Geometry Homework
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The answer of the given question based on quadratic equation is the value of c is -43.

An equation is mathematical statement that shows the two expressions are equal. It consists of two parts, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=). The LHS and RHS can be made up of numbers, variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

For example, the equation 2x + 3 = 7 shows that the expression 2x + 3 is equal to 7. The variable x represents an unknown quantity that can be solved for by manipulating the equation using algebraic techniques.

If the product of the roots of the quadratic equation y = 4x² + 16x + c is -43/4, we can use the fact that the product of the roots is equal to c/4 to solve for c.

So, we have:

[tex]\frac{c}{4} =-\frac{43}{4}[/tex]

Multiplying both sides by 4, we get:

c = -43

Therefore, the value of c is -43.

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From the given quadratic equation the value of c=-43.

What is quadratic equation?

Quadratic equations are second-degree algebraic expressions that is it an equation of degree 2 and are of the form ax² + bx + c = 0.

where a≠0.

Given quadratic equation is

 y= 4x² +16x+c

Comparing with the quadratic equation y= ax²+bx+c₁ we get,

a= 4, b= 16, c₁= c

Product of the roots is c₁/a

So for this problem product of the roots will be c/4

Here given the product of the roots of the given quadratic equation is -43/4

Equating both we get,

c/4= -43/4

c=-43

Hence, the value of c= -43.

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Answer:

5%

It is 5% because in the section, 3.0-3.5, it says 5, which also counts for 5% of the babies.

The length decreased by 6 cm (from 18 cm to 12 cm) and the width increased by 3 cm (from 9 cm to 12 cm) indeed form a square with an area of 135 cm², which is 27 cm² smaller than the area of the rectangle.

A quadrilateral with parallel sides that are equal to one another and four equal vertices is known as a rectangle. It is also known as an equiangular quadrilateral for this reason.

Let's start by defining the variables we will use:

Let L be the original length of the rectangle.

Let W be the original width of the rectangle.

Let A be the area of the rectangle.

From the problem statement, we can set up the following equations:

(L - 6) = (W + 3) (the length decreased by 6 cm is equal to the width increased by 3 cm, forming a square)

(L - 6)(W + 3) - 27 = A (the area of the square is 27cm² smaller than the area of the rectangle)

Simplifying the first equation, we get:

L - W = 9

Solving for one variable in terms of the other, we can express L in terms of W:

L = W + 9

Substituting this expression into the second equation, we get:

(W + 3 - 6)(W + 3) - 27 = A

W(W + 6) - 27 = A

Expanding and simplifying, we get:

W² + 6W - 27 = A

Now, we can use the first equation to substitute for L in terms of W, and express A solely in terms of W:

A = LW

A = (W + 9)W

A = W² + 9W

Substituting this expression for A into the previous equation, we get:

W² + 6W - 27 = W² + 9W

Simplifying, we get:

3W = 27

W = 9

Substituting this value for W into the expression for L, we get:

L = W + 9 = 9 + 9 = 18

Therefore, the original length and width of the rectangle are 18 cm and 9 cm, respectively, and the area of the rectangle is:

A = LW = 18 x 9 = 162 cm²

We can check that the length decreased by 6 cm (from 18 cm to 12 cm) and the width increased by 3 cm (from 9 cm to 12 cm) indeed form a square with an area of 135 cm², which is 27 cm² smaller than the area of the rectangle.

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Answer:

252

Step-by-step explanation:

lol I take rsm too and I just guessed and checked

Answer:

a. The x-coordinate of the minimum point of this parabola is halfway between the two roots (-1 and 2). So the x-coordinate here is 1/2, and we have:

[tex]f( \frac{1}{2} ) = { (\frac{1}{2} )}^{2} - \frac{1}{2} - 2 = - 2 \frac{1}{4} [/tex]

So the coordinate of the turning point is (1/2, -2 1/4), or (.5, -2.25).

b. The roots of this parabola are at (-1, 0) and (2, 0).

