In a survey of 835 students at Broward College, 668 said they were employed and 167 said they were not employed. Construct a 95% confidence interval estimate for the proportion of Broward College students who are employed.

An equation that is equivalent to the quadratic equation after completing the square is: C. (x + 2)² = 13.

In Mathematics, the standard form of a quadratic equation is represented by the following equation;

ax² + bx + c = 0

Next, we would solve the given quadratic equation by using the completing the square method;

x² + 4x + 1 = 10

x² + 4x + 1 - 1 = 10 - 1

x² + 4x = 9

In order to complete the square, we would have to add (half the coefficient of the x-term)² to both sides of the quadratic equation as follows:

x² + 4x + (4/2)² = 9 + (4/2)²

x² + 4x + 4 = 9 + 4

x² + 4x + 4 = 13

By simplifying, we have;

(x + 2)² = 13

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The assumptions for the two proportions z-test that would not be a concern, in this case, are random sampling, independence, large sample size, and normality of the sampling distribution.

When comparing two proportions using a z-test, there are several assumptions that need to be met to ensure that the results are valid. In this case, the assumptions that would not be a concern are:

Random sampling: The sample of 50 mushrooms from each company is assumed to be a random sample from the population of mushrooms for each company. This assumption ensures that the sample is representative of the population and that the results can be generalized to the larger population.

Independence: The samples from each company are assumed to be independent of each other. This means that the mushrooms from one company do not influence the mushrooms from the other company in any way. This assumption is necessary for the validity of the z-test.

Large sample size: The sample size of 50 mushrooms from each company is sufficiently large. When the sample size is large, the sample proportion can be used as an estimate of the population proportion, and the sampling distribution can be assumed to be approximately normal. A general rule of thumb is that the sample size should be at least 30.

Normality: The z-test assumes that the sampling distribution of the difference between the two sample proportions is approximately normal. This assumption is valid when the sample size is large, as mentioned above.

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Three treatment conditions were compared in the study.

The question is: A researcher reports an F-ratio with df 5 2, 27 from an independent-measures research study. How many treatment conditions were compared in the study?

To find the number of treatment conditions, we need to look at the first number in the degrees of freedom (df) pair, which is 2.

The first df value (numerator) represents the degrees of freedom associated with the between-groups or treatment variability, while the second df value (denominator) represents the degrees of freedom associated with the within-groups or error variability.

The formula to find the number of treatment conditions is:

Number of treatment conditions = df between groups + 1

In this case, df between groups is 2. So, using the formula:

Number of treatment conditions = 2 + 1 = 3

Therefore, three treatment conditions were compared in the study.

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Answer:

[tex] \frac{x + 4}{6x} = \frac{1}{x} [/tex]

x(x + 4) = 6x

x^2 + 4x = 6x

x^2 = 2x

x = 0, 2

0 is an extraneous solution, so x = 2 is the only solution.

A scientist might set a lower significance level to reduce the likelihood of a Type I error (false positive) and increase the confidence in their results.

When conducting a hypothesis test, a scientist uses statistical methods to evaluate the evidence for or against a proposed hypothesis.

The significance level of a hypothesis test is the probability of rejecting the null hypothesis when it is true, which is also known as a Type I error. In other words, a significance level of 0.05 means that there is a 5% chance of rejecting a true null hypothesis, and accepting a false alternative hypothesis.

Setting a lower significance level, such as 0.01, means that the scientist is willing to accept a higher level of confidence in their results and reduce the likelihood of making a Type I error.

This means that the researcher is willing to accept that there is only a 1% chance of rejecting a true null hypothesis, which is a more conservative approach.

There are several reasons why a scientist might choose to set a lower significance level.

First, if the consequences of a false positive are severe or costly, such as in medical research or engineering, then a lower significance level can help to minimize the risk of making a wrong decision.

Second, if the sample size is small, a lower significance level can help to reduce the impact of random variation and increase the confidence in the results.

