Use the dual simplex method to solve the following LP:max z = -2x1-x3st x1 + x2-x3=>5x1 -2x2+4x3=>8x1, x2,x3=>0

The probability that the person has health insurance given that they seek treatment is 0.851, or approximately 85.1%.

We can use Bayes' theorem to solve this problem. Let's define the events as follows:

H: the person has health insurance

T: the person seeks treatment

We are given:

P(H) = 0.70 (70% have health insurance)

P(T|H) = 0.97 (of those with health insurance, 97% seek treatment)

P(not T|not H) = 0.60 (of those without health insurance, 60% do not seek treatment)

We want to find P(H|T), the probability that the person has health insurance given that they seek treatment.

By Bayes' theorem:

P(H|T) = P(T|H) * P(H) / P(T)

To find P(T), we need to use the law of total probability:

P(T) = P(T|H) * P(H) + P(T|not H) * P(not H)

We are not given P(T|not H) directly, but we can find it using the complement rule:

P(T|not H) = 1 - P(not T|not H) = 1 - 0.60 = 0.40

Now we can substitute into the formula for P(T) and then into Bayes' theorem:

P(T) = P(T|H) * P(H) + P(T|not H) * P(not H) = 0.97 * 0.70 + 0.40 * 0.30 = 0.799

P(H|T) = P(T|H) * P(H) / P(T) = 0.97 * 0.70 / 0.799 = 0.851

Therefore, the probability that the person has health insurance given that they seek treatment is 0.851, or approximately 85.1%.

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Yes, it is possible for solutions to a separable ODE to "go missing" when both sides of the ODE are divided by a function of y.

When we have a separable ODE of the form

g(y) dy/dx = f(x)

we can integrate both sides with respect to their respective variables to obtain

∫ g(y) dy = ∫ f(x) dx

However, to perform this integration, we need to assume that g(y) is nonzero for all values of y in the domain of the solution. If g(y) has any zeros in the domain, then we need to treat those zeros as singularities and solve the ODE separately on each side of the singularity.

If we divide both sides of the ODE by a function of y, say h(y), to obtain

dy/dx = f(x)/h(y)

then we need to be careful to ensure that h(y) is nonzero for all y in the domain of the solution. If h(y) has any zeros in the domain, then dividing by h(y) can cause solutions to "go missing" at those points. This is because dividing by zero is undefined, and solutions can become singular or undefined at those points.

For example, consider the separable ODE

y' = 2x/(y-1)

which we can rewrite as

(y-1) y' = 2x

Dividing both sides by y-1, we get

y' = 2x/(y-1)

which is the same as the original ODE. However, this division by y-1 is not valid when y=1, since it makes the denominator zero. At y=1, the original ODE has a singularity, and we need to treat this point separately when finding the solution. If we fail to do so, we may miss a solution that exists only at y=1.

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This is an example of a theoretical probability.

Theoretical probability is calculated based on the possible outcomes and their likelihood without conducting any experiments or observations. In this case, the probability of getting a tail on the coin is 1/2 (since there are 2 sides) and the probability of getting a 2 on the six-sided die is 1/6 (since there are 6 sides).

To find the combined probability, you multiply the individual probabilities: (1/2) * (1/6) = 1/12, which equals approximately 8.3%.

This is an example of a theoretical probability, as it is based on the assumption of a fair six-sided die and a fair coin. The probability of getting a tail on the coin is 0.5, and the probability of rolling a 2 on the die is 1/6.

Multiplying these probabilities gives a theoretical probability of 0.5 ×1/6 = 1/12, which is equivalent to 8.3% (rounded to one decimal place).

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The lowest possible speed Wilmer needs to type to finish his report in 3 days is 27,000 words / 450 minutes = 60 words per minute.

To get the lowest possible speed in words per minute that Wilmer needs to type to finish his 27,000-word report in 3 days, follow these steps:
Step:1. Calculate the total time Wilmer has for typing: 2.5 hours/day * 3 days = 7.5 hours.
Step:2. Convert the total time to minutes: 7.5 hours * 60 minutes/hour = 450 minutes.
Sep:3. Divide the total word count by the total time in minutes: 27,000 words / 450 minutes.
The lowest possible speed Wilmer needs to type to finish his report in 3 days is 27,000 words / 450 minutes = 60 words per minute.

