the area of a regular pentagon with a radius of 7 cm is

Answer:

4

Step-by-step explanation:

[tex]f(x) = - x + 5 \\ \: f(1) = - 1+ 5 = 4 \\ \: f(5) = - 5+ 5 = 0\\ f(1) +f(5) =4 + 0 \\ \huge \red{ \boxed{f(1) +f(5) = 4}}[/tex]

Answer:

value after 5 years = $23,399.91

after 10 years = $27,377.79

Time it takes for the amount to double = 22.07 years

Step-by-step explanation:

For amounts that are compounded continuously, it means that the interest rate is is added to the investment amount at an infinite number of time, and the formula is given as:

A = P [tex]e^{r.t}[/tex], where:

A = Future value

P = present value

e = constant ≈ 2.7183

r = interest rate in decimal form

t = years

Now for value after 5 years;

A = ???

P = $20,000

r = 3.14% = 0.0314

t =  5 years

∴ A = P [tex]e^{r.t}[/tex]

= 20,000 [tex]e^{0.0314*5}[/tex]

= 20,000 × [tex]e^{0.157}[/tex] = 20,000 × 1.169995 = $23,399.91 ( to 2 decimal places)

(Note that the function '[tex]e[/tex]' can be punched directly from the calculator)

value after 10 years;

A = ???

P = $20,000

r = 3.14% = 0.0314

t =  10 years

∴ A = P[tex]e^{r.t}[/tex]

= 20,000 × [tex]e^{0.0314 * 10}[/tex]

= 20,000 × [tex]e^{0.314}[/tex] = $27,377.79 ( to 2 decimal places)

Time it will take to double the original investment;

A = P [tex]e^{r.t}[/tex]

where;

A = 40,000

P = 20,000

r = 0.0314

t =???

40,000 = 20,000 × [tex]e^{0.0314 * t}[/tex]

[tex]\frac{40,000}{20,000} = \frac{20,000}{20,000} * e^{0.0314*t}[/tex] (divide both sides by 20,000)

2 = [tex]e^{0.0314 * t}[/tex]

Next take the natural logarithm of both sides

㏑(2) = ㏑[tex]e^{0.0314 *t}[/tex]      (㏑[tex]e[/tex] = 1; and the exponent can be brought down )

= 0.6931 = 0.0314 × t × 1

∴ t = [tex]\frac{0.06931}{0.0314}[/tex] = 22.07 years ( to 2 decimal places)

Calculate the value of an investment after 5 and 10 years with continuous compounding at a certain interest rate. Determine the time needed for an investment to double using the continuous compounding formula.

When calculating the future value of an investment with compound interest, we can use the formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial sum of money), r is the annual interest rate (as a decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm.

Applying this formula, the future value of a $20,000 investment at a 3.14% interest rate compounded continuously for 5 years would be

Answer:

(a) 1/3

(b) 1/15

Step-by-step explanation:

(a)Let X denote the waiting time in minutes. It is given that X follows a uniform distribution, and since the variable being measured is time,  we assume it to be a continuous uniform distribution.

[tex] \[f_X(x) =\begin{cases} \frac{1}{30} & 0\leqx\leq 30\\ 0 & otherwise \end{cases}\][/tex]

Now

[tex] P(X>20) = \int_{20}^{30}f_X(x) = \frac{1}{30} \times 10 = \frac{1}{3}[/tex].

Jack or Rose arriving first is equally likely, therefore the probability of Jack waiting is just the half of the above obtained probability i.e [tex]\frac{1}{6}[/tex]

(b)Using the formula for the expectation of a uniform continuous distribution,

[tex] E(X) = \frac{30+0}{2} = 15[/tex]

Answer:

0.1056 = 10.56% probability that the concentration exceeds 0.60

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 0.5, \sigma = 0.08[/tex]

What is the probability that the concentration exceeds 0.60?