The value of x in the triangles in diagram A and B is 9.28 respectively.

Triangle is a plane shape that has three sides.

(A) To solve for x in the triangle in diagram A, we use the formula below

Formula:

Where:

Substitute these values into equation 1 and solve for x

(B) Similarly, to solve for x in the triangle in diagram B, we use the formula below

Formula:

Where:

Substitute these values into equation 2 and solve for x

Hence, the values of x in A and B is 9.28 respectively.

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Norma's expenses and earnings will both be $89, and she needs to teach 8 classes during her first month to earn back her certification cost.

To calculate Norma's expenses and earnings, we need to first determine how many classes she needs to teach to earn back her certification cost.
Let x be the number of classes Norma teaches in her first month.

Her earnings for the first month will be her base salary plus the additional amount per class, which can be represented as 57 + 4x.
To find out how many classes Norma needs to teach to cover her certification cost, we can set up the equation:
57 + 4x = 89
Now, we need to solve for x:

4x = 89 - 57
4x = 32
x = 8
So, Norma needs to teach 8 classes to earn back her certification cost.

Now, we can calculate Norma's total expenses and earnings for the first month.

Her expenses are the certification cost, which is $89.

Her earnings are the base salary plus the additional amount per class:
Earnings = 57 + 4 * 8
Earnings = 57 + 32
Earnings = $89.

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Depending on the given angle measures of the triangle, the angle measures of triangle ABC are either 30°, 60°, and 90° or 90°, 60°, and 30° or 60°, 90°, and 30°.

Triangle ABC with angle measures ∠A, ∠B, and ∠C given as:

Therefore, depending on the given angle measures of the triangle, the angle measures of triangle ABC are either 30°, 60°, and 90° or 90°, 60°, and 30° or 60°, 90°, and 30°.

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What are the angle measures of triangle abc?

a)∠a = 30 , ∠b = 60, ∠c= 90

b)∠a=90, ∠b= 60, ∠c= 30

c)∠a= 60, ∠b= 90, ∠c=30

d)∠a = 90, ∠b = 30, ∠c = 60

According to Benford's law, 5 will be the first digit of a random number set is d. less often than 4.

According to Benford's law, which is an empirical law about the frequency distribution of leading digits in many real-life collections of numerical data, the digit 5 is actually the second least frequent leading digit after 1 and is less common than 4 as a first digit. In actuality,

Benford's rule states that the likelihood of a number having four as its first digit is nearly thirty percent, whereas the likelihood of a number having 5 as its first digit is approximately 25%. This is due to the fact that numbers with smaller starting digits are more common than those with bigger leading digits in many naturally occurring datasets.

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There is a  there is a 3.125% chance that all five customers will give Steve a tip.

To calculate the probability of all five customers giving Steve a tip, we need to multiply the probability of one customer giving a tip (0.5) by itself five times since each customer's tip is an independent event. So, the probability is:

0.5 * 0.5 * 0.5 * 0.5 * 0.5 = 0.03125

So, there is a 3.125% chance that all five customers will give Steve a tip.

To simulate this situation, we could use a random number generator that generates either 0 or 1, with a 50% chance of each. We could then repeat this process five times and count the number of times we get five ones (representing all five customers giving a tip). We could then use the proportion of times we get five ones out of the total number of simulations to estimate the probability of all five customers giving a tip. We would want to repeat the simulation many times to get a more accurate estimate.

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From the 12 listed amount, the amount which are within 1.5 standard deviation of the mean are (E) Eleven.

The term "Standard-deviation" is defined as a statistical measure of amount of variation of a set of data points from mean value.

The amounts which are within 1.5 standard deviation means can eb calculated by the formula :

⇒ {mean - 1.5×(S.D.), mean + 1.5×(S.D.)}

Substituting the values,

We get,

⇒ {8.1 - 1.5×0.3,  8.1 + 1.5×0.3}

⇒ {7.65 , 8.55}.