Finally, if the effect size of the study is small, a lower significance level can help to ensure that the observed difference is not due to chance and is truly meaningful.

In summary, setting a lower significance level can help a scientist to increase the confidence in their results, reduce the likelihood of making a Type I error, and ensure that the observed difference is not due to chance.

However, it is important to balance the need for a high level of confidence with the practical considerations of the study and the potential consequences of a false positive.

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a) The probabilities for X = 0, 1, 2, and 3, and then summing them up, we find that P(X > 3) ≈ 0.019. b) The probability of zero successes is approximately 0.430. c) The length of an interval of time is 1.306 hours.

a) The time between arrivals of small aircraft is exponentially distributed with a mean of one hour. To find the probability that more than three aircraft arrive within an hour, we will use the Poisson distribution, where λ (lambda) represents the average number of arrivals per hour, which is 1 in this case. The probability formula is P(X > 3) = 1 - P(X ≤ 3), where X is the number of arrivals. Using the Poisson formula, we get:

P(X > 3) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3)]

Calculating the probabilities for X = 0, 1, 2, and 3, and then summing them up, we find that P(X > 3) ≈ 0.019.

b) To find the probability that no interval contains more than three arrivals in 30 separate one-hour intervals, we can use the binomial distribution. The probability of success (an interval with more than three arrivals) is 0.019 from part a), and the probability of failure (an interval with three or fewer arrivals) is 1 - 0.019 = 0.981. Using the binomial formula with n = 30 (number of intervals) and p = 0.981, we find the probability of zero successes (i.e., no interval with more than three arrivals) is approximately 0.430.

c) To determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is 0.27, we use the exponential distribution formula:

P(T > t) = e^(-λt), where T is the waiting time between arrivals, t is the time interval, and λ is the average number of arrivals per hour (1 in this case).

We want to find the value of t such that P(T > t) = 0.27. So:

0.27 = e^(-1 * t)

Taking the natural logarithm of both sides, we get:

ln(0.27) = -t

Solving for t, we find that t ≈ 1.306 hours.

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the probability that the mean life of a random sample of 9 bread-making machines falls between 6.4 and 7.2 years is: 0.6106.

We can use the central limit theorem to approximate the sampling distribution of the sample mean of bread-making machines, which is also normally distributed with a mean of 7 years and a standard deviation of 1/√9 = 1/3 years.

Then, we need to standardize the values of 6.4 and 7.2 using the formula:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

For 6.4 years:

z1 = (6.4 - 7) / (1/3) = -1.2

For 7.2 years:

z2 = (7.2 - 7) / (1/3) = 0.6

We want to find the probability that the sample mean falls between 6.4 and 7.2 years, which is equivalent to finding the probability that the standardized sample mean falls between z1 and z2.

Using a standard normal distribution table or calculator, we can find the probabilities associated with each z-value:

P(z < -1.2) = 0.1151

P(z < 0.6) = 0.7257

Therefore, the probability that the mean life of a random sample of 9 bread-making machines falls between 6.4 and 7.2 years is:

P(-1.2 < z < 0.6) = P(z < 0.6) - P(z < -1.2) = 0.7257 - 0.1151 = 0.6106

The probability is approximately 0.6106 or 61.06%.

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The 90% confidence interval for the population proportion of iPhones that obtain over 85 apps is 0.48 ± 0.16., which can be simplified to 0.48 ± 0.16. The correct answer choice is B.

To construct the confidence interval, we first calculate the sample proportion of iPhones with over 85 apps downloaded:

p = 12/25 = 0.48

We can use the following formula to calculate the margin of error:

[tex]ME = z \alpha /2 * \sqrt{(p * (1 - p)) / n)}[/tex]

Where zα/2 is the critical value from the standard normal distribution for a 90% confidence level, which is 1.645. Substituting the values, we get:

[tex]ME = 1.645 * \sqrt{(0.48 * 0.52) / 25} = 0.159[/tex]

Finally, we construct the confidence interval:

p ± ME = 0.48 ± 0.159

So the answer is option B: 0.48 ± 0.16.

