(i) If volume is high this week, then next week it will be high with a probability of 0.9 and low with a probability of 0.1.
(ii) If volume is low this week then it will be high next week with a probability of 0.4. The manager estimates that the volume is five times as likely to be high as to be low this week.

Assume that state 1 is high volume and that state 2 is low volume.

(1) Find the transition matrix for this Markov process.(2) If the volume this week is high, what is the probability that the volume will be high two weeks from now?

Answer:

The difference between mistrust and distrust comes down to nuances in meaning. Distrust is a withholding of trust based on evidence or informed opinion. Many people distrust salespeople working on commission, for instance, knowing that these salespeople personally benefit from their purchases.

Step-by-step explanation:

Answer:

six hunddered n twelve

Step-by-step explanation:

Answer: 150

Step-by-step explanation: cause it has six side in total of 720 and subtract it and you have 150

Answer:

The alternative hypothesis H0, should be rejected, if sample mean, X' < 25.051

Step-by-step explanation:

Given:

Sample size, n = 40

Mean, μ = 27

Significance level = 0.02

Standard deviation = 6

For null hypothesis :

H0 : μ ≥ 27

For alternative hypothesis :

H1 : μ < 27

At significance level, α = 0.02, from Z table, Zα = 2.054

This is a left tailed test

Solving for X' we have:

[tex] X' = u - Za \frac{\sigma}{\sqrt{n}}[/tex]

[tex] X' = 27 - 2.054 \frac{6}{\sqrt{40}}= 25.051[/tex]

The alternative hypothesis H0, should be rejected, if sample mean, X' < 25.051

The rejection rule is based on the value of for the test to determine whether the manufacturer's claim should be rejected is [tex]\mu<27[/tex].

Given :

The following steps can be used in order to determine the rejection rule based on the value of the test:

Step 1 - The Hypothesis test can be used in order to determine the rejection rule based on the value of the test.

The null hypothesis is given below:

[tex]H_0 : \mu\geq 27[/tex]

The alternate hypothesis is given below:

[tex]H_a : \mu<27[/tex]

Step 2 - Now, the formula of X' is given below:

[tex]X' = \mu-Z_\alpha \dfrac{\sigma}{\sqrt{n} }[/tex]

Step 3 - Now, substitute the values of the known terms in the above formula.

[tex]X' = 27-2.054 \dfrac{6}{\sqrt{40} }[/tex]

Step 4 - SImplify the above expression.

[tex]X' = 25.051[/tex]

From the above steps, it can be concluded that the null hypothesis is rejected.

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Answer:

42cm²

Step-by-step explanation:

b×h

6×7=42

area-To find the area of a rectangle multiply its height by its width. For a square you only need to find the length of one of the sides (as each side is the same length) and then multiply this by itself to find the area.

Answer:

42 sq cm

Step-by-step explanation:

6 x 7=42

Answer:

12 inches

Step-by-step explanation:

c=2*pi*r

75 = 2*3.14*r

r=75/(2*314)=75/6.28=11.9, which is close to 12

Answer:

A.   x = [  5/3   -2/3  ] [  -3  ]

      y = [  2        -1   ]  [  -9  ]

Step-by-step explanation:

got it correct on the unit test review on edge 2020

The matrix equation to solve the system of equations 3x - 2y = -3 and 6x - 5y = -9 is AX = B, where A is the coefficient matrix[tex]\(\begin{bmatrix}3 & -2 \\ 6 & -5\end{bmatrix}\)[/tex], X is the variable matrix[tex]\(\begin{bmatrix}x \\ y\end{bmatrix}\)[/tex], and B is the constant matrix[tex]\(\begin{bmatrix}-3 \\ -9\end{bmatrix}\)[/tex].

To solve the system of linear equations presented using matrices, we can set up a matrix equation of the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

The system of equations is:

From the system, we can identify the coefficient matrix A, the variable matrix X, and the constant matrix B as follows:

A =
[tex]\(\begin{bmatrix}3 & -2 \\ 6 & -5\end{bmatrix}\)[/tex]

X =
[tex]\(\begin{bmatrix}x \\ y\end{bmatrix}\)[/tex]

B =
[tex]\(\begin{bmatrix}-3 \\ -9\end{bmatrix}\)[/tex]

The matrix equation that can be used to solve the system is:

[tex]\(\begin{bmatrix}3 & -2 \\ 6 & -5\end{bmatrix}\) \(\begin{bmatrix}x \\ y\end{bmatrix}\) = \(\begin{bmatrix}-3 \\ -9\end{bmatrix}\)[/tex]

Answer:

[tex]-77\sqrt{30}[/tex]

Step-by-step explanation:

[tex](-7\sqrt{3})(11\sqrt{10})=-77\sqrt{30}[/tex]

Hope this helps!