The 12 listed amount are : {8.22, 7.51, 7.86, 8.36, 7.83, 8.30, 8.09, 8.01, 7.73, 8.53, 8.25, 7.96},

We can see that, from the 12 listed amounts, only 7.51 is out of this rage and the remaining 11 amounts are within this rage.

Therefore, the correct option is (E) Eleven.

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The given question is incomplete, the complete question is

A vending machine is designed to dispense 8 ounces of coffee into a cup. after a test that recorded the number of ounces of coffee in each of 1,000 cups dispensed by the vending machine, the 12 listed amounts, in ounces, were selected from the data. if the 1,000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3 ounce, how many of the 12 listed amounts are within 1.5 standard deviations of the mean?

7.51, 8.22, 7.86, 8.36, 8.09, 7.83, 8.30, 8.01, 7.73, 8.25, 7.96, 8.53.

(A) Four

(B) Six

(C) Nine

(D) Ten

(E) Eleven

The answer of the given question based on the Probability is,  the experimental probability that the card selected was a K or 6 is 19/100 or 0.19.

What is  probability ?

Experimental probability is a method of determining the probability of an event occurring based on experimental data. It involves conducting an experiment or observation and recording the number of times the event of interest occurs, then calculating the ratio of the number of times the event occurred to the total number of trials or observations.

Experimental probability is often used in fields such as science, engineering, and social sciences, where empirical data is collected through experimentation or observation. It is a useful tool for making predictions and testing hypotheses based on real-world data.

The number of times a K or 6 was drawn is the sum of their respective frequencies, which is:

Frequency of K or 6 = Frequency of K + Frequency of 6 = 12 + 7 = 19

The total number of cards drawn is 100, so the experimental probability of drawing a K or 6 is:

Experimental probability = Frequency of K or 6 / Total number of cards drawn = 19 / 100

Therefore, the experimental probability that the card selected was a K or 6 is 19/100 or 0.19, which is approximately equal to 4/21 or 0.1905 when expressed as a fraction or rounded to four decimal places.

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The experimental likelihood that the chosen card was either a K or a 6 is 13/100, or 0.13 or 13%.

The likelihood of results occurring is examined in the study of probability, which is based on the ratio of likely to improbable scenarios.

It is necessary to add the frequencies of cards K and 6 and divide by the total number of draws in order to determine the experimental probability of drawing a K or 6:

Frequency οf K οr 6 = 7 + 6 = 13

Tοtal number οf draws = 4 + 7 + 5 + 6 + 7 + 6 + 8 + 10 + 7 + 10 + 8 + 12 + 10 = 100

Experimental prοbability οf drawing a K οr 6 = Frequency οf K οr 6 / Tοtal number οf draws = 13 / 100

Therefore, the experimental likelihood that the chosen card was a K or 6 is 13/100, or 0.13 or 13%.

The correct response is 8 over 13, as that is the easiest answer to the 13/100 question.

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X, the number of correct guesses out of all the attempts for members in the group, follows a binomial distribution

The theoretical probability of one correct guess is the total number of correct guesses divided by the total number of guesses, which in this case is 11/15 or 0.73.

X represents the number of correct guesses out of all the attempts for members in the group. Since each member's guess is independent of the others, and each guess has the same probability of being correct, X follows a binomial distribution.

The binomial distribution is characterized by two parameters: the number of trials (n) and the probability of success on each trial (p). In this case, n is the total number of guesses, which is 15, and p is the theoretical probability of one correct guess, which is 0.73.

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The given question is incomplete, the complete question is:

Theoretical Probability of one correct guess # of Correct Total # of Group Member Guesses Guesses Jay 3 Amanda 4 Nicholas 4 Overall 11 5 5 15 2.) Let X represent the number of correct guesses out of all the attempts for members in your group. What kind of distribution does X have? Justify your response, indicating values for any of the variables that describe the distribution.

the correct expressions to represent the current value of Gabriella's stock accounts are 1.05d for stock A, 0.95d for stock B, d(1-0.05) for stock B, and d(1+0.05) for stock A.