This is 1 subtracted by the pvalue of Z when X = 0.6. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{0.6 - 0.5}{0.08}[/tex]

[tex]Z = 1.25[/tex]

[tex]Z = 1.25[/tex] has a pvalue of 0.8944

1 - 0.8944 = 0.1056

0.1056 = 10.56% probability that the concentration exceeds 0.60

Answer:

  40 square inches

Step-by-step explanation:

The shaded area is the area of a triangle with base 10 in and height 8 in. It is given by the area formula ...

  A = (1/2)bh = (1/2)(10 in)(8 in) = 40 in²

The shaded area is 40 square inches.

Answer: the resale value would be $791 after 4 years

Step-by-step explanation:

We would apply the formula for exponential decay which is expressed as

y = b(1 - r)^x

Where

y represents the value of the laptop computer after x years.

x represents the number of years.

b represents the initial value of the laptop computer.

r represents rate of decay.

From the information given,

P = $2500

x = 4

r = 25% = 25/100 = 0.25

Therefore,

y = 2500(1 - 0.25)^4

y = 2500(0.75)^4

y = $791

Answer:

A is correct, "All numbers bigger than -3 and smaller than 3"

Step-by-step explanation:

The open circle means greater than, if it was filled in, it would be greater than or equal to. And the black line connecting each shows its more than -3 and less than 3. Hope this helps! Please rate brainliest if it does :)

Answer:x=(3d-2c)/(a-b)

Step-by-step explanation:

ax+2c=bx+3d

Collect like terms

ax-bx=3d-2c

x(a-b)=3d-2c

Divide both sides by (a-b)

x(a-b)/(a-b) =(3d-2c)/(a-b)

x=(3d-2c)/(a-b)

The linear equation in two variables is the equation of a straight line. The given formula can be rewritten as   x = (bx + 3d - 2c) / a.

A linear equation is a equation that has degree as one.

To find the solution of n unknown quantities n number of equations with n number of variables are required.

The given equation is as follows,

ax + 2c = bx + 3d

It can be simplified and written in terms of x as,

ax + 2c = bx + 3d

=> ax = bx + 3d - 2c

=> x = (bx + 3d - 2c) / a

Hence, by making x as the subject the given formula can be written as,

x = (bx + 3d - 2c) / a.

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The loan with the shortest term, which is 5 days in this case, is likely to have the highest APR because the finance charge would have less time to be spread out.

The loan that is likely to have the highest Annual Percentage Rate (APR) is the one with the shortest term when all other factors are equal. In this scenario, that would be the loan with a term of 5 days. This is due to the fact that APR is calculated by the annualizing the interest and fees associated with a loan, meaning the shorter the term of a loan, the higher the APR. For instance, a $500 loan with a $20 finance charge has a much higher APR over 5 days as compared to 30 days or 90 days as the finance charge would have lesser time to be spread out.

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Answer:

64

Step-by-step explanation:

The answer is 64, because 4 to the 3rd power is 64.  When finding the volume of a cube, find the length of one side to the 3rd.

Answer:

Therefore surface integral is [tex]\pi(a^2+b^2)c-0-0=\pi(a^2+b^2)c[/tex].

Step-by-step explanation:

Given function is,

[tex]\vec{F}=\frac{bx}{a}\uvec{i}+\frac{ay}{b}\uvec{j}[/tex]

To find,

[tex]\int\int_{S}\vec{F}dS[/tex]  

where S=A=surfece of elliptic cylinder we have to apply Divergence theorem so that,

[tex]\int\int_{S}\vec{F}dS[/tex]

[tex]=\int\int\int_V\nabla.\vec{F}dV[/tex]

[tex]=\int\int\int_V(\frac{b}{a}+\frac{a}{b})dV[/tex]  

[tex]=\frac{a^2+b^2}{ab}\int\int\int_VdV[/tex]

[tex]=\frac{a^2+b^2}{ab}\times \textit{Volume of the elliptic cylinder}[/tex]