The simplified form of the expression is  -77√33

Given the surd function (-7√3)(11√10)

Multiply the surd functions together. To do this, you multiply both the integers and the surd functions separately  as shown:

Hence the simplified form of the expression is  -77√33

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Answer:

a. Gwen expect to make 3.75 sets

b. The variance for the number of sets Gwen will make is 0.9375

c. Gwen expect to make 2 shots

Step-by-step explanation:

a. According to the given data we have the following:

Here this follows negative binomial distribution with parameter r =3 and p=0.8

To calculate how many sets does Gwen expect to make we have to calculate the following formula:

expected number of sets =r/p

expected number of sets =3/0.8=3.75

Gwen expect to make 3.75 sets.

b. In order to calculate the variance for the number of sets Gwen will make we have to use the following formula:

variance for the number of sets=σ∧2=r(1-p)/p∧2

variance for the number of sets=3*(1-0.8)/0.8^2

variance for the number of sets=0.9375

The variance for the number of sets Gwen will make is 0.9375

c. To calculate how many shots does Gwen expect to make, we have to calculate first the probability she shoots all the three in the set as follows:

probability she shoots all the three in the set=0.8∧3=0.512

if E(X)=1/p, therefore, 1/p=1/0.512=1.95

Gwen expect to make 2 shots

Answer:

choice c. 46

Step-by-step explanation:

Find a b if a = 10i + 4j and b = 3i + 4%

a = <10, 4>

b = <3, 4>

a*b = <10, 4> * <3, 4> = 10*3 + 4*4 = 30 + 16 = 46

Answer:

C. 46

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

Answer:

It is 45m2

Step-by-step explanation:

Just took did the question of the topic calculating the Base of Area of a Cone

Answer:

its 42 m^2

Step-by-step explanation: did it on edge

Answer:

Step-by-step explanation:

Corresponding heights of presidents and height of their main opponents form matched pairs.

The data for the test are the differences between the heights.

μd = the​ president's height minus their main​ opponent's height.

President's height. main opp diff

191. 166. 25

180. 179. 1

180. 168. 12

182. 183. - 1

197. 194. 3

180. 186. - 6

Sample mean, xd

= (25 + 1 + 12 - 1 + 3 + 6)/6 = 5.67

xd = 5.67

Standard deviation = √(summation(x - mean)²/n

n = 6

Summation(x - mean)² = (25 - 5.67)^2 + (1 - 5.67)^2 + (12 - 5.67)^2+ (- 1 - 5.67)^2 + (3 - 5.67)^2 + (- 6 - 5.67)^2 = 623.3334

Standard deviation = √(623.3334/6 sd = 10.19

For the null hypothesis

H0: μd ≥ 0

For the alternative hypothesis

H1: μd < 0

The distribution is a students t. Therefore, degree of freedom, df = n - 1 = 6 - 1 = 5

The formula for determining the test statistic is

t = (xd - μd)/(sd/√n)

t = (5.67 - 0)/(10.19/√6)

t = 1.36

We would determine the probability value by using the t test calculator.

p = 0.12

Since alpha, 0.05 < than the p value, 0.12, then we would fail to reject the null hypothesis.

Therefore, at 5% significance level, we can conclude that for the population of heights for presidents and their main​ opponents, the differences have a mean greater than 0 cm.

The null hypothesis in this case would be that there is no average height advantage for presidents over their main opponents (µd ≤ 0), while the alternative hypothesis is that presidents are taller on average (µd > 0). A paired t-test with a significance level of 0.05 is usually employed in testing these hypotheses using the p-value and t-score.

In hypothesis testing, the goal is to determine the validity of a claim made. In this case, the claim is that the mean difference in height, where the difference is calculated as the president's height minus their main opponent's height, is greater than 0 cm. This represents the theory that taller presidential candidates have an advantage.

For setting up a null hypothesis and an alternative hypothesis, we consider the following parameters:

To test these hypotheses, we would typically use a one-sample t-test for paired differences with a significance level (alpha) of 0.05. A p-value less than this would allow us to reject the null hypothesis in favor of the alternative hypothesis that presidents are on average taller than their main opponents. Use of p-value and t-score is essential in conducting such a test.

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Answer:

D.