How to estimate sell?

To calculate the value of Gabriella's stocks, we need to consider the changes in the stock prices of A and B over the year.

We know that stock A has increased by 5%, so its current value is 1.05 times its original value, which means the value of Gabriella's investment in stock A is now 1.05d.

On the other hand, stock B has decreased by 5%, which means its current value is 0.95 times its original value. Thus, the value of Gabriella's investment in stock B is now 0.95d.

Therefore, the expressions that could be used to represent the current value of Gabriella's stock accounts are 1.05d for stock A and 0.95d for stock B.

The expression d+0.05d is incorrect because it assumes that both stocks have increased by 5%, which is not the case. The expressions d-0.5d and d+0.5d are incorrect because they do not take into account the percentage changes in the stock prices. The expression d-0.05d is incorrect because it assumes that both stocks have decreased by 5%, which is not the case.

The expression 0.95d is correct but only represents the value of stock B, while the expression d(1-0.05) is correct for stock B, but d(1+0.05) is correct for stock A.

In summary, the correct expressions to represent the current value of Gabriella's stock accounts are 1.05d for stock A, 0.95d for stock B, d(1-0.05) for stock B, and d(1+0.05) for stock A.

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15(x+3) = 2x(12)

15x+45 = 24x

45 = 9x

x = 5

The value of the given term in which it represent  in standard notation and scientific notation is (a) 0,000914 mole per lit (b) [tex]4.2[/tex] x [tex]10^{5}[/tex] pound

What about standard notation?

In mathematics, standard notation refers to the conventional way of writing mathematical expressions and equations using commonly accepted symbols, signs, and rules. This notation is widely recognized and used by mathematicians, scientists, engineers, and other professionals in the field.

Define scientific notation:

Scientific notation is a way of representing numbers that are either very large or very small. It is a shorthand way of writing numbers using powers of 10. In scientific notation, a number is expressed as the product of a coefficient and a power of 10.

For example, the number 3,000,000 can be written in scientific notation as 3 x 10^6. The coefficient 3 is multiplied by 10 raised to the power of 6 (or 1,000,000), which represents the number of zeros in the original number.

According to the given information:

(a) Given that there is [tex]9.14[/tex] x [tex]10^{-4}[/tex] mole per lit.

So, obtain it in standard notation we have that,

⇒ [tex]9.14[/tex] x [tex]10^{-4}[/tex] = [tex]9.14[/tex] x [tex]\frac{1}{10000}[/tex]

                      = [tex]0.000914[/tex] mole per lit.

(b) Given that there is 420000 pounds

So, obtain it in scientific notation we have that,

⇒ [tex]420000[/tex] = [tex]42[/tex] x [tex]10^{4}[/tex]

                = 4.2 x [tex]10^{5}[/tex] pounds.

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Answer:  

vertex= 4,-25

y coordinate of vertex = (x+4)2-25

x coordinate of vertex= x2-8x-9

Step-by-step explanation:

a. The middle 99.7% of students between 0.04 and 0.10.
b. The range for the 16% of lowest BACs is 0.06 or below.

c. 2.28% of students have a BAC above 0.09.

To find the range, use the properties of the normal distribution and the given mean (0.07) and standard deviation (0.01).

The middle 99.7% of students between 0.04 and 0.10.

To find the range use the z-score corresponding to the 16th percentile (z ≈ -1).

The range for the 16% of lowest BACs is 0.06 or below.

To find the percentage, find the z-score for 0.09 first.

Approximately 2.28% of students have a BAC above 0.09.

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We can't make any inferece of any definitive conclusions about the entire populations of students in the two classes.

What is population in Mathematics?

