to home = 40 miles per hour

Time to home = 1.25-t

Total Time = T + t = 1.25

Distance for trip to home

Speed = Distance/Time

40 = Total Distance/1.25-t

Total Distance = 40(1.25-t)

Therefore, 40(1.25-t) is the correct answer.

!!

Answer:

The fact that "b varies directly with a" means that there exists a constant k such that b = ka. Since "a" and "b" are located the same distance from 0 in opposite directions, we know that a and b have the same absolute value but different signs. Therefore, we can write a = -b or b = -a.

Substituting -a for b in the equation b = ka, we get:

-a = ka

Solving for k, we get:

k = -a/a = -1

Substituting k = -1 back into the equation b = ka, we get:

b = -a

Therefore, the equation that represents the direct variation between a and b is:

b = -a

In all cases, we use hypothesis testing to determine if the sample mean is significantly different from the hypothesized population mean of 40. The results of the test will help us make conclusions about the population mean based on the information gathered from the sample.

(a) State the null hypothesis:
The null hypothesis (H0) is the claim that there is no significant difference between the population mean and the value specified. In this case:
H0: µ = 40

(b) State the alternate hypothesis if you have no information regarding how the population means might differ from 40:
The alternate hypothesis (H1) is the claim that the population mean is different from the value specified. In this case:
H1: µ ≠ 40

(c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may exceed 40:
If you believe that the population mean may be greater than 40, the alternate hypothesis would be:
H1: µ > 40

(d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be less than 40:
If you believe that the population mean may be less than 40, the alternate hypothesis would be:
H1: µ < 40

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∠1 ≅ ∠2 ⇒ given

∠1 ≅ ∠3 ⇒ Vertical angles theorem.

∠2 ≅ ∠4 ⇒ Vertical angles theorem.

∠3 ≅ ∠4 ⇒ Transitive property of congruence

Given, ∠1 ≅ ∠2

∠2 ≅ ∠3 ⇒ Transitive property of congruence

We have to prove that ∠3 ≅ ∠4.

For that, here given the flow chart.

We have to complete the flow chart by using the given statements.

That is,

∠1 ≅ ∠2 ⇒ given

∠1 ≅ ∠3 ⇒ Vertical angles theorem.

∠2 ≅ ∠4 ⇒ Vertical angles theorem.

∠3 ≅ ∠4 ⇒ Transitive property of congruence

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The square root of 7 is classified as an irrational number.

Rational numbers can also be classified as follows:

The square root of 7 is a non-exact square root, hence it is classified as an irrational number.

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Answer:

There should be 33 girls in the class.

Step-by-step explanation:

Take the total amount of students, 132, and multiply it by one-fourth, since three-fourths of the students are boys.

If triangle has opposite side as "B", adjacent side as "A" and hypotnuse as 10 units, then Sine is B/10 and Cosine is A/10.

The Sine is defined as the ratio of the opposite side to the hypotenuse:

So, sin(θ) = opposite/hypotenuse

In this case, the opposite-side is = B and the hypotenuse is = 10 units,

So we can write : sin(θ) = B/10,

Similarly, cosine is defined as the ratio of the adjacent-side to the hypotenuse : cos(θ) = adjacent/hypotenuse

In this case, the adjacent side is = A and the hypotenuse is = 10 units,

So, we can write : cos(θ) = A/10,

Therefore, the sine of the triangle is B/10 and the cosine of the triangle is A/10.

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The given question is incomplete, the complete question is

Find the Sine and Cosine of a triangle with the opposite side=B, the adjacent side=A and the hypotnuse=10 units.

The probability that the mean wait time for a random sample of 45 wait times is between 185.7 and 206.5 seconds is approximately 95.53%.

To calculate the probability that the mean wait time for a random sample of 45 wait times is between 185.7 and 206.5 seconds, we can use the z-score formula.

First, we need to find the standard error of the mean (SEM): SEM = standard deviation / √sample size = 29.5 / √45 ≈ 4.39 seconds.