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The main effects are room illumination and room temperature. Therefore, 2 main effects are there.

Generally speaking, room temperature refers to a range of air temperatures that people favor indoors. When someone is dressed in regular indoor attire, they feel at ease. Depending on humidity, air circulation, and other factors, human comfort can go beyond this range. Neither heated nor chilled, food or beverages may be served at room temperature.

Temperature ranges are defined as room temperature for certain products and processes in industry, science, and consumer goods.

In a study examining the effect of room illumination (low, medium, high) and room temperature (cold, warm, hot) on test performance, there are 2 main effects possible. The main effects are room illumination and room temperature.

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When two variables are correlated, it means that there is a statistical relationship between them. However, correlation does not necessarily imply causation.

In other words, just because two variables are correlated does not mean that one variable causes the other. There may be other factors or variables that contribute to the relationship between the two variables. To establish causation, a researcher would need to conduct further studies to rule out any confounding variables and establish a clear temporal sequence between the variables. Therefore, researchers cannot be completely sure that one variable causes the other simply based on correlation.                                                                                                                                                            A variable is a quantity that may change within the context of a mathematical problem or experiment. Typically, we use a single letter to represent a variable. The letters x, y, and z are common generic symbols used for variables.

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The harmonic series diverges with two different tests namely divergence test and comparison test.

Using the formula of harmonic test -

Hn = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n

where n is a number that is positive and bigger than 1. The harmonic series is a well-known illustration of a divergent series, which lacks a finite sum. We can see that for the harmonic series, the terms 1/n do not move closer to zero as n increases. In reality, 1/n approaches 0, but never truly achieves it, as n approaches infinity.

Whereas, as per comparison test, if two series, Σan and Σbn, are such that for every n, 0 < a_n <b_n, and b_n converges, then Σan must likewise converge. On the other hand, if b_n diverges, Σan must likewise diverge. The harmonic series with another series can be compared to apply the comparison test to it. Using the p-series test with p = 2, we may determine that the series 1/n² converges.

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Answer:

.2 Cm

Step-by-step explanation:

So, the latitude closest to the equator at which the total eclipse will be visible is approximately 16.665 degrees south.

To find the latitude closest to the equator at which the total eclipse will be visible, we need to minimize the function f(t) = 0.00212t² - 0.473t + 32.391. This function represents a parabola with a positive coefficient for the t² term, so its minimum value will occur at the vertex.

The formula to find the t-coordinate of the vertex for a parabola in the form of f(t) = at² + bt + c is:

t vertex = -b / (2a)

In our case, a = 0.00212 and b = -0.473. Plugging these values into the formula, we get:

t vertex = -(-0.473) / (2 * 0.00212) ≈ 111.3208

Now, we can find the latitude at this time by plugging t_vertex back into the function f(t):

f(111.3208) = 0.00212(111.3208)² - 0.473(111.3208) + 32.391 ≈ 16.665

So, the latitude closest to the equator at which the total eclipse will be visible is approximately 16.665 degrees south

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The concentration of the nutrient that yields the largest population density of this organism is c ≈ (1/5000)^(1/3) moles/liter. The population density of this organism is at this optimal concentration is  P(c) ≈ 0.9772 number/cm².

To find the optimal nutrient concentration that yields the largest population density, we need to analyze the given equation:
P(c) = 100c / (1 + 2500c²)
To maximize the population density P(c), we can find the critical points by taking the first derivative of P(c) with respect to the concentration c, and then setting it equal to 0.
P'(c) = dP(c)/dc
Using the Quotient Rule, the first derivative is:
P'(c) = ( (1 + 2500c²) * (100) - 100c * (5000c) ) / (1 + 2500c²)²
Simplify the expression:
P'(c) = (100 - 500000c³) / (1 + 2500c²)²
Now, set P'(c) = 0 to find the critical points:
0 = (100 - 500000c³) / (1 + 2500c²)²
Since the denominator can't be equal to zero, we focus on the numerator:
100 - 500000c³ = 0
Rearrange the equation to solve for c:
500000c³ = 100
c³ = 100 / 500000
c³ = 1 / 5000
c = (1/5000)^(1/3)
Now, we can find the population density at this optimal concentration:
P(c) = 100 * (1/5000)^(1/3) / (1 + 2500 * (1/5000)^(2/3))
P(c) ≈ 0.9772 (in number/cm²)
So, the optimal nutrient concentration is approximately c ≈ (1/5000)^(1/3) moles/liter, which yields the largest population density of approximately P(c) ≈ 0.9772 number/cm².