In mathematics, population refers to a group of individuals, objects, or events that share a common characteristic or feature. This group is the target of a study, and the goal is to make generalizations about the entire population based on a sample from that population.

For example, in a study on the average height of adult males in a particular country, the population would be all adult males in that country. However, it is not practical to measure the height of every single adult male in the country, so a sample is taken instead. The sample is a subset of the population and should be chosen in such a way that it is representative of the entire population.

To make an inference about the quiz scores of all the students in class A compared to all the students in class B, we can compare the medians of the two classes. The median is the middle score when the scores are arranged in order.

The scores of the 20 students in class A are

85, 86, 88, 89, 90, 90, 91, 92, 93, 94, 94, 94, 95, 96, 97, 98, 99, 99, 100, 100

And the scores of the 20 students in class B are

75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94

To find the median score for each class, we need to find the middle score. For class A, the middle score is 94 (since there are an even number of scores, we take the average of the middle two). For class B, the middle score is also 87.

Comparing the medians of the two classes, we see that the median score for class A (94) is higher than the median score for class B (87). This suggests that the quiz scores for class A may be higher overall than the quiz scores for class B. However, it's important to note that this is only based on a sample of 20 students from each class, so we can't make any definitive conclusions about the entire populations of students in the two classes.

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Correct question is "Two math classes took the same quiz. The scores of 20 randomly selected students from each class are listed below. Based on the medians of the scores for each class, what inference would you make about the quiz scores of all the students in class a compared to all the students in class b?

Here the scores of the 20 students in class A are 85, 86, 88, 89, 90, 90, 91, 92, 93, 94, 94, 94, 95, 96, 97, 98, 99, 99, 100, 100

And the scores of the 20 students in class B are

75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94"

For the given similar figures, the area of larger parallelogram is calculated to be 1470 ft².

In Euclidean geometry, a parallelogram is known to be a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. Opposite or opposite sides of a parallelogram have equal length, and opposite angles of a parallelogram are equal.

Parallelograms can be divided into different types according to different properties. They are mainly divided into three special types.

Given,

For the smaller parallelogram:

Area  = 30 ft²

Base = 5 ft

Height = ?

Area = Base × Height

30 = 5 × Height

Height = 30/5

Height = 6 ft

Since the parallelograms are similar, so the proportion of their base and height will be similar as well:

Given, that the base of larger parallelogram is 35 ft,

Base/Height = 5/6 = 35/Height

Height = (35 × 6)/5

Height = 42 ft

Now the area of larger parallelogram will be:

Area = Base × Height

Area = 35 × 42

Area = 1470 ft²

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Answer:

  radius : height = 7 : 3

Step-by-step explanation:

You want the ratio of the base radius to the height of a cylinder that has 30% of its surface area as lateral area.

The lateral area of a cylinder is ...

   LA = 2πrh

The base area of a cylinder is ...

  A = 2(πr²) = 2πr·r

Then the ratio of lateral area to total area is ...

  [tex]\dfrac{\text{LA}}{\text{LA+A}}=\dfrac{2\pi rh}{2\pi rh+2\pi rr}=\dfrac{h}{h+r}=\dfrac{30}{100}[/tex]

Inverting this relation lets us solve for r/h:

  [tex]\dfrac{h+r}{h}=\dfrac{100}{30}\\\\1+\dfrac{r}{h}=1+\dfrac{70}{30}\\\\\boxed{\dfrac{r}{h}=\dfrac{7}{3}}[/tex]

Answer:

Step-by-step explanation:

So we know by Vertical Angles the opposite is contgruent and since j shares many angles one line its supplementry so we have

j + 76 + 90 = 180 suppl. angles there for j equals 14 degrees ( dont let the h fool you)

Answer:

(a) = 0.92
(b) = 0.7613

Step-by-step explanation:

Here are some information to help us answer these problems:

Given That:

P(A) = 0.69
P(B) = 0.23

Then,

Problem (a): P(A ∪ B) when, A ∩ B = ∅:

P(A ∪ B) = 0.69 + 0.23 - 0 = 0.92



Problem (b): P(A ∪ B) when, A ∩ B ≠ ∅:

P(A ∪ B) = 0.69 + 0.23 - (0.69 × 0.23) = 0.7613




I hope you find this easy to understand....