Step-by-step explanation:

Answer:

D.He divided the exponents in the parentheses instead of subtracting.

Step-by-step explanation:

Edge 2022

Answer:

D, E, F

Step-by-step explanation:

The first step I would do is distribute the original equation. After distributing, the equation is now 8x² + 16xy. The first answer I see that matches this is D.

Then, after already eliminating A, B, and C, I look at E. I distribute the x and find out it is also equal to 8x² + 16xy.

Then, I look at F. After distributing again, it is also equal to 8x² + 16xy.

Answer:

A, D, and the choice that says the volume is ~261.33 metres cubed

Step-by-step explanation:

The volume of a triangular prism is denoted by: V = (1/3) * Bh, where B is the base area and h is the height.

Here, we know that the base is a triangle with base 8 and height 14, and the overall height is 7. The first step is to find the area of the base. The area of a triangle is denoted by:

A = (1/2) * b * h, where b is the base and h is the height, so A is correct.

Plug values in:

A = (1/2) * 8 * 14 = 56 metres squared, so the D is correct.

Then use this and the height of 14 to find the volume:

V = (1/3) * Bh

V = (1/3) * 56 * 14 = 784/3 metres cubed (I'm assuming you missed an answer choice when copying the problem on here, so the correct last option is the one that says the volume is 784/3 or ~261.33 metres cubed)

Answer:

Use the formula A = one-half b h to find the area of the base.

The area of the base, A, is One-half (8) (14) = 56 meters squared.

The volume of the prism, V is (56) (7) = 392 meters cubed.

Step-by-step explanation:

Volume of prism:

Base area × height

Base area:

½ × 8 × 14 = 56

Volume:

56 × 7 = 392

Answer:x-2y

Step-by-step explanation:

-1/2(-2x+4y)

Open the brackets

2x/2 - 4y/2

x - 2y

Answer:

452.16

Step-by-step explanation:

Area of a circle = pi*radius squared

A= 3.14(12)^2

=3.14*144

=452.16

Final answer:

The area of a circle with a radius of 12 cm can be calculated using the formula A = \u03C0r^2. By applying the radius to this formula with pi approximated to 3.14, we obtain an area of 452.16 cm^2, which corresponds to option D.

Explanation:

To calculate the area of the circle with a radius of 12 cm, we use the formula: A = \\u03C0r^2\

Where (pi) is approximately 3.14 and r is the radius of the circle.

Plugging the radius into the formula:

A = 3.14 * (12 cm)^2

A = 3.14 * 144 cm^2

A = 452.16 cm^2

Thus, the correct answer is D. 452.16 cm^2.

Answer:

the answer is 262 days

Step-by-step explanation:

To find the second time after the hottest day of the year that the daily high temperature is 20 degrees Celsius, you need to solve the equation T(t) = 20. This involves finding the inverse cosine of a specific value, setting up an equation, and adding one year to the solution. After performing these steps, you can find the value of t that corresponds to the second time.

To find the second time after the hottest day of the year that the daily high temperature is 20 degrees Celsius, we need to solve the equation T(t) = 20. We can rewrite this equation as 7.5cos(2π/365t) + 21.5 = 20. Subtracting 21.5 from both sides gives us 7.5cos(2π/365t) = -1.5. Dividing both sides by 7.5 and simplifying further, we have cos(2π/365t) = -0.2. To find the second time, we need to find the value of t that satisfies this equation.

To find the value of t, we need to use the inverse cosine function (also known as arccosine). The inverse cosine function (cos^(-1)) gives us the angle whose cosine is a specific value. In this case, we want to find t such that cos(2π/365t) = -0.2. We can use a calculator or math software to find the inverse cosine of -0.2. Let's assume the inverse cosine of -0.2 is x.

Now we can set up an equation: 2π/365t = x. Solving for t, we get t = (365x)/(2π). However, we need to find the second time after the hottest day, so we need to find the value of t that satisfies the equation after adding one year (365 days) to the original value. Therefore, the second time after the hottest day of the year that the daily high temperature is 20 degrees Celsius is t = (365x)/(2π) + 365.

Answer:

Best predicted value of y' = 86.16 kg

Step-by-step explanation:

Given,

n = 100

Range of heights = 138 - 190cm

Range of weight = 39 to 150 kg

x' =167.46 cm

y' = 81.44 kg

r = 0.108

p value = 0.285

y = - 105 + 1.08x

Significance level = 0.05

We reject H0 since pvalue, 0.285 is less than significance level of 0.05.