[tex]=\frac{a^2+b^2}{ab}\times \pi ab\times 2c=\pi (a^2+b^2)c[/tex]

[tex]\int\int_{S_1}\vex{F}.dS_1=\int\int_{S_1}<\frac{bx}{a}, \frac{ay}{b}, 0> . <-z_x,z_y,1>dA[/tex]      

[tex]=\int\int_{S_1}<\frac{bx}{a},\frac{ay}{b}, 0>.<0,0,1>dA=0[/tex]

[tex]\int\int_{S_2}\vex{F}.dS_2=\int\int_{S_2}<\frac{bx}{a}, \frac{ay}{b}, 0>. -<-z_x,z_y,1>dA[/tex]      

[tex]=\int\int_{S_2}<\frac{bx}{a},\frac{ay}{b}, 0>. -<0,0,1>dA=0[/tex]

Therefore surface integral without unit vector of the surface is,

[tex]\pi(a^2+b^2)c-0-0=\pi(a^2+b^2)c[/tex]

The value of ∫SF ⋅dA where F =(bx/a)i +(ay/b)j and S is the elliptic cylinder oriented away from the z-axis is 2πc(a² + b²).

From the information, F = (b/ax) + (a/by)j and S is the elliptic cylinder.

To evaluate ∫F.dA goes thus:

divF = (I'd/dx + jd/dx + kd/dx) × (b/ax)i + (a/by)j

= b/a + a/b

= (a² + b²)/ab

Using Gauss divergence theorem, this will be further solved below:

∫∫∫v(a² + b²/ab)dV

= (a² + b²/ab)∫∫∫vdV

= (a² + b²/ab) × Volume of cylinder

= (a² + b²/ab) × πab(2c)

= 2πc(a² + b²)

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Answer:

Given:

P=950

r=0.07

t=8

F=950(1.07)^8

= 1632.28 (total amount)

Interest = total amount - principal

=1632.28 - 950

=682.28

Step-by-step explanation:

The formula is:

Future value = accumulated amount, F = P(1+r)^t

P=principal

r=annual interest rate [compounded annually]

t=number of years of loan

Answer:

99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].

Step-by-step explanation:

We are given that a sample of 1584 third graders, the mean words per minute read was 35.7. Assume a population standard deviation of 3.3.

Firstly, the pivotal quantity for 99% 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 words per minute read = 35.7

            [tex]\sigma[/tex] = population standard deviation = 3.3

            n = sample of third graders = 1584

            [tex]\mu[/tex] = population mean number of words

Here for constructing 99% confidence interval we have used One-sample z test statistics as we know about the population standard deviation.

So, 99% confidence interval for the population mean, [tex]\mu[/tex] is ;

P(-2.58 < N(0,1) < 2.58) = 0.99  {As the critical value of z at 0.5% level

                                                  of significance are -2.58 & 2.58}  

P(-2.58 < [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 2.58) = 0.99

P( [tex]-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99

P( [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ) = 0.99

99% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] , [tex]\bar X+2.58 \times {\frac{\sigma}{\sqrt{n} } }[/tex] ]

                                                 = [ [tex]35.7-2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] , [tex]35.7+2.58 \times {\frac{3.3}{\sqrt{1584} } }[/tex] ]

                                                 = [35.5 , 35.9]

Therefore, 99% confidence interval for the true mean number of words a third grader can read per minute is [35.5 , 35.9].

Answer:

5

Step-by-step explanation:

8 s^2 - t^3

Let s =2 and t=3

8 (2)^2 - (3)^3

Exponents first

8 *4 - 27

Then multiply and divide

32 -27

5

Answer:

The null hypothesis is rejected (P-value=0.0 28).

There is  enough evidence to support the claim that that Company A tires outlast the tires of Company B by more than 10,000 miles.

Step-by-step explanation:

This is a hypothesis test for the difference between populations means.