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The probability that a randomly chosen value from this normal population will be greater than 57 is approximately 0.0297, or 2.97%.

To find the probability that a randomly chosen value will be greater than 57 from a normal population with a mean (μ) of 40 and a standard deviation (σ) of 9, you will need to follow these steps:

1. Calculate the z-score:

The z-score represents the number of standard deviations a value is away from the mean.

To calculate the z-score, use the formula:

z = (X - μ) / σ, where X is the value in question (57 in this case).

2. In this case, z = (57 - 40) / 9 = 17 / 9 ≈ 1.89.

3. Look up the z-score in a standard normal distribution table (or use a calculator or software) to find the probability of obtaining a z-score less than 1.89.

The table value for a z-score of 1.89 is approximately 0.9703.

4. Since we want the probability that the value is greater than 57, we need to find the probability of obtaining a z-score greater than 1.89.

To do this, subtract the table value from 1:

1 - 0.9703 = 0.0297.

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The probability that the conveyor belt will function for a month without a breakdown is approximately 0.061 (accurate to 3 decimal places after the decimal point).

To find the probability that the conveyor belt will function for a month without a breakdown, given that the number of monthly breakdowns follows a Poisson distribution with λ = 2.8, we will use the Poisson probability formula:

P(X = k) = (e^(-λ) * (λ^k)) / k!

In this case, k = 0 (no breakdowns) and λ = 2.8. Plug these values into the formula:

P(X = 0) = (e^(-2.8) * (2.8^0)) / 0!

P(X = 0) = (e^(-2.8) * 1) / 1

Now, use a calculator or software to compute e^(-2.8) and multiply it by 1:

P(X = 0) ≈ 0.06078

So, the required probability is approximately 0.061 (accurate to 3 decimal places after the decimal point).

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The given series is convergent under the condition that[tex]\sqrt{n^{4-1}/ n^{5 +3}}[/tex]

To determine whether the series is convergent or divergent, we can use the limit comparison test.
Let's compare the given series with the series 1/n^2.

lim n→∞ (√[tex]n^{4-1}/ n^{5 +3})[/tex]  / (1/n²)

= lim n→∞ (√ [tex]n^{4-1 }* n^2[/tex] / ([tex]n^{5 +3}[/tex]))

= lim n→∞ (√ 1 - 1/n⁴)

= 1
Here, the limit is finite and positive, both series converge or diverge together. Since the series 1/n^2 converges (p-series with p = 2 > 1), the given series also converges.

Therefore, the given series is convergent.

The limit comparison test is a convergence test applied in calculus to determine the convergence or divergence of a series. This test involves comparing the given series, (sum a_ {n}), to a preferred convergent series, (sum b_ {n}), through the limit of the ratio (a_ {n} / b_ {n}) as n approaches infinity. If the limit is finite and positive, then both series converge or diverge together.
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In multiple testing situations, it's important to control the family-wise error rate (FWER) to avoid making false conclusions. If the p-value is below the adjusted significance level, we reject the null hypothesis for that test. Overall, the Bonferroni Correction is a useful tool for controlling FWER in multiple testing situations, and can help ensure that our conclusions are reliable and accurate.

Explanation of the terms "significance," "Bonferroni Correction," and "null hypothesis," and how they relate to controlling the Family-Wise Error Rate (FWER) in a series of tests. Here's a concise explanation:
1. Significance: Significance is the probability of rejecting the null hypothesis when it is true. In hypothesis testing, it is denoted by the Greek letter alpha (α), which is the significance level. A common value used for α is 0.05, meaning there's a 5% chance of rejecting the null hypothesis when it's true.
2. Null Hypothesis (H0): The null hypothesis is the statement being tested in a hypothesis test. It is usually a claim about a population parameter, such as a mean or proportion, and assumes that there is no effect or difference between groups being compared.
3. Bonferroni Correction: The Bonferroni Correction is a method used to control the FWER when performing multiple hypothesis tests. It adjusts the significance level (α) by dividing it by the number of tests (m) conducted, i.e., α/m.
To control the FWER, we can use the Bonferroni Correction by setting the significance level of each individual test to α/m. We then reject the null hypothesis (H0) of each test (i) when the corresponding p-value is less than or equal to the adjusted significance level, which is α * pi ≤ m. This ensures that the overall FWER is controlled at the desired level (e.g., α = 0.05).

