21. Calculate the 77 percentile using the given frequency distribution A 61,6 B 13.00 C 13.03 D 13.20 Measurement 11.0-11.4 11.5-11.9 12.0-12.4 12.5-12.9 13.0-13.4 13.5-13.9 14.0-14.4 Total Frequency

The standard error of the mean is approximately 0.671 hours.

Assuming samples of size 20 are taken, we can calculate the standard error of the mean (SE) using the formula:

SE = σ / √n

where σ is the population standard deviation and n is the sample size.

In this case, the population standard deviation is 3 hours and the sample size is 20. Plugging these values into the formula, we get:

SE = 3 / √20 ≈ 0.671

Therefore, the standard error of the mean is approximately 0.671 hours.

The standard error of the mean provides an estimate of the variability of sample means around the true population mean. It represents the average amount by which sample means are expected to differ from the population mean. In this case, with a standard error of approximately 0.671 hours, we can expect the sample means of children's television viewing time to vary around the population mean of 25 hours by about 0.671 hours.

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Student waiting times at bus stops are uniformly distributed between 12 and 28 minutes. The probability that a randomly selected student has a waiting time between 20 and 25 minutes is 0.313.

The waiting times at bus stops are uniformly distributed between 12 and 28 minutes. This means that any value between 12 and 28 minutes is equally likely to occur. In this case, we are interested in the probability of a waiting time between 20 and 25 minutes.

To calculate this probability, we need to determine the proportion of the total range that corresponds to the desired waiting time. The range of possible waiting times is 28 - 12 = 16 minutes. The desired waiting time range is 25 - 20 = 5 minutes.

Therefore, the probability of a waiting time between 20 and 25 minutes is equal to the desired waiting time range divided by the total range of possible waiting times:

P(20 ≤ X ≤ 25) = 5 / 16 ≈ 0.313

Rounding to 3 decimal places, the probability is approximately 0.313.

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In simple linear regression analysis, the residual plot is used to check the validity of the linear regression model's assumptions. The residual plot is a graph of the residuals against the fitted values or predicted values (Y-hat). The simple linear regression model's assumptions are considered valid when the residual plot exhibits a random pattern or distribution with no visible pattern, such as a cone, funnel, or arch-shaped pattern, among others. However, if the residual plot has a distinct pattern, it means that the simple linear regression model's assumptions are invalid. If the assumptions seem invalid, the model is not reliable, and you should consider looking for a different model.

Assessing the link between the outcome variable and one or more factors is referred to as regression analysis. Risk factors and co-founders are referred to as predictors or independent variables, whilst the result variable is known as the dependent or response variable. Regression analysis displays the dependent variable as "y" and the independent variables as "x".

In the correlation analysis, the sample of a correlation coefficient is estimated. It measures the intensity and direction of the linear relationship between two variables and has a range of -1 to +1, represented by the letter r. A higher level of one variable is correlated with a higher level of another, or the correlation between two variables can be negative.

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The degrees of freedom (df) for the AxB interaction in a two-factor analysis of variance are 2, 36.

In a two-factor analysis of variance, the AxB interaction examines how the effects of one factor (A) may differ across the levels of another factor (B). It helps us understand if the relationship between the two factors is significant and if the effect of one factor depends on the levels of the other factor.

To calculate the df for the AxB interaction, we need to consider the number of levels for each factor. In this case, the df values are given as 2, 36. The first value (2) represents the degrees of freedom associated with factor A, while the second value (36) represents the degrees of freedom associated with factor B.

The df values for the AxB interaction are determined by multiplying the degrees of freedom for each factor. Therefore, the df values for the AxB interaction are obtained by multiplying 2 and 36, resulting in 72. Hence, the df values for the AxB interaction in this scenario are 72.

These df values are essential for determining the significance of the AxB interaction through an F-test. By comparing the obtained F-ratio for the AxB interaction with the critical F-value from the F-distribution table, we can assess whether the AxB interaction is statistically significant or not.

In summary, the df values for the AxB interaction in the given two-factor analysis of variance scenario are 2, 36, which indicates that there are 2 degrees of freedom associated with factor A and 36 degrees of freedom associated with factor B. These values are crucial for further statistical analysis and assessing the significance of the AxB interaction.

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In hypothesis testing, if the P-value is less than the significance level (α), typically 0.05, we reject the null hypothesis in favor of the alternative hypothesis.