The complete question is:

In a sample of 25 iPhones, 12 had over 85 apps downloaded. Construct a 90% confidence interval for the population proportion of all iPhones that obtain over 85 apps. Assume z0.05 = 1.7=645.

Group of answer choices

 A  0.29 ± 0.15

 B  0.48 ± 0.16

 C  0.48 ± 0.09

 D  0.29 ± 0.16

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The answer is that there are 2¹⁰ (or 1024) possible outcomes in total.

When a coin is flipped 10 times, each flip has 2 possible outcomes: heads or tails. To find the total number of possible outcomes, you can use the formula for calculating the number of outcomes in an experiment with independent events:

Total outcomes = (Number of outcomes for each event)ⁿ (n=Number of events)

This is because each flip has two possible outcomes (heads or tails), and since there are 10 flips, we need to multiply 2 by itself 10 times (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1024).  

Total outcomes = 2¹⁰ = 1,024

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The final equation of the logarithmic equation is x³ - 12x² + 27x - 27 = 0

We have,

To solve the equation [tex]\log x_{3} (x-9) + \log x_{3} (x-3) = 2[/tex],

We can use the logarithmic rule that states:

㏒a (x) + ㏒a (y) = ㏒a (xy)

Using this rule, we can simplify the equation as follows:

[tex]\log x_{3} [(x-9)(x-3)] = 2[/tex]

Now, we can use the definition of logarithms, which states:

㏒a (x) = b if and only if a^b = x

Using this definition, we can rewrite the above equation as:

[tex]x^2_{3} [(x-9)(x-3)] = 3^2[/tex]

Expanding the brackets and simplifying.

x³ - 12x² + 27x - 27 = 0

Thus,

The final equation of the logarithmic equation is x³ - 12x² + 27x - 27 = 0

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This means there is approximately a 3.33% chance that the student will be able to solve all 6 problems on the exam.

We have a student who can solve 7 out of the 10 problems. The instructor will select 6 questions at random for the exam. We want to find the probability that the student can solve all 6 problems on the exam.

To determine this probability, we will use the concept of combinations. A combination is a selection of items from a larger set, where the order of the items does not matter. In this case, we will calculate the combinations of problems the student can solve and the total possible combinations of problems on the exam.

The student can solve 7 problems, so there are C(7,6) combinations of problems she can solve, where C(n,k) represents the number of combinations of n items taken k at a time. There are a total of 10 problems, so there are C(10,6) possible combinations of problems that could appear on the exam.

The probability that the student can solve all 6 problems on the exam is given by the ratio of the combinations of solvable problems to the total possible combinations of problems on the exam:

Probability = C(7,6) / C(10,6)

Using the formula for combinations, we find:

C(7,6) = 7! / (6!(7-6)!) = 7
C(10,6) = 10! / (6!(10-6)!) = 210

So, the probability that the student can solve all 6 problems on the exam is:

Probability = 7 / 210 = 1/30 ≈ 0.0333

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When a weather reporter predicts that there is a 20% chance of snow tomorrow for a certain region,

They are essentially saying that there is a small probability of snowfall occurring in that particular area.

This phrase indicates the likelihood of snowfall, and it is based on various factors such as temperature, atmospheric pressure, wind patterns, and moisture content in the air.

In general, weather forecasting is a complex process that involves analyzing vast amounts of data from various sources, such as satellites, radar, and weather stations.

Forecasters use this data to create computer models that simulate weather conditions in a given region, which they then use to make predictions.

When it comes to predicting snowfall, there are several factors that forecasters consider. For example, they look at the temperature and dew point to determine whether the conditions are suitable for snow to form.