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Null Hypothesis is 4 and Alternative Hypothesis is greater than 34.

Let's first state the null hypothesis (H0) and the alternative hypothesis (Ha) using the information given.

Null Hypothesis (H0): The average score on the Attributional Complexity Scale is equal to the midpoint (4). In symbols, this can be written as H0: μ = 4.

Alternative Hypothesis (Ha): The average score on the Attributional Complexity Scale is greater than the midpoint (4). In symbols, this can be written as Ha: μ > 4.

Now, let's analyze the data provided:

- The sample mean (M) is 5.35
- The sample standard deviation (SD) is 0.54
- The sample size (n) is 9 (since there are 9 participants)

To test the hypothesis, you would typically perform a one-sample t-test, comparing the sample mean to the midpoint of 4. Based on the given information, the sample mean is higher than the midpoint (5.35 > 4), which supports the alternative hypothesis that people, on average, score above the midpoint on the Attributional Complexity Scale. However, to draw a valid conclusion, you would need to calculate the t-value, degrees of freedom, and compare the result to the critical value or obtain a p-value to determine the statistical significance of the findings.

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For the given box plot: The median of the second trip is higher than the median of the first trip.

Box plots, often called box-and-whisker plots, are graphical representations of data sets that highlight essential characteristics and summarise the distribution of the data. A box plot consists of a rectangle (the box) that spans the middle value of the data from the lower quartile (Q1) to the upper quartile (Q3), and a vertical line (the median) inside the box. Any data points outside of this range are displayed as separate dots (outliers), and whiskers (lines) extend from the box to the lowest and highest data points within 1.5 times the IQR (interquartile range). Box plots make it simple to compare several sets of data, and they can highlight variations in central tendency, variability, and skewness.

From the description of the box plots we observe that:

The first trip's range is greater than the second trip's range because the first trip's whiskers are longer than the second trip's whiskers.

Due to the second trip's box being relocated to the right of the first trip's box, the median of the second trip is greater than the median of the first.

Because the box for the first trip is wider than the box for the second trip, the interquartile range (IQR) of the second trip is less than the IQR of the first trip.

Since both trips have a data point at 22, they both have the same highest value.

Hence, The median of the second trip is higher than the median of the first trip.

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There are 6 parts that correspond to these numbers, so the probability of getting a number less than 7 is 6/8 or 3/4.

So, the correct answer is option (a) 3/4.

According to the knowledge or evidence that is currently available, likelihood describes the possibility or probability of an occurrence or result occurring.

When describing the likelihood of an outcome given a collection of observable facts or information, it is frequently employed in statistics.

The likelihood in statistical analysis is frequently described as a probability distribution or function, which expresses the possibility of a certain collection of data given a particular set of assumptions or characteristics.

A statistical model's unknown parameters, such as the mean and variance of a normal distribution or the odds ratio of a logistic regression model, are estimated using the likelihood function.

Since the spinner is divided into 8 equal parts and we want to find the probability of getting a number less than 7, we need to count the number of parts that correspond to numbers less than 7, which are 1, 2, 3, 4, 5, and 6.

These numbers are divided into 6 parts, hence the likelihood of receiving a result that is less than 7 is 6/8 or 3/4.

The solution is therefore (a) 3/4.

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If you change the confidence level from 99% to 95%, the size of the confidence interval will become smaller. Therefore, we can see that the size of the confidence interval has been reduced from 24 (10 to 34) to 22.26 (10.87 to 33.13) when we change from a 99% CI to a 95% CI.