Next, we calculate the z-scores for the lower and upper bounds:
z1 = (185.7 - 193.2) / 4.39 ≈ -1.71
z2 = (206.5 - 193.2) / 4.39 ≈ 3.03

Now, we can look up these z-scores in a standard normal table or use a calculator to find the probabilities. The probability for z1 is approximately 0.0436, and for z2, it is approximately 0.9989.

Finally, to find the probability that the mean wait time is between 185.7 and 206.5 seconds, we subtract the probabilities: 0.9989 - 0.0436 ≈ 0.9553.

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Answer:

Area: 201.06 cm^2

Circumference: 50.265 cm

Step-by-step explanation:

Area of a circle: A=πr²

π≅3.1415

A=π(8)²

Area≅201.06 cm²

Circumference of a circle: C=2π²

C=2π(8)

C=50.265 cm

Answer:

Area = 50.27cm² , Circumference = 25.13cm

Step-by-step explanation:

First off, to find the area of a circle the formula is [tex]\pi[/tex]r², and in the equation you gave a diameter, so to get a radius divide the diameter by 2.

So therefore [tex]\pi[/tex]4² is your equation and the answer is 50.27cm²

Now to find the circumference. The formula to find circumference is 2[tex]\pi[/tex]r.

Again if you're given a diameter divide it by 2 to get the radius.

So 2[tex]\pi[/tex]4 is the equation, and the answer is 25.13 cm.

Please let me know if you have any more questions, or need any extra assistance!

Step-by-step explanation:

.6 of the animals are sheep    .6 * 80 = 48 are sheep

    the rest of the 80 are goats   = 32  goats

Answer:

32 goats

Step-by-step explanation:

If the probability of selecting a sheep is 0.6, then the probability of selecting a goat is 1 - 0.6 = 0.4.
This means that 40% of the animals are goats.

To find the actual number of goats, we can set up an equation:

0.4 x 80 = number of goats

Simplifying this equation, we get:

32 = number of goats

Therefore, there are 32 goats in the petting zoo, and the remaining animals (48) are sheep.

there is another way to solve this problem.

Since we know that the total number of animals in the petting zoo is 80 and that 60% of them are sheep, we can calculate the number of sheep as follows:

Number of sheep = 0.6 x 80 = 48

We can then calculate the number of goats by subtracting the number of sheep from the total number of animals:

Number of goats = 80 - 48 = 32

So, we get the same answer as before, which is that there are 32 goats in the petting zoo.

The requried magnitude of 3v is 90 and the direction of 3v is 225°.

To find the magnitude and direction of 3v, we can start by calculating the magnitude of 3v:

|3v| = 3|v| = 3(30) = 90

This means that the magnitude of 3v is 90.

To find the direction of 3v, we need to add 180° to the direction of v, since 3v is in the opposite direction to v. The direction of v is given as θ = 45°, so the direction of 3v is:

[tex]\theta_3v = \theta_v + 180^o = 45^o+ 180^o = 225^o[/tex]

Therefore, the magnitude of 3v is 90 and the direction of 3v is 225°.

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If the p-value is less than α, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean price of used cars at this particular dealership is statistically different from the national average. If the p-value is greater than α, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference.

To test whether the mean price of used cars at this particular dealership differs from the national mean, we can use a hypothesis test.

Let μ be the population mean price for used cars at the dealership, and let μ0 be the national mean price for used cars, which is given as $10,192.

We want to test the null hypothesis H0: μ = μ0 against the alternative hypothesis Ha: μ ≠ μ0, at a significance level of α = 0.05.

We can use a two-tailed t-test for the mean to test this hypothesis, assuming that the population standard deviation is unknown and using the sample standard deviation s as an estimate. The test statistic can be calculated as:

t = (X - μ0) / (s / √(n))

where X is the sample mean, s is the sample standard deviation, and n is the sample size.

Under the null hypothesis, the test statistic follows a t-distribution with n-1 degrees of freedom. We can then calculate the p-value associated with the observed test statistic, and compare it with the significance level α.

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