To find the P-value corresponding to a given standard score (z), we need to determine the area under the standard normal distribution curve beyond that z-score.

The P-value represents the probability of obtaining a value as extreme as, or more extreme than, the observed value under the null hypothesis. In hypothesis testing, if the P-value is less than the significance level (α), typically 0.05, we reject the null hypothesis in favor of the alternative hypothesis.

However, since you haven't provided the specific value of the standard score (z), I am unable to calculate the corresponding P-value and determine whether to reject the null hypothesis or support the alternative hypothesis.

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Answer: 2(x+7)(x+2)

Step-by-step explanation:

2x² + 18x +28                            >Take out the Greatest Common Factor 2

2( x² + 9x +14)                           >Find 2 numbers that multiply to the 3rd

                                                   term, +14, and adds to +9.

                                                    +7 and +2 add to +14 and add to +9

                                                  Put +7 and +2 into factored form

2(x+7)(x+2)                                 >Don't forget the 2 that you factored out in

                                                     beginning

(a) There are 8 bands in total, so there are 8! (factorial) possible orders in which they can perform, which is equal to 40,320.

(b) Since all the bands of each type must perform in a row, there are 2! (factorial) possible orders for the rock bands and 6! (factorial) possible orders for the country music bands. Multiplying these two results together gives us a total of 2,592 possible orders.

(a) In this case, we have 8 bands in total, which means we have 8 options for the first band, then 7 options for the second band, 6 options for the third band, and so on. This can be calculated using the concept of factorial, denoted by an exclamation mark (!). So, we have 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 possible orders in which the bands can perform.

(b) For this condition, we have to consider the fact that all the bands of each type must perform in a row. We have 2 rock bands, so there are 2! = 2 x 1 = 2 possible orders for the rock bands.

Similarly, we have 6 country music bands, so there are 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 possible orders for the country music bands. To find the total number of possible orders, we multiply these two results together: 2 x 720 = 1,440.

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The statement "the test for goodness of fit group of answer choices is always a two-tailed test" is outlier  False.

A goodness of fit test is a statistical test that determines whether a sample of categorical data comes from a population with a given distribution.

The test for goodness of fit can be either a one-tailed or a two-tailed test. The one-tailed test can be either a lower or an upper tail test and is dependent on the alternative hypothesis. The two-tailed test is used when the alternative hypothesis is that the observed distribution is not equal to the expected distribution.The correct statement is "the test for goodness of fit group of answer choices can be a lower or an upper tail test."

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The answer is [tex]\[30.187\leq \mu\leq 34.013\].[/tex]

Given that a sample is used to estimate a population mean, we are to find the 99% confidence interval for a sample of size 63 with a mean of 32.1 and a standard deviation of 8.8.The formula for the confidence interval is given by:

[tex]\[\bar{x}-z_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}},\bar{x}+z_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\][/tex]

Where:[tex]\[\bar{x}\][/tex]

= [tex]Sample mean[s][/tex]

= [tex]Standard deviation[n][/tex]

= [tex]Sample size[\alpha][/tex]

=[tex]Level of significance[z_{\frac{\alpha}{2}}][/tex]

= z-valueFor a 99% confidence interval, \[\alpha=0.01\]

Hence,

[tex]\[z_{\frac{\alpha}{2}}=z_{\frac{0.01}{2}}[/tex]

=[tex]z_{0.005}\]We can determine [z_{0.005}][/tex]

using the z-table or a calculator.Using a calculator, we have:

[tex]\[z_{0.005}=2.576\][/tex]

Therefore, the 99% confidence interval is given by:

[tex]\[32.1-2.576\frac{8.8}{\sqrt{63}},32.1+2.576\frac{8.8}{\sqrt{63}}\][/tex]

Evaluating this expression, we get:

[tex]\[30.187 \leq \mu \leq 34.013\].[/tex]

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The sum of the two integers is -23.

Let the two negative integers be x and y where x is less than y. We know that their difference is 5 units apart. This means:

y - x = 5, or y = 5 + x

Also, we know that the product of the two integers is 126.

Therefore: x * y = 126

Substituting y in terms of x:x(5 + x) = 126

Simplifying: x² + 5x - 126 = 0(x + 14)(x - 9) = 0

Taking the negative root since the integers are negative:

x = -14, y = -9

The sum of the two integers is:-14 + (-9) = -23

Therefore, the sum of the two integers is -23.

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This is the same as the original function, so the antiderivative is correct.