To see why, let's first calculate the margin of error for the 99% CI:

Margin of error = (upper bound - lower bound) / 2 = (34 - 10) / 2 = 12

The point estimate for M is M - 22, so the 99% confidence interval can be written as:

M - 22 ± 12

To calculate a 95% confidence interval, we need to find the z-score for the 97.5th percentile of the standard normal distribution, which is 1.96. Using this value, the margin of error for the 95% CI can be calculated as:

Margin of error = 1.96 * standard error

where the standard error is the standard deviation of the sample divided by the square root of the sample size:

standard error = s / sqrt(N)

We do not have the standard deviation of the sample, so we cannot calculate the standard error. However, we can make an assumption that the standard deviation of the population is equal to the standard deviation of the sample. Using the given 99% CI, we can estimate the standard deviation of the sample as:

12 = 2.58 * s / sqrt(15)

Solving for s, we get:

s = 14.14

Using this value for s, we can calculate the margin of error for the 95% CI as:

Margin of error = 1.96 * s / sqrt(N) = 1.96 * 14.14 / sqrt(15) ≈ 9.13

So the 95% CI is:

M - 22 ± 9.13 = 10.87 to 33.13

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The intercept in the context of the data is the value of $95,000. This represents the base salary that a professor would earn with zero years of experience at the college. However, the value of the intercept may not be meaningful as it implies that a professor with zero years of experience would still earn a salary of $95,000, which is unlikely in most real-world scenarios.

The given equation represents a linear regression model where Salary is the dependent variable and Years is the independent variable. The intercept, $95,000, is the value of Salary when Years is equal to zero. In other words, it represents the base salary that a professor would earn with zero years of experience at the college.

However, it's important to note that the intercept may not be meaningful in this context. A base salary of $95,000 for a professor with zero years of experience may not be realistic, as it implies that a professor would earn a significant salary even without any experience. In most real-world scenarios, it's unlikely that a professor with no years of experience would start with such a high salary.

Therefore, the intercept in this case may not hold much meaning and should be interpreted with caution when considering the actual salary of a professor with zero years of experience at the college. It's important to consider other factors and data points to determine a more realistic base salary for professors

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When analyzing the data, as this will provide a more reliable understanding of the trends and patterns within the groups.

Describing and comparing data from three or more groups using the StudentSurvey.mtw datafile collected from a sample of college students. To analyze the data, follow these steps:

Remember to be thorough and accurate when analyzing the data, as this will provide a more reliable understanding of the trends and patterns within the groups.

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The probability that a randomly selected observation is between 13 and 65 is 0.867. Therefore, the correct option is B.

We are required to determine the probability that a randomly selected observation of the uniform random variable x is between 13 and 65 with c = 10 and d = 70.

1. Calculate the range of the variable: d -

c = 70 - 10 = 60

2. Calculate the length of the interval of interest:

65 - 13 = 52

3. Divide the length of the interval of interest by the range of the variable:

52 / 60 = 0.867

So, the probability that a randomly selected observation of the uniform variable x lies between 13 and 65 is 0.867, which corresponds to option B.

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The right triangles which are similar because if their same size of angle have the ratio q/s of the smaller triangle equivalent to x/z of the bigger triangle.

A ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. It can be used to express one quantity as a fraction of the other ones.

For the bigger triangle;

cos α = x/z {adjacent/hypotenuse}

Also for the smaller triangle;

cos α = q/s {adjacent/hypotenuse}

Therefore by comparison, the ratio q/s of the smaller triangle is equivalent to x/z of the bigger triangle.

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When applying a primitive procedure to its parameters in programming, the procedure to be applied and the arguments it should be applied to are normally specified using the syntax of the programming language.

Depending on the programming language being used, the precise syntax for applying a primitive procedure may differ, but generally speaking, it entails writing the name of the procedure followed by the inputs that it to be applied to, contained in parentheses.

For instance, in the Python programming language, you might use the syntax shown below to apply the primitive procedure print to the string argument "Hello, world!". The syntax would be:

print("Hello, world!")

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(1) This means that there is a 91.92% chance that a randomly selected value from a standard normal distribution will be less than or equal to 1.40.
(2) This means that there is a 98.50% chance that a randomly selected value from a standard normal distribution will be less than 2.17.