Therefore,

Given height of adult male, x = 177 cm

y = - 105 + 1.08x

The best predicted value of y' =

y' = - 105 + 1.08(177)

y' = 86.16 kg

The best predicted value of y' is 86.16kg

Answer:

Yes. At this significance level, there is evidence to support the claim that there is a difference in the ability of the brands to absorb water.

Step-by-step explanation:

The question is incomplete:

The significance level is 0.05.

The data is:

Brand X: 91, 100, 88, 89

Brand Y: 99, 96, 94, 99

Brand Z: 83, 88, 89, 76

We have to check if there is a significant difference between the absorbency rating of each brand.

Null hypothesis: all means are equal

[tex]H_0:\mu_x=\mu_y=\mu_z[/tex]

Alternative hypothesis: the means are not equal

[tex]H_a: \mu_x\neq\mu_y\neq\mu_z[/tex]

We have to apply a one-way ANOVA

We start by calculating the standard deviation for each brand:

[tex]s_x^2=30,\,\,s_y^2=6,\,\,s_z^2=35.33[/tex]

Then, we calculate the mean standard error (MSE):

[tex]MSE=(\sum s_i^2)/a=(30+6+35.33)/3=71.33/3=23.78[/tex]

Now, we calculate the mean square between (MSB), but we previously have to know the sample means and the mean of the sample means:

[tex]M_x=92,\,\,M_y=97,\,\,M_z=84\\\\M=(92+97+84)/3=91[/tex]

The MSB is then:

[tex]s^2=\dfrac{\sum(M_i-M)^2}{N-1}\\\\\\s^2=\dfrac{(92-91)^2+(97-91)^2+(84-91)^2}{3-1}\\\\\\s^2=\dfrac{1+36+49}{2}=\dfrac{86}{2}=43\\\\\\\\MSB=ns^2=4*43=172[/tex]

Now we calculate the F statistic as:

[tex]F=MSB/MSE=172/23.78=7.23[/tex]

The degrees of freedom of the numerator are:

[tex]dfn=a-1=3-1=2[/tex]

The degrees of freedom of the denominator are:

[tex]dfd=N-a=3*4-3=12-3=9[/tex]

The P-value of F=7.23, dfn=2 and dfd=9 is:

[tex]P-value=P(F>7.23)=0.01342[/tex]

As the P-value (0.013) is smaller than the significance level (0.05), the null hypothesis is rejected.

There is evidence to support the claim that there is a difference in the ability of the brands to absorb water.

Answer:

Yes. We have evidence to support the claim that there is a difference in the use of the four entrances.

Step-by-step explanation:

The question is incomplete:

Entrance Frequency

Main Street 81

Broad Street 129

Cherry Street 72

Walnut Street 119

Total: 401

The building maintenance supervisor wants to know if the entrances are equally utilized.

This problem can be solved using the Chi-square goodess of fit test.

The expected value for each door is

[tex]E=401/4=100.25[/tex]

The degrees of freedom are equal to the number of categories (4 doors) minus one:

[tex]df=n-1=4-1=3[/tex]

Then, the value of the chi-square statistic can be calculated as:

[tex]\chi^2=\sum \dfrac{(O_i-E)^2}{E}\\\\\\\chi^2=\dfrac{(81-100.25)^2}{100.25}+\dfrac{(129-100.25)^2}{100.25}+\dfrac{(72-100.25)^2}{100.25}+\dfrac{(119-100.25)^2}{100.25}\\\\\\\chi^2=\dfrac{370.5625+826.5625+798.0625+351.5625}{100.25}=\dfrac{2346.75}{100.25}=23.41[/tex]

The P-value for this test statistic χ^2=23.41 and df=3 is:

[tex]P-value=P(\chi^2_3>23.41)=0.00003[/tex]

This P-value is much smaller than the significance level (0.01), so the effect is significant.

We have evidence to support the claim that there is a difference in the use of the four entrances.

Answer:

The answer is constant multiplicative and it is 1.05 times larger.

There is constant rate of change in an exponential relationship.

The value of the account is 1.05 times.

In the given statement, it states that there is a [ ] rate of change in an exponential relationship. The missing word in this case would be "constant."

In an exponential relationship, the rate of change between consecutive terms is constant.

Now, after each year, the value of the account is [. ] times as large as the previous year. The missing value in this case would be "1.05."

This indicates that the value of the account increases by a factor of 1.05 each year, which corresponds to a 5% annual growth rate.

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Answer:

A.