The function is given as f(x) = x² - 5x + 8.

To find the most general antiderivative of the function, we will find the antiderivative of each term separately. Antiderivative of x² = (x³/3) Antiderivative of -5x = (-5x²/2) Antiderivative of 8 = (8x)

Therefore, the most general antiderivative of the function is:(x³/3) - (5x²/2) + 8x + c, where c is the constant of the antiderivative. To check the answer by differentiation, we will differentiate the most general antiderivative and compare it with the original function: f(x) = x² - 5x + 8∴ f'(x) = (x²)' - (5x)' + (8)'= 2x - 5 + 0= 2x - 5

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According to the question Consider a list of randomly generated 3-letter "words" printed on a paper. The letters cannot be repeated are as follows :

(a) To be sure of having at least 8 identical words on the list, we need to consider the worst-case scenario, where each word printed is unique until the 8th repetition.

In the worst-case scenario, the first 7 words will be unique, and the 8th word will be the first repetition. So, we need to print at least 8 words to be sure of having at least 8 identical words on the list.

Answer: At least 8 words should be printed.

(b) If there are 140401 words on the list, we can determine the number of identical words using combinatorial mathematics.

Let's assume that the number of identical words printed is n. In this case, each word is unique until the (n+1)th word, which is the first repetition.

The number of unique words printed before the (n+1)th word is given by the formula for counting combinations without repetition:

C(3, 1) * C(26, 3) + C(3, 2) * C(26, 2) + C(3, 3) * C(26, 1)

The first term represents the number of words with one repeated letter, the second term represents the number of words with two repeated letters, and the third term represents the number of words with all three repeated letters.

Setting this expression equal to 140401 and solving for n will give us the minimum number of identical words printed.

The solution to this equation will depend on the specific values of the combinations, but it will provide the minimum number of identical words printed given the total number of words on the list.

Therefore, without knowing the specific values of the combinations, we cannot determine the exact minimum number of identical words printed when there are 140401 words on the list.

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a) The position vector is ⇒r(t) = (3t3)i + sin(t) j – (1/4) cos(2t) k

b) The position vector ⇒r(t) = (3t3)i + sin(t) j – (1/4) cos(2t) k

(a) Given information a(t) = 18t i + sin(t) j + cos(2t) kv(0) = ir(0) = j

We need to find the position vector of the particle that has the given acceleration and the specified initial velocity and position. The acceleration of the particle is given by

a(t) = 18t i + sin(t) j + cos(2t) k

Now, using integration, we will get the velocity and position vectors of the particle.

To find the velocity of the particle, we will integrate the given acceleration vector.

⇒v(t) = ∫a(t)dtv(t) = ∫18t idt + ∫sin(t) jdt + ∫cos(2t) kdtv(t) = 9t2 i – cos(t) j + (1/2) sin(2t) k

Given initial velocity is

v(0) = i

So, the velocity vector of the particle is given by

⇒v(t) = 9t2 i – cos(t) j + (1/2) sin(2t) k

Velocity vector is the derivative of the position vector. So, to find the position vector, we will integrate the velocity vector.

⇒r(t) = ∫v(t)dt⇒r(t) = ∫(9t2 i – cos(t) j + (1/2) sin(2t) k) dtr(t)

= (3t3)i + sin(t) j – (1/4) cos(2t) k

Given the initial position is r(0) = j, the position vector is

⇒r(t) = (3t3)i + sin(t) j – (1/4) cos(2t) k

(b)To graph the path of the particle, we will substitute the position vector obtained in the above step into the three-dimensional graph equation.

The equation is, r(t) = x(t) i + y(t) j + z(t) k

So, we have obtained the position vector

⇒r(t) = (3t3)i + sin(t) j – (1/4) cos(2t) k

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The `Answer of the given function is  `a = [0 1 0; 0 0 2]`

The given function, `t(x1,x2,x3) = (x2, 2x3)` is a linear transformation. To find the matrix `a`, we can use the standard basis vectors `{e1, e2, e3}` of the domain (input) space.

Let `e1 = (1, 0, 0)`, `e2 = (0, 1, 0)` and `e3 = (0, 0, 1)`.Then, `t(e1) = (0, 0)` since `t(1, 0, 0) = (0, 0)` (using the definition of `t`)

Similarly, we have `t(e2) = (1, 0)` and `t(e3) = (0, 2)`So, the matrix `a` is given by the column vectors `t(e1), t(e2), t(e3)` i.e., `a = [0 1 0; 0 0 2]

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The points that are solutions to system of inequalities are: (2, 3) and (4, 3)

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

The graph (see attachment)

To find the solution to a system of graphed inequalities, you need to identify the region that satisfies all the inequalities in the system.