1. To determine P(Z ≤ 1.40) for a normal random variable Z with a mean of 0 and a standard deviation of 1, follow these steps:

Step 1: Identify the given information:
Mean (μ) = 0
Standard deviation (σ) = 1
Z-score = 1.40

Step 2: Use a standard normal distribution table or calculator to find the probability:
P(Z ≤ 1.40) ≈ 0.9192

2. To find P(Z < 2.17) for a standard normal random variable Z, follow these steps:

Step 1: Identify the given information:
Z-score = 2.17

Step 2: Use a standard normal distribution table or calculator to find the probability:
P(Z < 2.17) ≈ 0.9850

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Answer:

$115

Step-by-step explanation:

shirt: $15 each

pants: $10 each

5 shirts + 4 pants =

= 5 × $15 + 4 × $10

= $75 + $40

= $115

Answer: $115

The mean of the distribution is 0.55, and the standard error is 0.0363.

To find the mean and standard error of the sampling distribution for the proportion of supporters with sample size n=169, we use the given true proportion (p) and the sample size (n).

1. Calculate the mean: The mean of the sampling distribution is equal to the true proportion, which is p=0.55.

2. Calculate the standard error: Use the formula SE=sqrt[p(1-p)/n]. Plug in the values: SE=sqrt[0.55(1-0.55)/169] ≈ 0.0363.

So, the mean is 0.55, and the standard error is 0.0363.

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The statement 'two functions in F(S, F) are equal if and only if they have the same value at each element of S' is true as functions are considered equal when their domain (S) and co-domain (F) are the same, and they produce the same output values for each input element in their domain.

If two functions in F(S, F) have the same value at each element of S, then they are equal. This is because a function maps each element of the domain (S) to a unique element of the range (F). Therefore, if two functions have the same output for every input, then they are mapping each element of S to the same corresponding element in F, which means that they are the same function.

Conversely, if two functions are equal, then they have the same value at each element of S, since a function's value is uniquely determined by its input.

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The matrix which represents R with respect to standard coordinates is -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

Given is that R : R → R* be the rotation with the properties. The axis of rotation is the line L, spanned and oriented by the vector v = (3,-1,3). R is rotated about L through the angle t = 17 according to the Right Hand Rule

We have θ = 17°.

The given cartesian vector is -

3i - j + 3k

We can write the matrix as -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

So, the matrix which represents R with respect to standard coordinates is -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

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The null hypothesis (H0) states that there is no significant difference between the population parameters being compared. In symbols, this can be represented as H0: μ1 = μ2. The alternative hypothesis (H1) states that there is a significant difference between the population parameters. In symbols, this can be represented as H1: μ1 ≠ μ2.

First, let's state the null and alternative hypotheses:

b. The null hypothesis (H0) states that there is no significant difference between the population parameters being compared. In symbols, this can be represented as H0: μ1 = μ2.

The alternative hypothesis (H1) states that there is a significant difference between the population parameters. In symbols, this can be represented as H1: μ1 ≠ μ2.

c. To identify the critical value and draw the rejection regions, we need to know the alpha level (α) and degrees of freedom (df). The alpha level is the criterion used to determine the significance of the result. For example, a common alpha level is 0.05, which means there is a 5% chance of rejecting the null hypothesis when it is true.

Degrees of freedom (df) is a measure of the number of independent pieces of information used to calculate a statistic. In the case of a two-sample t-test, the degrees of freedom can be calculated as:

df = (n1 - 1) + (n2 - 1), where n1 and n2 are the sample sizes of the two groups being compared.

Once you have the alpha level and degrees of freedom, you can use a t-distribution table or calculator to find the critical value (t*). The rejection regions are the areas in the tails of the distribution that correspond to the alpha level. If the calculated t-value is greater than t* (or less than -t*), you would reject the null hypothesis in favor of the alternative hypothesis.

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The score on the remaining test was 85.

Let's denote the score on the remaining test by x. Then, we know that the average of all seven tests is 78, so we can use the formula for the mean:

(72 + 82 + 84 + 66 + 68 + 89 + x) / 7 = 78

Multiplying both sides by 7, we get:

72 + 82 + 84 + 66 + 68 + 89 + x = 546

Adding up the six scores we know, we get:

(72 + 82 + 84 + 66 + 68 + 89) = 461

Substituting this into the previous equation, we have:

461 + x = 546

Subtracting 461 from both sides, we get:

x = 85

Therefore, the score on the remaining test was 85.

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