Have any questions? Write in the comments.

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The range interval of the given data is {73,80]

and range for term 2 is [74,80]

You may have studied the techniques for determining a representative value for the provided data in statistics, or the measure of central tendency. Remember that there are three ways to measure central tendency: mean, median, and mode.

Spatial Statistics

Due to the fact that the measurements of central tendency do not provide sufficient detail about a given set of data. Another factor that must be investigated in statistics is variability. We require a single number to characterize variability, much like measures of central tendency. This one number has demonstrated a certain amount of dispersion. You will discover more about range, one of the dispersion measures, in this post.

According to the given problem

the range of any date is equal to the interval of lower and upper limits,

So

Form term 1

Our data start from 73, 74,,75,76,78,79,80

So according to the given data 73 is the lower limit and 80 is the upper limit

The range interval of the given data is {73,80]

and range for 2

in  term 2 the lower limit is 74 so our range  is [74,80]

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Therefore there is graph below and whole process of graph

g(x) = -1/x (x).

g(x) = 1/(-x + 5).

Describe equation?

A mathematical equation is a formula that uses the equals symbol (=) to denote the equality of two expressions.

A formula would be 3x - 5 = 16,

for instance. We may determine the value of the variable x by solving this equation: x = 73.

We can apply a number of transformations on f to convert f(x) to g(x) (x).

A . Starting with the graph of f(x) = 1/x for function a, we can perform a vertical reflection to obtain the graph of g(x) = -1/x (x).

B . Starting with the graph of f(x) = 1/x, we can apply a horizontal shift to the function b's graph, g(x) = 3/(x + 2).

We can apply a number of transformations on f to convert f(x) to g(x) (x).

For instance, we can use the following transformations to change f(x) = x2 into g(x) = (x - 3)2 + 2:

- Replace x with to move the graph 3 units to the right (x - 3)

- Apply a horizontal shift to the left of two units and a vertical stretch of three factors to f f(x) = 1/x.

Starting with the graph of f(x) = 1/x, we may apply a horizontal shift to the right by 5 units and a vertical reflection to obtain the graph of function c, g(x) = 1/(-x + 5).

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The  expression that makes the equation true is A)40 ÷ (5 x 2) x 17.55.

What is expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. Unknown variables, integers, and arithmetic operators are the components of an algebraic expression. There are no symbols for equality or inequality in it.

Here first solve given expression then,

=> 13.5 x (1.2 + 4) = 13.5×1.2+13.5×4 = 70.2

Now solving option expressions then,

A) 40 ÷ (5 x 2) x 17.55 = 40÷10×17.55 = 4×17.55 = 70.2

B) 36 ÷ (4.8 − 2.6) x 3 = 36÷2.4×3 = 45

C) 22 x (3.6 − 1.2) + 4 = 22×(2.4)+4 = 56.8

D) 18 ÷ (3 x 3) x 35.2 = 18÷ 9×35.2 = 2×35.2 = 70.4

Hence the  expression that makes the equation true is A)40 ÷ (5 x 2) x 17.55.

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The approximate probability that Maurice will obtain a sample proportion less than 0.06 is 0.2119, rounded to three decimal places.

The population proportion is p = 0.08 since we are aware that the actual percentage of students who have not declared a major is 0.08. We are asked to estimate the likelihood that Maurice will acquire a sample fraction smaller than 0.06 from a simple random sampling of n = 250 students.