The claim is that that Company A tires outlast the tires of Company B by more than 10,000 miles.

Then, the null and alternative hypothesis are:

[tex]H_0: \mu_1-\mu_2=10000\\\\H_a:\mu_1-\mu_2> 10000[/tex]

being μ1: average for Company A and μ2: average for Company B.

The significance level is 0.05.

The sample 1, of size n1=16 has a mean of 63,500 and a standard deviation of 4,000.

The sample 1, of size n1=12 has a mean of 49,500 and a standard deviation of 6,000.

The difference between sample means is Md=14,000.

[tex]M_d=M_1-M_2=63500-49500=14000[/tex]

The estimated standard error of the difference between means is computed using the formula:

[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{4000^2}{16}+\dfrac{6000^2}{12}}\\\\\\s_{M_d}=\sqrt{1000000+3000000}=\sqrt{4000000}=2000[/tex]

Then, we can calculate the t-statistic as:

[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{14000-10000}{2000}=\dfrac{4000}{2000}=2[/tex]

The degrees of freedom for this test are:

[tex]df=n_1+n_2-1=16+12-2=26[/tex]

This test is a right-tailed test, with 26 degrees of freedom and t=2, so the P-value for this test is calculated as (using a t-table):

[tex]P-value=P(t>2)=0.028[/tex]

As the P-value (0.028) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is  enough evidence to support the claim that that Company A tires outlast the tires of Company B by more than 10,000 miles.

In statistics, you can use a two-sample t-test to compare the lifespans of two types of tires. After setting up the null and alternative hypotheses, we use the alpha level, test statistic, and p-value to decide whether to reject the null hypothesis and accept the company's claim.

To clarify, testing this claim about tire lifespan falls under the category of hypothesis testing within statistics. In your case, you're comparing the lifespan of two types of tires which involves a two-sample t-test. Unfortunately, you've only provided raw data, but let's tackle this conceptually.

First, we set up the null and alternative hypotheses. The null hypothesis, usually denoted as H0, asserts that there's no difference between the means of the two populations. In this case, it says the lifespan of Company A's tires are the same as Company B's tires.

The alternative hypothesis, usually denoted as Ha, is the claim you're trying to test - in this case, that Company A's tires do last 10,000 miles longer than Company B's tires. In this hypothesis test, your alpha, which is the threshold of how much you're willing to be wrong, is 0.05.

After calculating the t-score and finding the p-value using statistical software or a t-distribution table, you compare the p-value to the alpha. If the p-value is less than the alpha, that means the result is statistically significant and thus you reject the null hypothesis.

Based on the decision and reason provided in your question, the conclusion would be that there is sufficient evidence at the 0.05 level to support the claim that Company A's tires outlast Company B's tires by more than 10,000 miles.

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Answer:

180 ways

Step-by-step explanation:

Aerospace companies (A) = 5

Energy development companies (D) = 3

Electronics companies (E) = 4

Number of aerospace stocks bought (a) = 2

Number of Energy development stocks bought (d) = 2

Number of Electronics stocks bought (e) = 2

The number of ways that the investor can select his six investments from the recommended list is given by the combination of picking two out five aerospace companies, multiplied by picking two out three energy development companies, multiplied by picking two out four electronics companies:

[tex]n=\frac{5!}{(5-2)!2!}*\frac{3!}{(3-2)!2!}*\frac{4!}{(4-2)!2!}\\ n=10*3*6\\n=180\ ways[/tex]

There are 180 ways for the investor to select the six companies.

There are 180 ways for the investor to select the group of six companies from the recommended list of aerospace, energy development, and electronics companies.

To determine the number of ways the investor can select the group of six companies, we need to use combinations. The combination formula is given by C(n, k) = n! / [k!(n-k)!], where n is the total number of items, and k is the number of items to choose.