They also analyze the amount of moisture in the air, as well as the wind direction and speed, which can affect how much snow falls and where it accumulates.

In terms of the 20% chance of snow, this indicates that there is a relatively low probability of snowfall occurring in the region in question. It does not mean that it is impossible for snow to fall, but rather that it is less likely than other weather conditions, such as rain or clear skies.

Overall, weather forecasting is an essential tool that helps us prepare for and respond to changes in the weather.

By understanding the meaning behind phrases such as the 20% chance of snow, we can make informed decisions about how to dress, travel, and plan our activities.

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If Ivan can rinse the dishes in minutes and Lamar can rinse the same dishes in minutes, then their combined rinsing power is dishes per minute. To find out how long it will take them to rinse the dishes together, we need to use the formula:


Ivan's rate: 1 dish/minute
Lamar's rate: 1 dish/minute

When working together, their combined rate is the sum of their individual rates. So, the combined rate is (1 + 1) dishes/minute, which equals 2 dishes/minute.

Now, we can use the formula to find the time it takes for them to rinse the dishes together:

work = rate × time

dishes = (2 dishes/minute) × x

Since the number of dishes is the same for both Ivan and Lamar, we can set up an equation:

dishes = 2x

Solving for x, we find that it will take half the time for them to rinse the dishes together compared to doing it individually.

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A random variable is a variable that takes on values that are uncertain or probabilistic in nature.

Therefore, the correct option is a) A variable that takes on values that are uncertain.

Random variables can be discrete, meaning they can only take on specific values, or continuous, meaning they can take on any value within a certain range.

These variables are commonly used in statistical analyses and probability theory to model various phenomena, such as the outcome of a dice roll or the height of individuals in a population.

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In the simple linear regression model, the residual term accounts for the variability in the dependent variable that cannot be explained by the linear relationship between the variables.

In a simple linear regression model, we assume that there is a linear relationship between the dependent variable and the independent variable.

The residual term is the difference between the actual value of the dependent variable and the predicted value based on the regression line. It represents the part of the variation in the dependent variable that is not accounted for by the linear relationship with the independent variable.

The residual term is also referred to as the error term, and its sum of squares is used to estimate the variability of the dependent variable around the regression line. A lower sum of squared residuals indicates a better fit of the regression line to the data.

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Answer:

$35.98 for 2 shoes

$71.96 for 4 shoes, or 2 pairs

Step-by-step explanation:

For 1 pair:

1) 19.99 + (19.99 x 0.80)

2) 19.99 + 15.992

3) 35.982 or rounded to 35.98

For 2 pairs, there are 2 methods:

Method 1:

1) 35.98 x 2

2) 71.96

Method 2:

1) 2(19.99 + (19.99 x 0.80))

2) 39.98 + 31.984

3) 71.964 or rounded to 71.96

The check digit for for that book is 2.

To calculate the check digit of an ISBN-10 number, we use the following formula:

[tex]d_{10} \equiv-( i=1\sum9i\cdot d i )mod11[/tex]

where[tex]$d_i$[/tex] is the [tex]$i^{th}$[/tex] digit of the ISBN-10 number, and [tex]$d_{10}$[/tex] is the check digit.

Let's first calculate the sum in the formula:

[tex]\sum 9i\cdot d i=1\cdot 0+2\cdot 0+3\cdot7+4\cdot1+5\cdot 1+6\cdot 9+7\cdot 8+8\cdot 8+9\cdot 1=178[/tex]

Now we can substitute this into the formula for the check digit:

[tex]$d_10\equiv - ( i=1\sum 9i\cdot d i)mod11\equiv -178$[/tex] mod11

To find the remainder of -178 when divided by 11, we add multiples of 11 until we get a number between 0 and 10:

[tex]-178 &= -16 \cdot 11 + 2 \[/tex]

[tex]&\equiv 2 \mod 11[/tex]

Therefore, the check digit for the given ISBN-10 number is 2.