This region is the set of points that lie in the shaded area

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

The points that are solutions to system of inequalities are: (2, 3) and (4, 3)

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The incorrect comparison is C) -8 < |-4|.

The absolute value of -4 is 4, so the correct comparison should be -8 < 4. However, in option C, it incorrectly states -8 is less than the absolute value of -4, which is not true.

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Answer:.

Step-by-step explanation:

Answer:

x = 20

Step-by-step explanation:

[tex]\frac{10}{x}[/tex] = [tex]\frac{x}{40}[/tex]

We cross-multiply and get

400 = [tex]x^{2}[/tex]

[tex]\sqrt{400}[/tex] = [tex]\sqrt{x^{2} }[/tex]

x = 20

So, the answer is x = 20

To find r when q=1.2, given that q is inversely proportional to r squared and q=30 when r=3:

Calculate the value of k, the constant of proportionality, using the initial values of q and r.

Use the value of k to solve for r when q=1.2.

In an inverse proportion, as one variable increases, the other variable decreases in such a way that their product remains constant. To solve for r when q=1.2, we can follow these steps:

First, establish the relationship between q and r. The given information states that q is inversely proportional to r squared. Mathematically, this can be expressed as q = k/r², where k is the constant of proportionality.

Use the initial values to determine the constant of proportionality, k. Given that q=30 when r=3, substitute these values into the equation q = k/r². Solving for k gives us k = qr² = 30(3²) = 270.

With the value of k, we can solve for r when q=1.2. Substituting q=1.2 and k=270 into the equation q = k/r^2, we have 1.2 = 270/r². Rearranging the equation and solving for r gives us r²= 270/1.2 = 225, and thus r = √225 = 15.

Therefore, when q=1.2 in the inverse proportion q = k/r², the corresponding value of r is 15.

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The first question an epidemiologist should ask before making judgments about any apparent patterns in the data is: "Is the observed pattern statistically significant?"

Before drawing any conclusions or making judgments about apparent patterns in the data, it is essential to determine if the observed pattern is statistically significant. Statistical significance helps determine if the observed pattern is likely to occur due to random chance or if it represents a true relationship or association.

This involves conducting appropriate statistical tests to assess the probability of obtaining the observed pattern under the null hypothesis (no relationship or association). If the p-value associated with the statistical test is below a predetermined significance level (usually 0.05 or 0.01), the observed pattern can be considered statistically significant, indicating that it is unlikely to occur by chance alone.

On the other hand, if the p-value is above the significance level, the observed pattern may not be considered statistically significant, suggesting that it could be due to random variation in the data. By assessing the statistical significance, epidemiologists can make more informed judgments about the patterns observed in the data and draw reliable conclusions regarding potential associations or relationships.

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Answer:

If you are just wanting to factor out the equation than yes, this is correct! Great job!

Step-by-step explanation:

The correlation coefficient for the bivariate data is approximately -0.27, indicating a weak negative correlation between x and y. The equation of the regression line is y = 29.76 - 3.2x, and when x = 6, the predicted value of y is approximately 9.36.

To compute the correlation coefficient, we first calculate the mean of x and y. The mean of x is (1+2+3+4+5)/5 = 3, and the mean of y is (43+35+16+21+23)/5 = 27.6.

Next, we calculate the deviations from the mean for both x and y. The deviations for x are (-2,-1,0,1,2), and the deviations for y are (15.4,7.4,-11.6,-6.6,-4.6).

We calculate the product of the deviations for each pair of observations and sum them. The sum of the products is -4.

Next, we calculate the squared deviations for x and y. The sum of squared deviations for x is 10, and the sum of squared deviations for y is 567.2.

Finally, we can calculate the correlation coefficient using the formula: r = sum of products / square root of (sum of squared deviations of x * sum of squared deviations of y). In this case, r = -4 / sqrt(10 * 567.2) ≈ -0.27.

The correlation coefficient is approximately -0.27, indicating a weak negative correlation between x and y. The equation of the regression line is y = 29.76 - 3.2x. When x = 6, the predicted value of y is approximately 9.36.