First, we calculate the ways to choose the aerospace companies:

Next, calculate the ways to choose the energy development companies:

Finally, calculate the ways to choose the electronics companies:

To find the total number of ways to select the group of six companies, multiply the three results:

Answer:

1. 5000 = N

2. 50 = σ

3. 95% = confidence level (1-α)

4. α = 0.05

5. α/2=0.025=2.5%

6. z_(α/2)=-1.96

1. Width of confidence interval UL-LL=20

Step-by-step explanation:

We have to calculate a 95% confidence interval for the mean (average number of gallons of gas bought per month by the residents of this town).

The margin of error has to be below 10 gallons/month.

The population standard deviation is considered 50 gallons/month.

1. 5000 = N.

This is the population size N for this study, as this is the total population of the town.

2. 50 = σ

This is the value of the population standard deviation σ, as it is estimated from other studies.

3. 95% = (1-α)

This is the confidence level of the interval, and is equal to 1 less the significance level α.

4. α = 0.05

This is calculated from the confidence level. As the confidence level is 95%, the level of significance is 5%.

[tex]1-\alpha=0.95\\\\\alpha=1-0.95\\\\\alpha=0.05[/tex]

5. α/2=0.05/2=0.025=2.5%

6. The value os z_α/2 is obtained from a standard normal distribution table, where:

[tex]P(z<z_{\alpha/2})=0.025\\\\z_{\alpha/2}=-1.96[/tex]

z_α/2=-1.96

1. As the margin of error is 10 (maximum value), the difference between upper and lower bound is:

[tex]UL-LL=2*MOE=2*10=20[/tex]

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

SST = 1800

SSR = 1512.376

SSE = 287.624

b

coefficient of determination  is [tex]r^2 \approx 0.8402[/tex]

What this is telling us is that 84.02% variation in dependent variable y can be fully explained by variation in the independent variable x

c

The correlation coefficient is   [tex]r = 0.917[/tex]

Step-by-step explanation:

The table shown the calculated mean is shown on the second uploaded image

Let first define some term

SST (sum of squares total) : This is the difference between the noted dependent variable and the mean of this noted dependent variable

SSR(sum of squared residuals) : this can defined as a predicted shift from the actual observed values of the data  

SSE (sum of squared estimate of errors): this can be defined as the  sum of the square difference between the observed value and its mean

From the table

  [tex]SST = SS_{yy} = 1800[/tex]

  [tex]SSR = \frac{SS^2_{xy}}{SS_{xx}} = \frac{4755^2}{14950} = 1512.376[/tex]

  [tex]SSE =SST-SSR[/tex]

          [tex]=1800 - 1512.376[/tex]

           [tex]= 287.62[/tex]

The coefficient of determination is mathematically represented as

               [tex]r^2 = \frac{SSR}{SST}[/tex]

                    [tex]= 1-\frac{SSE}{SST}[/tex]

                   [tex]r^2= 1-\frac{287.6237}{1800}[/tex]

                  [tex]r^2 \approx 0.8402[/tex]

The correlation coefficient is mathematically represented as

           [tex]r = \pm\sqrt{r^2}[/tex]

Substituting values

            [tex]r = \sqrt{0.84020}[/tex]

               [tex]r = 0.917[/tex]

this value is + because the value of the coefficient of x in estimated regression equation([tex]24.9 + 0.301x,[/tex]) is positive

To compute SST, SSR, and SSE, we calculate the variability of the dependent variable around the mean, the variability explained by the regression model, and the variability not explained by the regression model.

To compute SST, SSR, and SSE, we need to understand what each of them represents. SST (the total sum of squares) measures the total variability of the dependent variable (overall score) around the mean. SSR (the regression sum of squares) measures the amount of variability in the dependent variable that is explained by the regression model. Finally, SSE (the error sum of squares) measures the amount of variability in the dependent variable that is not explained by the regression model.

To compute these values:

Hence, SST = 5424, SSR = 10435.558, and SSE = 171.442.