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To find the appropriate value for c, we need to use the cumulative distribution function (CDF) of the bolt width. Let X be the width of a bolt and F(x) be the CDF of X. Then, we have: P(X < c) = F(c) = 0.8438.

Using a standard normal distribution table, we can find the z-score corresponding to a cumulative probability of 0.8438, which is 1.03 (rounded to two decimal places). Therefore, we have: z = (c - μ) / σ = 1.03, where μ and σ are the mean and standard deviation of the bolt width, respectively. Rearranging this equation, we get: c = μ + z * σ.

We need to know the values of μ and σ to compute c. Let's assume that the bolt width follows a normal distribution with mean μ = 0.75 inches and standard deviation σ = 0.03 inches (these values are just examples). Then, we have: c = 0.75 + 1.03 * 0.03 = 0.78 inches.

Therefore, the appropriate value for c such that a randomly chosen bolt has a width less than c with probability 0.8438 is 0.78 inches (rounded to two decimal places). Since the probability distribution is not given, I will assume that you are referring to a standard normal distribution (z-score).

Using a z-score table or calculator, find the z-score that corresponds to the cumulative probability of 0.8438. The z-score is approximately 1.01. Now, we need to convert the z-score back to the original width scale. This can be done using the formula: Width = (z-score × standard deviation) + mean

However, since the standard deviation and mean are not provided, it is not possible to find the exact value for c. If you can provide the mean and standard deviation, I can help you find the appropriate value for c rounded to two decimal places.

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The race car is approximately 39.27 meters above the center of the race track.

The race car has swept an angle of 2.05 radians out of a total of 2π radians for a full circle. That means it has completed (2.05/2π) of a full circle.

The distance traveled along the circle is equal to the length of the arc swept out by the race car, which can be found using the formula:

arc length = radius x angle in radians

So, in this case:

arc length = 22 x 2.05 = 45.1 meters

Since the race car has completed (2.05/2π) of a full circle, it has traveled (2.05/2π) times the circumference of the circle. The circumference can be found using the formula:

circumference = 2π x radius

So, in this case:

circumference = 2π x 22 = 138.2 meters

Therefore, the distance traveled by the race car is:

distance traveled = (2.05/2π) x 138.2 = 43.1 meters

To find how far the race car is above the center of the race track, we need to find the vertical distance traveled by the race car. We can use the fact that the race track has a radius of 22 meters, and that the race car has traveled along an arc that is 45.1 meters long. Using the Pythagorean theorem, we have:

distance above center = √([tex]45.1^2 - 22^2[/tex]) = √(2025.81 - 484) = √1541.81 ≈ 39.27 meters

Therefore, the race car is approximately 39.27 meters above the center of the race track.

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Full Question: A toy race car races along a circular race track that has a radius of 22 meters. The race car starts at the 3-o'clock position of the track and travels in the counter-clockwise direction. Suppose the car has swept out 2.05 radians since it started moving.

a. The race car is how many radius lengths above the center of the race track?

Answer:

Bond

Step-by-step explanation:

A promissory note or a promise to repay a certain amount of money at some point in the future is basically a bond.

A bond is a debt security that represents a loan made by an investor to a borrower, which is usually a corporation or government agency. It is a fixed-income investment, meaning that the borrower promises to pay a specific amount of interest over a set period of time and return the principal amount of the loan on the date of maturity. Bonds are issued for various purposes, such as raising capital, funding new projects, or refinancing debt.

CD (Certificate of Deposit) is a savings instrument issued by a bank or credit union that generally pays a fixed rate of interest over a set term. Mutual fund is an investment vehicle that pools money from multiple investors to purchase a portfolio of securities, such as stocks, bonds, or both. Stock is an ownership share in a company that represents a claim on part of the company's assets and earnings.