The sample proportion p' is a random variable with a mean equal to the population proportion p and standard deviation σ = √(p(1-p)/n). In this case, we have:

p = 0.08

n = 250

σ = √(0.08×(1-0.08)/250)

  = 0.025

To find the probability that p' < 0.06, we can standardize the variable using the standard normal distribution:

z = (p' - p) / σ

z = (0.06 - 0.08) / 0.025

 = -0.8

Using a standard normal distribution table or a calculator, we find that the probability of getting a z-score less than -0.8 is approximately 0.2119.

Therefore, the approximate probability that Maurice will obtain a sample proportion less than 0.06 is 0.2119, rounded to three decimal places.

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1. the equation of the quadratic function is: y = (8/65)(x + 5)(x - 13)

2. The axis of symmetry is the line x = 0 (the y-axis).

What is quadratic function?

The quadratic equation with roots -5 and 13 can be written as:

y = a(x + 5)(x - 13)

where a is a constant that depends on the coefficient of the quadratic term.

To find the value of a, we can use the fact that the coefficient of the quadratic term is the sum of the roots, multiplied by -1:

a = -(sum of roots) / (product of roots)

a = -(-5 + 13) / (-5 * 13)

a = 8 / 65

Therefore, the equation of the quadratic function is:

y = (8/65)(x + 5)(x - 13)

What is symmetry?

2. The axis of symmetry is the line x = 0 (the y-axis).

To find the axis of symmetry, we can use the formula:

x = -(b / 2a)

where a and b are the coefficients of the quadratic function. In this case, a = 8/65 and b = 0, since there is no linear term.

Substituting these values, we get:

x = -(0 / (2 * (8/65)))

x = -0

Therefore, the axis of symmetry is the line x = 0 (the y-axis).

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The test statistic for the given hypothesis test is 2.31, indicating a significant difference between the sample mean and the hypothesized mean. The p-value is less than 0.025, and at a significance level of 0.05, the null hypothesis is rejected, concluding that the true population mean is greater than 12. The rejection rule using the critical value is to reject the null hypothesis if the test statistic is greater than 1.645.

To compute the test statistic, we first need to calculate the standard er ror of the mean:

SE = /sqrt(n) = 4.32/sqrt(25) = 0.864

Then, we can calculate the test statistic:

z = (sample mean - hypothesized mean)/SE = (14 - 12)/0.864 = 2.315

Rounding to 2 decimals, the test statistic is 2.31.

The degrees of freedom for this test are 24 (n-1), so we can find the range for the p-value using a t-distribution table with 24 degrees of freedom. For a two-tailed test at a significance level of 0.05, the range is between 2.064 and 2.492. Since the test statistic (z = 2.31) falls outside this range, the p-value is less than 0.025.

At a significance level of 0.05, since the p-value is less than 0.05, we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that the true population mean is greater than 12.

The rejection rule using the critical value at a significance level of 0.05 is to reject the null hypothesis if the absolute value of the test statistic is greater than 1.96 (for a two-tailed test) or if the test statistic is greater than 1.645 (for a one-tailed test).

In this case, the test is one-tailed and the critical value is 1.645 (obtained from a t-distribution table with 24 degrees of freedom). Since the test statistic (z = 2.31) is greater than 1.645, we reject the null hypothesis.

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We need to find the probability that the two winners line in Pike.

We know that 2 out of the 10 customers who entered the drawing live in Pike.

Thus, the probability of the first winner line in Pike is:

[tex]\dfrac{2}{10}[/tex]

Then, considering the first winner live in Pike, there are left 9 customers, and 1 of them live in Pike. Thus, the probability that the second winner lives in Pike is:

[tex]\dfrac{1}{9}[/tex]

Now, the probability that the first one lives in Pike and the second one also lives in Pike is the product of the two probabilities we found:

[tex]\dfrac{2}{10}\times\dfrac{1}{9}=\dfrac{2}{10\times9} =\dfrac{1}{10\times4.5} =\dfrac{1}{45}[/tex]

Therefore, the probability that both winners live in Pike is:

[tex]\dfrac{1}{45}[/tex]