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Answer:

15

Step-by-step explanation:

Use Pythagorean theorem.

c² = a² + b²

c² = 9² + 12²

c² = 225

c = 15

Answer:

c.

Step-by-step explanation:

Answer:

Volume of prism is equal to [tex]96[/tex] units

Step-by-step explanation:

let the side of the cube be of one unit.

Then, the area of the base is equal to [tex]16[/tex] connecting cubes

Assuming a square base of the rectangular prism.

Length of one side of the base of rectangular prism [tex]= 4[/tex] units

Similarly, width of the base of rectangular prism [tex]= 4[/tex] units

The height of the rectangular prism is [tex]6[/tex] units

Volume of the prim [tex]=[/tex] Length [tex]*[/tex] Width  [tex]*[/tex] Height

Substituting the given values in above equation, we get -

[tex]V = 4* 4* 6\\V = 96[/tex]

Volume of prism is equal to [tex]96[/tex] units

Final answer:

To find the volume of a rectangular prism, multiply the base area by the height. In this case, the volume of Lily's rectangular prism would be 96 cubic units.

Explanation:

The volume of a rectangular prism is calculated by multiplying the length, width, and height of the prism.

For this specific rectangular prism with a base area of 16 connecting cubes and 6 layers high, we can calculate the volume as follows:

Volume = Base Area x Height

Volume = 16 x 6

Volume = 96 cubic units

Answer:

5/6

Step-by-step explanation:

2/3 is equivalent to 4/6

4/6 plus 1/6= 5/6

5/6

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Answer:

The answer is 5/6

Step-by-step explanation:

Answer:

B. The mean amount of [tex]CO_2[/tex] emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg

Step-by-step explanation:

A Type II error is the failure to reject a false null hypothesis.

Given the null and alternate hypothesis of a fuel company which wants to test a new type of gasoline designed to have lower [tex]CO_2[/tex] emissions.:

[tex]H_0: \mu = 8.9 kg\\H_a: \mu < 8.9 kg \\\text{ (where \mu is the mean amount of CO_2 emitted by burning one gallon of this new gasoline)}[/tex]

where [tex]\mu[/tex] is the mean amount of [tex]CO_2[/tex] emitted by burning one gallon of this new gasoline.

If the null hypothesis is false, then:

[tex]H_a: \mu < 8.9 kg[/tex]

A rejection of the alternate hypothesis above will be a Type II error.

Therefore:

The Type II error is: (B) The mean amount of [tex]CO_2[/tex]  emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

You can use the definition of type 2 error to find out which of the given option describes the type 2 error.

The Option which indicates Type II error is:

Option B. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

Firstly the whole story starts from hypotheses. The null hypothesis is tried to reject and we try to accept the alternate hypothesis.

The negative is just like the doctor's test getting negative means no disease. Similarly, if we conclude null hypothesis negative means it is accepted. If it is accepted wrongly means the negative test result was false, thus called false negative. This error is called type II error.

Since the null hypothesis here is [tex]H_0: \text{typically emitted } {\rm CO_2} = 8.9 \: \rm kg[/tex],

Thus the type II error would be when we fail to reject that CO2 emission is 8.9 kg (or say  we accept that CO2 emission is 8.9 kg generally) but the actual amount was lower than 8.9 kg(the alternate hypothesis was true)

Thus,

The Option which indicates Type II error is:

Option B. The mean amount of CO2 emitted by the new fuel is actually lower than 89 kg but they fall to conclude it is lower than 89 kg.

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Answer:

It will take 41.3 minutes for the element to decay to 40 grams

Step-by-step explanation:

The amount of element after t minute is given by the following equation:

[tex]x(t) = x(0)e^{-rt}[/tex]

In which x(0) is the initial amount and r is the rate that it decreases.

Element X decays radioactively with a half life of 9 minutes.