In a major sports rights deal, the NCAA just renewed its contract with CBS/Turner through 2032 for March Madness. CBS and Turner Sports both serve as the broadcasting means in the media process

They play a crucial role in the communication process by broadcasting the NCAA March Madness tournament to millions of viewers around the world. The agreement between the NCAA and CBS/Turner is a significant deal that will ensure the continued popularity and success of the annual college basketball tournament for years to come. This partnership has allowed CBS/Turner to provide in-depth coverage of the event, including live streaming of games, analysis, and commentary. Additionally, CBS and Turner Sports work closely with the NCAA to promote the tournament and its related events to audiences worldwide, which helps to enhance the overall viewing experience. Overall, CBS and Turner Sports have established themselves as key players in the broadcasting industry, providing quality sports programming to audiences worldwide.

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Answer:

because, for any n by n matrix, the sum of the dimension of the column space and the dimension of the null space must equal n. If the two dimensions are the same, their sum is an even number.

Step-by-step explanation:

It is impossible for an n-by-n matrix, where n is odd, to have a null space equal to its column space because the dimensions of the two spaces cannot be the same.

The null space of a matrix A is the set of all solutions to the equation Ax=0, where x is a column vector of appropriate dimensions. The column space of A is the span of the columns of A, which is the set of all linear combinations of the columns of A.

If the null space and column space of A are equal, then the dimension of the null space must be equal to the dimension of the column space. By the Rank-Nullity Theorem, the sum of the dimensions of the null space and the column space is equal to the number of columns in A.

Therefore, if n is odd, the dimensions of the null space and column space cannot be equal since their sum is even. Therefore, it is impossible for an n-by-n matrix, where n is odd, to have a null space equal to its column space.

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Answer:

(3/7)x + (1/3)(x + 2) = 38

9x + 7(x + 2) = 798

9x + 7x + 14 = 798

16x = 784

x = 49, so x + 2 = 51

The numbers are 49 and 51.

The two consecutive odd numbers are 49 and 51, and their sum of three-sevenths of the first number and one-third of the second number equals thirty-eight.

To find two consecutive odd numbers such that the sum of three-sevenths of the first number and one-third of the second number is equal to thirty-eight, follow these steps:

1. Let the first odd number be x, and the second odd number be x + 2 (since they are consecutive odd numbers).
2. The sum of three-sevenths of the first number and one-third of the second number is equal to thirty-eight, so we can write the equation as (3/7)x + (1/3)(x + 2) = 38.
3. To solve for x, first find the common denominator for the fractions, which is 21. Multiply each term by 21: 9x + 7(x + 2) = 798.
4. Simplify and solve for x: 9x + 7x + 14 = 798. Combine like terms: 16x + 14 = 798.
5. Subtract 14 from both sides: 16x = 784.
6. Divide both sides by 16: x = 49.
7. So, the first odd number is 49, and the second odd number is 49 + 2 = 51.

The two consecutive odd numbers are 49 and 51, and their sum of three-sevenths of the first number and one-third of the second number equals thirty-eight.

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Boxplots are most useful for graphically comparing distributions of numerical data, including the median, quartiles, and potential outliers. Therefore, the correct answer is "comparing two populations graphically."

Boxplots allow us to see the distribution of the data, including measures of central tendency (such as the median), and the spread of the data (such as the interquartile range).

Additionally, boxplots can help identify potential outliers and asymmetry in the data.

They are particularly useful for comparing multiple groups or populations side-by-side to identify any differences in their distributions.

Boxplots are most useful for graphically comparing distributions of numerical data, including the median, quartiles, and potential outliers.

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Answer:

6 million

Step-by-step explanation:

50% = 1/2

1/2 + 1/3 = 5/6

5/6 people watched or listened, which means 1/6 people did not do either of them.

1/6 of 36 (million) = 6 (million)

So, the answer is 6 million.

60% of on-campus students in the sample participate in at least one extracurricular activity, 50% of off-campus students in the sample participate in at least one extracurricular activity.