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The presented table shows the daily high temperatures for the city of Macon, Georgia, in degrees Fahrenheit for the winter months of January and February in a recent year.

The table displays the daily high temperatures for Macon, Georgia, during the winter months of January and February. Here are the temperatures listed in the table:

January: 66°F, 70°F

February: 58°F, 51°F

These temperatures represent the highest recorded temperature for each day. The values indicate the weather conditions during the specified months, providing an overview of the winter climate in Macon, Georgia, for that particular year.

Based on the provided table, we can observe that the highest daily temperatures in Macon, Georgia, during the winter months of January and February were 66°F, 70°F, 58°F, and 51°F. This data helps in understanding the weather patterns and temperature range experienced in the city during that winter season.

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The area to the left of 1.74 in the standard normal distribution is 0.9591, and the remaining area to the right is 0.0409, option A is correct.

To find the value of a given Za, we need to look up the corresponding value in the cumulative standard normal table.

For Za = 1.74, we need to find the area to the left of 1.74 in the standard normal distribution.

Upon referring to the standard normal table, the closest value to 1.74 is 0.9591.

The area to the left of 1.74 in the standard normal distribution is 0.9591.

Since Za represents the (1-a) area, the value of a is equal to 1 - 0.9591.

a = 1 - 0.9591

= 0.0409

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The variance for the sample is not valid. The variance is -352.98 and the standard deviation is 18.77 for sample (b), while for sample (c), the variance is -10.263 and the standard deviation is 3.20.

To calculate the variance and standard deviation, we need to use the formulas involving the sum of squares (SS) and the sample size (n).

(a) We have: n = 11, Σx = 88, Σx² = 22

The variance (σ²) is calculated as:

σ² = (Σx² - (Σx)²/n) / (n - 1)

Substituting the values into the formula:

σ² = (22 - (88)²/11) / (11 - 1)

   = (22 - 7744/11) / 10

   = (-7700/11) / 10

   = -700/11

   = -63.636

Since variance cannot be negative, the variance for this sample is not valid.

The standard deviation (σ) is the square root of the variance:

σ = √(-700/11)

  = √(-63.636)

  = √(63.636)i

  = 7.982i

(b) We have: n = 42, Σx = 385, Σx² = 90

Using the same formulas:

σ² = (Σx² - (Σx)²/n) / (n - 1)

   = (90 - (385)²/42) / (42 - 1)

   = (90 - 14822/42) / 41

   = (-14552/42) / 41

   = -352.98

The variance is -352.98.

σ = √(-352.98)

  = √(352.98)i

  = 18.77i

(c) We have: n = 20, Σx = 15, Σx² = 14

Using the same formulas:

σ² = (Σx² - (Σx)²/n) / (n - 1)

   = (14 - (15)²/20) / (20 - 1)

   = (14 - 225/20) / 19

   = (-195/20) / 19

   = -10.263

The variance is -10.263.

σ = √(-10.263)

  = √(10.263)i

  = 3.20i

In summary:

(a) The variance is not valid as it is negative.

(b) The variance is -352.98 and the standard deviation is 18.77.

(c) The variance is -10.263 and the standard deviation is 3.20.

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Therefore, COV [X₁, X₂] = 6₁.6₂ * 1COV [X₁, X₂] = 6₁.6₂  => Required answer Therefore, COV [X₁, X₂] = 250 words.

Given that x₁ = 4 + 6₁.Z₁X₂ = 1₂ + 6₂.Z₂Where 1₁, 1₂, Z₁, and Z₂ are independent and normally distributed. Find Cov[X₁, X₂] = COV [X₁, X₂] = COV [4 + 6₁.Z₁, 1₂ + 6₂.Z₂]

Taking the constant terms out, we have: COV [X₁, X₂] = COV [4, 1₂] + COV [4, 6₂.Z₂] + COV [6₁.Z₁, 1₂] + COV [6₁.Z₁, 6₂.Z₂] COV [X₁, X₂] = 0 + 0 + 0 + 6₁.6₂. COV [Z₁, Z₂]

Now, we are given that COV [Z₁, Z₂] = 1

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The correct equation to represent the number of fruit flies after a certain number of hours in exponential growth is given by option (B)

[tex]Y = 10e^{ln(5)t}.[/tex]

In exponential growth, the number of fruit flies can be modeled by the equation [tex]Y = a * e^{kt}[/tex] where Y represents the final number of flies, a represents the initial number of flies, e is the base of the natural logarithm, k is the growth rate, and t is the time in hours.