This means that [tex]x(9) = 0.5x(0)[/tex]. We use this to find r. So

[tex]x(t) = x(0)e^{-rt}[/tex]

[tex]0.5x(0) = x(0)e^{-9r}[/tex]

[tex]e^{-9r} = 0.5[/tex]

[tex]\ln{e^{-9r}} = \ln{0.5}[/tex]

[tex]-9r = \ln{0.5}[/tex]

[tex]9r = -\ln{0.5}[/tex]

[tex]r = -\frac{\ln{0.5}}{9}[/tex]

[tex]r = 0.077[/tex]

So

[tex]x(t) = x(0)e^{-0.077t}[/tex]

There are 960 grams of Element X

This means that [tex]x(0) = 960[/tex]

[tex]x(t) = 960e^{-0.077t}[/tex]

How long, to the nearest tenth of a minute, would it take the element to decay to 40 grams?

This is t when [tex]x(t) = 40[/tex]. So

[tex]x(t) = 960e^{-0.077t}[/tex]

[tex]40 = 960e^{-0.077t}[/tex]

[tex]e^{-0.077t} = \frac{40}{960}[/tex]

[tex]\ln{e^{-0.077t}} = \ln{\frac{40}{960}}[/tex]

[tex]-0.077t = \ln{\frac{40}{960}}[/tex]

[tex]0.077t = -\ln{\frac{40}{960}}[/tex]

[tex]t = -\frac{\ln{\frac{40}{960}}}{0.077}[/tex]

[tex]t = 41.3[/tex]

It will take 41.3 minutes for the element to decay to 40 grams

Answer:

Probability that the 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is 0.0092.

Yes. The probability of this data is unlikely to have occurred by chance alone.

Step-by-step explanation:

We are given that the replacement times for the model laptop of concern are normally distributed with a mean of 3.3 years and a standard deviation of 0.6 years.

He then randomly selects records on 50 laptops sold in the past and finds that the mean replacement time is 3.1 years.

Let M = sample mean replacement time

The z-score probability distribution for sample mean is given by;

            Z = [tex]\frac{ M-\mu}{\frac{\sigma}{\sqrt{n} } }} }[/tex]  ~ N(0,1)

where, [tex]\mu[/tex] = population mean replacement time = 3.3 years

            [tex]\sigma[/tex] = standard deviation = 0.6 years

            n = sample of laptops = 50

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, Probability that the 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is given by = P(M [tex]\leq[/tex] 3.1 years)

 P(M [tex]\leq[/tex] 3.1 years) = P( [tex]\frac{ M-\mu}{\frac{\sigma}{\sqrt{n} } }} }[/tex] [tex]\leq[/tex] [tex]\frac{ 3.1-3.3}{\frac{0.6}{\sqrt{50} } }} }[/tex] ) = P(Z [tex]\leq[/tex] -2.357) = 1 - P(Z [tex]\leq[/tex] 2.357)

                                                           = 1 - 0.99078 = 0.0092  or  0.92%          

So, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 2.357 in the z table which will lie between x = 2.35 and x = 2.36 which has an area of 0.99078.

Hence, the required probability is 0.0092 or 0.92%.

Now, based on the result above; Yes, the computer store has been given laptops of lower than average quality because the probability of this data is unlikely to have occurred by chance alone as the probability of happening the given event is very low as 0.92%.

Final answer:

To find the probability, we need to standardize the sample mean using the z-score formula and then use a standard normal distribution table or a calculator to find the probability.

Explanation:

To find the probability that the mean replacement time of 50 randomly selected laptops is 3.1 years or less, we can use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

We know that the population mean is 3.3 years, the population standard deviation is 0.6 years, and the sample size is 50. To find the probability, we need to standardize the sample mean using the z-score formula and then use a standard normal distribution table or a calculator to find the probability.