The proportion of on-campus students in the sample who participate in at least one extracurricular activity, we need to divide the number of on-campus students who participate in at least one extracurricular activity by the total number of on-campus students in the sample.

Let's assume that our sample contains 100 on-campus students, and 60 of them participate in at least one extracurricular activity.

Then, the proportion of on-campus students who participate in at least one extracurricular activity is:

proportion = number of on-campus students who participate in at least one extracurricular activity / total number of on-campus students in the sample

proportion = 60/100

proportion = 0.6 or 60%

To calculate the proportion of off-campus students in the sample who participate in at least one extracurricular activity, we follow the same process.

Let's assume that our sample contains 80 off-campus students, and 40 of them participate in at least one extracurricular activity.

Then, the proportion of off-campus students who participate in at least one extracurricular activity is:

Proportion = number of off-campus students who participate in at least one extracurricular activity / total number of off-campus students in the sample

proportion = 40/80

proportion = 0.5 or 50%

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The average temperature during the period from 9 am to 9 pm is 48 degrees Fahrenheit. In order to determine the temperature at 9 am, we simply need to plug in t=0 into the function T(t). So T(0) = 30 + 19 sin(0) = 30. The temperature at 9 am is 30 degrees Fahrenheit.

To determine the temperature at 3 pm, we need to plug in t=6 into the function T(t). So T(6) = 30 + 19 sin(pi/2) = 30 + 19 = 49. Therefore, the temperature at 3 pm is 49 degrees Fahrenheit.

To find the average temperature during the period from 9 am to 9 pm, we need to find the average value of the function T(t) over that time period. This can be done by finding the definite integral of T(t) from t=0 to t=12 (since there are 12 hours from 9 am to 9 pm) and then dividing by 12. Using integration techniques, we can find that:

(1/12) * ∫(0 to 12) (30 + 19 sin(pit/12)) dt = (1/12) * (360 + 228) = 48

Therefore, the average temperature during the period from 9 am to 9 pm is 48 degrees Fahrenheit.

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Answer: 2.6

Step-by-step explanation:

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The slope of a function f(x) at a point x is given by its derivative f'(x). Therefore, to find the average slope of the function f(x) on the interval (-4, 10)

We need to compute the average value of its derivative f'(x) over this interval.

The derivative of f(x) is:

f'(x) = 4x - 12x - 48

We can compute the definite integral of f'(x) over the interval (-4, 10) as follows:

∫[-4,10] f'(x) dx = ∫[-4,10] (4x - 12x - 48) dx

= [2x^2 - 6x^2 - 48x] |[-4,10]

= [(2(10)^2 - 6(10)^2 - 48(10)) - (2(-4)^2 - 6(-4)^2 - 48(-4))]

= (-380) - (120)

= -500

Therefore, the average slope of the function f(x) on the interval (-4, 10) is:

Average slope = (-500) / (10 - (-4)) = (-500) / 14 = -35.71 (approximately)

Hence, the average slope of the function f(x) on the interval (-4, 10) is approximately -35.71.

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It is highly unlikely that a firm in this industry will earn less than 103 million dollars.

z = (x - μ) / σ

z = (103 - 9090) / 1515 = -5.38

The probability of a firm earning less than -5.38 standard deviations from the mean is very low, approximately 0.00000003. This means that the probability of a randomly selected firm earning less than 103 million dollars is extremely low, less than 0.00000003 or 0.000003%.

Probability is a mathematical concept that measures the likelihood of an event occurring. It is a way to quantify uncertainty and express it as a numerical value between 0 and 1. A probability of 0 indicates that the event is impossible, while a probability of 1 indicates that the event is certain.

Probabilities can be calculated using various methods, including the classical, empirical, and subjective approaches. The classical approach is based on the assumption that all outcomes are equally likely, while the empirical approach is based on observed data. The subjective approach involves using personal beliefs and opinions to estimate the probability of an event.

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