I say this is the answer because if she has gotten into a habait of buying and breaking glasses,shes just very careless

Answer:

hope she won

t

Step-by-step explanation:

We have been given that Layana’s house is located at [tex](2\frac{2}{3}, 7\frac{1}{3})[/tex] on a map. The store where she works is located at [tex](-1\frac{1}{3}, 7\frac{1}{3})[/tex].

We are asked to find the distance from Layana’s home to the store

We will use distance formula to solve our given problem.

Let us convert our given coordinates in improper fractions.

[tex]2\frac{2}{3}\Rightarrow \frac{8}{3}[/tex]

[tex]7\frac{1}{3}\Rightarrow \frac{22}{3}[/tex]

[tex]-1\frac{1}{3}\Rightarrow -\frac{4}{3}[/tex]

Now we will use distance formula to solve our given problem.

[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Upon substituting coordinates of our given point in above formula, we will get:

[tex]D=\sqrt{(\frac{22}{3}-\frac{22}{3})^2+(\frac{8}{3}-(-\frac{4}{3}))^2}[/tex]

[tex]D=\sqrt{(0)^2+(\frac{8}{3}+\frac{4}{3})^2}[/tex]

[tex]D=\sqrt{0+(\frac{8+4}{3})^2}[/tex]

[tex]D=\sqrt{(\frac{12}{3})^2}[/tex]

[tex]D=\sqrt{(4)^2}[/tex]

[tex]D=4[/tex]

Therefore, the distance from Layana's home to the store is 4 units and option A is the correct choice.

Answer:

its A ^3^

Step-by-step explanation:

Answer:

  see the bullet list below

Step-by-step explanation:

Given the expression: p² -3 +3p -8 +p +p³

The following statements are true:

_____

Terms are generally separated by + or - signs. (The sign is considered to be part of the term.) In the context of a polynomial, terms may be constants, or may be a product with factors that are constants or variables.

_____

Further comments on "term"

In other contexts, the word "term" is used for various purposes. It can designate a member of a sequence, the left or right side of an equation, the numerator or denominator of a rational expression, or just about any identifiable expression that can be considered as a unit. Whereas "coefficient" or "factor" may apply to just about any subset of the (prime) factors of a product, the word "term" is generally restricted to consideration of the product as a whole.

In the given expression, -3 and -8 are like terms, while 3p and p are also like terms. The expression contains six terms and like terms have the same variables raised to the same powers. However, not all terms with variables are like terms in this instance.

The expression given is p squared minus 3 + 3p minus 8 + p + p cubed. When we look into it, we can see a couple of true statements.

However, the terms p squared, 3p, p, and p cubed are not like terms since the powers of p in each term are different. Similarly, the terms p squared and p cubed are not like terms since the powers of p are 2 and 3, which are not the same.

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Answer:

"The probability that the amount of time spent on leisure activities per day for a randomly selected adult from the population of interest is less than 6 hours per day" is about 0.8749.

Step-by-step explanation:

We have here a random variable that is normally distributed, namely, the time spent on leisure activities by adults living in a household with no young children.

The normal distribution is determined by two parameters: the population mean, [tex] \\ \mu[/tex], and the population standard deviation, [tex] \\ \sigma[/tex]. In this case, the variable follows a normal distribution with parameters [tex] \\ \mu = 4.5[/tex] hours per day and [tex] \\ \sigma = 1.3[/tex] hours per day.

We can solve this question following the next strategy:

We use the standard normal distribution because we can "transform" any raw score into standardized values, which represent distances from the population mean in standard deviations units, where a positive value indicates that the value is above the mean and a negative value that the value is below it. A standard normal distribution has [tex] \\ \mu = 0[/tex] and [tex] \\ \sigma = 1[/tex].

The formula for the z-scores is as follows

[tex] \\ z = \frac{x - \mu}{\sigma}[/tex] [1]

Solving the question

Using all the previous information and using formula [1], we have

x = 6 hours per day (the raw score).

[tex] \\ \mu = 4.5[/tex] hours per day.

[tex] \\ \sigma = 1.3[/tex] hours per day.

Then (without using units)

[tex] \\ z = \frac{x - \mu}{\sigma}[/tex]

[tex] \\ z = \frac{6 - 4.5}{1.3}[/tex]

[tex] \\ z = \frac{1.5}{1.3}[/tex]

[tex] \\ z = 1.15384 \approx 1.15[/tex]

We round the value of z to two decimals since most standard normal tables only have two decimals for z.

We can observe that z = 1.15, and it tells us that the value is 1.15 standard deviations units above the mean.