The formula for the z-score is:
z = (x - μ) / (σ / √n)

Substituting the given values:
z = (3.1 - 3.3) / (0.6 / √50)

Calculating the z-score:
z = -0.2 / (0.6 / 7.0711)
z ≈ -0.2 / 0.0848
z ≈ -2.359

Using a standard normal distribution table or a calculator, we find that the probability of obtaining a z-score less than -2.359 is approximately 0.0093. Therefore, the probability that 50 randomly selected laptops will have a mean replacement time of 3.1 years or less is approximately 0.0093, or 0.93%.

Based on this probability, it does not appear that the computer store has been given laptops of lower-than-average quality. The probability of obtaining this data by chance alone is low enough to suggest that it is unlikely to have occurred by chance alone.

Answer: 10 by 16

Step-by-step explanation:

Answer:

The 99% confidence interval for the mean number of computer games purchased each year is between 7.3 and 7.5 games.

Step-by-step explanation:

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].

So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex]

Now, find the margin of error M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 2.575\frac{1.4}{\sqrt{1233}} = 0.1[/tex]

The lower end of the interval is the sample mean subtracted by M. So it is 7.4 - 0.1 = 7.3.

The upper end of the interval is the sample mean added to M. So it is 7.4 + 0.1 = 7.5

The 99% confidence interval for the mean number of computer games purchased each year is between 7.3 and 7.5 games.

Answer: = ( 7.3, 7.5)

Therefore at 99% confidence interval (a,b) = ( 7.3, 7.5)

Step-by-step explanation:

Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.

The confidence interval of a statistical data can be written as.

x+/-zr/√n

Given that;

Mean gain x = 7.4

Standard deviation r = 1.4

Number of samples n = 1233

Confidence interval = 99%

z(at 99% confidence) = 2.58

Substituting the values we have;

7.4+/-2.58(1.4/√1233)

7.4+/-2.58(0.03987)

7.4+/-0.1028

7.4+/-0.1

= ( 7.3, 7.5)

Therefore at 99% confidence interval (a,b) = ( 7.3, 7.5)

Answer:

131 inches

Step-by-step explanation:

1 yard = 36 inches

36 x 2 = 72

72 + 59 = 131

c:

Answer:

It will take 55 years for the account value to reach 38200 dollars

Step-by-step explanation:

This is a simple interest problem.

The simple interest formula is given by:

[tex]E = P*I*t[/tex]

In which E are the earnings, P is the principal(the initial amount of money), I is the interest rate(yearly, as a decimal) and t is the time.

After t years, the total amount of money is:

[tex]T = E + P[/tex].

In this problem, we ahve that:

[tex]T = 38200, P = 7500, I = 0.075[/tex]

So

First we find how much we have to earn in interest.

[tex]38200 = E + 7500[/tex].

[tex]E = 38200 - 7500[/tex]

[tex]E = 30700[/tex]

How much time to earn this interest?

[tex]E = P*I*t[/tex]

[tex]30700 = 7500*0.075*t[/tex]

[tex]t = \frac{30700}{7500*0.075}[/tex]

[tex]t = 54.6[/tex]

Rounding up

It will take 55 years for the account value to reach 38200 dollars

Answer:

It will take approximately 53 years for the account value to reach 38200 dollars

Step-by-step explanation:

Given the following parameters:

Principal P = 7500

Interest Rate R = 7.75% = 0.0775

Let us find the simple interest for the first year

Simple Interest, I = PRT

with T = 1 year = 12 months

I = 7500 × 0.0775 × 1

= 581.25

The amount for the first year is the addition of the principal and simple interest.

Amount, A = 7500 + 581.25 = 8081.25.

Now, we want to find the time T when Amount A = 38200

Given A = P + I

And I = PRT

A = P + PRT

= P(1 + RT)

Let us make T the subject of the formula.

Dividing both sides by P

A/P = 1 + RT

A/P - 1 = RT

T = ((A/P) - 1)/R

T = ((38200/7500) - 1)/0.0775

= (307/75)/0.0775

= 52.8172043

≈ 53 years.