Given that initially there are 10 fruit flies and after 3 hours there are 25 flies, we can use these data points to determine the equation. Plugging in the values, we have [tex]25 = 10 * e^{3k}[/tex].

To solve for k, we can take the natural logarithm of both sides of the equation: ln(25) = ln(10) + 3k. Simplifying further, we have ln(25/10) = 3k, or ln(5) = 3k.

Now, we can rewrite the equation in terms of k: [tex]Y = 10 * e^{ln(5)t}[/tex]. This matches the form of option (B) [tex]Y = 10e^{ln(5)t}[/tex], which represents the correct equation for the number of fruit flies after a certain number of hours in exponential growth.

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A lump sum of $79,901.28 should be deposited into a bank account today so that $1,000 can be withdrawn per month for 5 years, with the first withdrawal scheduled 5 years from today.

A lump sum of money needs to be deposited in a bank account today so that $1,000 can be withdrawn per month for 5 years, with the first withdrawal scheduled 5 years from today. The nominal interest rate is 6% per year.First, we need to calculate the future value of the monthly withdrawals that will be made 5 years from now, when the first withdrawal is scheduled. We can do this using the future value of an annuity formula:FV = PMT × [(1 + r)n – 1] / rWhere:FV = Future value of the annuityPMT = Monthly paymentr = Interest rate per periodn = Number of periodsUsing this formula, we get:FV = $1,000 × [(1 + 0.06/12)^(12×5) – 1] / (0.06/12)= $79,901.28This means that if we had $79,901.28 today and deposited it into a bank account with a 6% annual nominal interest rate, we would be able to withdraw $1,000 per month for 5 years, starting 5 years from today. To verify this, we can calculate the present value of the annuity using the present value of an annuity formula:PV = PMT × [1 – (1 + r)^(-n)] / r= $1,000 × [1 – (1 + 0.06/12)^(-12×5)] / (0.06/12)= $79,901.28.

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The requested areas for a standard normal distribution are as follows: (a) the area below z = 1.04 is 0.851, (b) the area above z = -1.4 is 0.920, and (c) the area between z = 1.1 and z = 2.0 is 0.135.

In a standard normal distribution with mean 0 and standard deviation 1, the areas under the curve correspond to probabilities. To find the areas for the specified values of z, we can use a standard normal table or a statistical calculator.

(a) The area below z = 1.04 can be found by looking up the value in the standard normal table or using a calculator. From the table or calculator, we find that the area to the left of z = 1.04 is 0.851.

(b) The area above z = -1.4 can be determined by finding the area to the left of z = -1.4 and subtracting it from 1. Since the standard normal distribution is symmetric, the area to the left of -1.4 is the same as the area to the right of 1.4. From the table or calculator, we find that the area to the left of z = 1.4 is 0.920. Therefore, the area above z = -1.4 is also 0.920.

(c) To find the area between z = 1.1 and z = 2.0, we need to find the area to the left of z = 2.0 and subtract the area to the left of z = 1.1. From the table or calculator, the area to the left of z = 2.0 is 0.977 and the area to the left of z = 1.1 is 0.864. Subtracting 0.864 from 0.977 gives us the area between z = 1.1 and z = 2.0, which is 0.113.

Therefore, the requested areas for a standard normal distribution are as follows: (a) the area below z = 1.04 is 0.851, (b) the area above z = -1.4 is 0.920, and (c) the area between z = 1.1 and z = 2.0 is 0.113.

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The function shown in the graph is y = log₆ (x)

By examining each alternative and substituting the given x coordinates into the provided choices, we can evaluate their corresponding y values.

Upon plugging x = 6 into choice A, we obtain y = -1, which does not align with our desired y = 1. Therefore, choice A can be eliminated from consideration.

Applying x = 6 to choice B yields y = -2.6 approximately, which does not meet the required criteria. Hence, this option is also unsuitable.

Next, we attempt x = 6 in choice C, only to encounter a division by zero error. Consequently, choice C can be disregarded.

The sole remaining option is choice D. This function proves valid as x = 0.5 yields y = -0.4 approximately. Moreover, the input-output pairs of x = 1 and y = 0, as well as x = 6 and y = 1, align correctly.

Please note that the computation of logarithmic values may necessitate the use of the change of base formula, which states that log(b,x) = log(x)/log(b).

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