With this value for z, we can consult the cumulative standard normal table, and for this z = 1.15, we have a cumulative probability of 0.8749. That is, this table gives us P(z<1.15).  

We can describe the procedure of finding this probability in the next way: At the left of the table, we have z = 1.1; we can follow the first line on the table until we find 0.05. With these two values, we can determine the probability obtained above, P(z<1.15) = 0.8749.

Notice that the probability for the z-score, P(z<1.15), of the raw score, P(x<6) are practically the same,  [tex] \\ P(z<1.15) \approx P(x<6)[/tex]. For an exact probability, we have to use a z-score = 1.15384 (without rounding), that is, [tex] \\ P(z<1.15384) = P(x<6) = 0.8757[/tex]. However, the probability is approximated since we have to round z = 1.15384 to z = 1.15 because of the use of the table.

Therefore, "the probability that the amount of time spent on leisure activities per day for a randomly selected adult from the population of interest is less than 6 hours per day" is about 0.8749.

We can see this result in the graphs below. First, for P(x<6) in [tex] \\ N(4.5, 1.3)[/tex] (red area), and second, using the standard normal distribution ([tex] \\ N(0, 1)[/tex]), for P(z<1.15), which corresponds with the blue shaded area.

Final answer:

The question seeks to find the probability that an adult spends less than 6 hours per day on leisure activities, using the given mean and standard deviation for a normal distribution. The z-score is calculated and then used to determine the probability using the cumulative normal distribution function.

Explanation:

The student's question asks to find the probability that a randomly selected adult from a certain population spends less than 6 hours per day on leisure activities, given that the distribution of time spent is normally distributed with a mean of 4.5 hours and a standard deviation of 1.3 hours.

To solve this, you can use the z-score formula:

z = (X - μ) / σ

where X is the value of interest (6 hours), μ is the mean (4.5 hours), and σ is the standard deviation (1.3 hours).

Using this, we calculate:

z = (6 - 4.5) / 1.3

= 1.15 / 1.3

= 0.8846

Now, we look up this z-score in a standard normal distribution table or use a calculator with the normal distribution function to find the corresponding probability.

Assuming normal CDF is the function for cumulative normal distribution:

probability = CDF(-∞, 0.8846, 0, 1)

This will give us the probability that the adult spends less than 6 hours per day on leisure activities. Remember, the cumulative distribution function gives the area to the left of the z-score, which corresponds to the probability of obtaining a value less than the one of interest.

Answer:

22 cm

Step-by-step explanation:

The perimeter is the distance all the way around a shape. To find the perimeter, add up all the sides

The side lengths are 5, 8 and 9

5+8+9=22

So, the perimeter is 22 centimeters.

Answer:

y = 41/6

Step-by-step explanation:

This means that  y = kx.    k is a constant.

6 = k*72

and.

y = k*82 ...

k = 6/72 = 1/12

y = 82/12

y = 41/6

Answer:

95​% confidence interval for the mean time required to earn a bachelor’s degree by all college students is [5.10 years , 5.20 years].

Step-by-step explanation:

We are given that the National Center for Education Statistics surveyed a random sample of 4400 college graduates about the lengths of time required to earn their bachelor’s degrees. The mean was 5.15 years and the standard deviation was 1.68 years respectively.

Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by;

                              P.Q. =  [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]  ~ N(0,1)

where, [tex]\bar X[/tex] = sample mean time = 5.15 years

            [tex]\sigma[/tex] = sample standard deviation = 1.68 years

            n = sample of college graduates = 4400

            [tex]\mu[/tex] = population mean time

Here for constructing 95% confidence interval we have used One-sample z test statistics although we are given sample standard deviation because the sample size is very large so at large sample values t distribution also follows normal.

So, 95% confidence interval for the population mean, [tex]\mu[/tex] is ;

P(-1.96 < N(0,1) < 1.96) = 0.95  {As the critical value of z at 2.5%

                                               level of significance are -1.96 & 1.96}  

P(-1.96 < [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.96) = 0.95

P( [tex]-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.95

P( [tex]\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.95

95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ]

                                              = [ [tex]5.15-1.96 \times {\frac{1.68}{\sqrt{4400} } }[/tex] , [tex]5.15+1.96 \times {\frac{1.68}{\sqrt{4400} } }[/tex] ]

                                             = [5.10 , 5.20]

Therefore, 95​% confidence interval for the mean time required to earn a bachelor’s degree by all college students is [5.10 years , 5.20 years].