A publisher wants to estimate the mean length of time (in minutes) all adults spend reading newspapers. To determine this estimate, the publisher takes a random sample of 15 people and obtains the following results:
11, 9, 8, 10, 10, 9, 7, 11, 11, 7, 6, 9, 10, 8, 10
Assume that the population of times is normally distributed.
a. Find the point estimate of the population mean.
b. Construct the 95% confidence interval for the mean population length reading time and interpret it.
c. What is the margin of error?
d. How can we reduce the margin of error?

Answer:

d = √(k/R)

Step-by-step explanation:

Eliminate fractions by multiplying by the denominator, then divide by the coefficient of d². Finally, take the square root.

[tex]R=\dfrac{k}{d^2}\\\\Rd^2=k \quad\text{multiply by $d^2$}\\\\d^2=\dfrac{k}{R} \quad\text{divide by $R$}\\\\d=\sqrt{\dfrac{k}{R}} \quad\text{take the square root}[/tex]

Answer:

The expected value of the safe bet equal $0

Step-by-step explanation:

If  

[tex]S=\left\{s_1,s_2,...,s_n\right\}[/tex]

is a finite numeric sample space and

[tex]P(X=s_k)=p_k[/tex] for k=1, 2,..., n

is its probability distribution, then the expected value of the distribution is defined as

[tex]E(X)=s_1P(X=s_1)+s_2P(X=s_2)+...+s_nP(X=s_n)X) [/tex]

What is the expected value of the safe bet?

In the safe bet we have only two possible outcomes: head or tail. Woodrow wins $100 with head and “wins” $-100 with tail So the sample space of incomes in one bet is

S = {100,-100}

Since the coin is supposed to be fair,  

P(X=100)=0.5

P(X=-100)=0.5

and the expected value is

E(X) = 100*0.5 - 100*0.5 = 0

The expected value of the safe bet is $0, which means that you would neither gain nor lose money on average if you played this game repeatedly.

The expected value of the safe bet can be calculated by multiplying the possible outcomes by their respective probabilities and summing them up. In this case, the safe bet has two possible outcomes: winning $100 with a 50% probability and losing $100 with a 50% probability. So, the expected value can be calculated as follows:

Expected value = (100 * 0.5) + (-100 * 0.5) = $0

Therefore, the expected value of the safe bet is $0, which means that you would neither gain nor lose money on average if you played this game repeatedly.

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The 90% confidence interval using the same sample values is E. 65 to 85.

The 90% confidence interval would be narrower than the 95% confidence interval but the middle point always remains the same.

The middle point there should be (60 + 90)/2 = 75 for the confidence interval. The confidence interval width for a 95% confidence interval width is 30.

For the 65 to 85 confidence interval, the width is 20, therefore this can be true because 20 < 30, therefore 65 to 85 could be the possible confidence interval required here.

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Answer:

L=0

Step-by-step explanation:

[tex]L=\lim\limits_{x \rightarrow \frac{\pi}{2}}3secx-3tanx[/tex]

Replacing the value of x we get ∞ - ∞ which is an indetermined expression

We must transform the limit so it can be shown as a fraction and the L'Hopital's rule can be applied:

[tex]L=\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{3-3sinx}{cosx}=\frac{0}{0}[/tex]

Now we can take the derivative in both parts of the fraction

[tex]L=\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{-3cosx}{-sinx}=3\lim\limits_{x \rightarrow \frac{\pi}{2}}\frac{cosx}{sinx}=3\times 0=0[/tex]

Answer:

We accept H₀, we dont have evidence to say that the family size container

of ketchup has smaller quantity

Step-by-step explanation:

Population mean  μ₀ =  20 ounces

sample size  n  =  10   df = n -1  df = 10-1   df= 9

n < 30   use of t-student distribution

sample mean  μ  = 19.86

sample standard deviation  s  =  0,22

One tail-test ( left tail)

1.-Test Hypothesis

H₀     null hypothesis                μ₀ =  20

Hₐ  Alternative hypothesis       μ₀ < 20

2.- α  =  0,01   and one test tail

3.- Compute

t(s)  =  [ ( μ  -   μ₀ ) ] / s/√n           t(s)  = [( 19.86  -  20 )* √10 ] / 0.22

t(s)  =  - ( 0,14 * 3.16) / 0,22  

t(s)  =  - 2.01

4.- We go to  t-student table  t(c) for df = 9  and 0,01 = α  

and find t(c) =  - 2.821

5.-Compare   t (s)   and  t (c)

t (c)  <   t (s)      -  2.821  < - 2.01

6. t(s)  is inside de acceptance region  we accept H₀

Answer:

0.546 is the probability that a randomly selected smartphone users in the age range 11 to 55+ is between 30 and 54 years old.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ =  34.8 years

Standard Deviation, σ = 14.1 years

We are given that the distribution of ages of smartphone is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P( age range is between 30 and 54 years old)

[tex]P(30 \leq x \leq 54) = P(\displaystyle\frac{30 - 34.8}{14.1} \leq z \leq \displaystyle\frac{54-34.8}{14.1}) = P(-0.3404 \leq z \leq 1.3617)\\\\= P(z \leq 1.3617) - P(z < -0.3404)\\= 0.913 - 0.367 = 0.546 = 54.6\%[/tex]

[tex]P(30 \leq x \leq 54) = 54.6\%[/tex]

0.546 is the probability that a randomly selected smartphone users in the age range 11 to 55+ is between 30 and 54 years old.

Answer:

The assumption that must be made for this study to follow the probabilities of a binomial experiment is that there must be only two outcomes of each trail in this study (meaning that it is either they prefer grey suits or they do not prefer grey suits). There must be no other option apart from those two options and each of the independent trails must be mutually exclusive, meaning that the two required options cannot occur together. It is either the first option (prefer grey suits) or the second option (do not prefer grey suits).

Step-by-step explanation:

Answer:

14.488 amperes

Step-by-step explanation:

The amplitude I of the current is given by

[tex]\large I=\displaystyle\frac{E_m}{Z}[/tex]

where

[tex]\large E_m[/tex] = amplitude of the energy source E(t).

Z = Total impedance.

The amplitude of the energy source is 120, the maximum value of E(t)  

The total impedance is given by

[tex]\large Z=\sqrt{R^2+(X_L-X_C)^2}[/tex]

where

R= Resistance

L = Inductance

C = Capacitance

w = Angular frequency

[tex]\large X_L=wL[/tex] = inductive reactance

[tex]\large X_C=\displaystyle\frac{1}{wC}[/tex] = capacitive reactance

As E(t) = 120sin(12t), the angular frequency w=12

So

[tex]\large X_L=12*0.37=4.44\\\\X_C=1/(12*7)=0.012[/tex]

and

[tex]\large Z=\sqrt{7^2+(4.44-0.012)^2}=8.283[/tex]

Finally

[tex]\large I=\displaystyle\frac{E_m}{Z}=\frac{120}{8.283}=14.488\;amperes[/tex]

Answer:

Step-by-step explanation:

The null hypothesis is that the mean weekly food budget for all households in the community is equal to the national average, while the alternative hypothesis is that it is higher. The test statistic is calculated using the sample mean and the standard deviation of the population. To determine if the results are significant, we compare the test statistic to the critical value.

The null hypothesis for this test is that the mean weekly food budget for all households in the community is equal to the national average, which is $99. The alternative hypothesis is that the mean weekly food budget for all households in the community is higher than the national average.

The test statistic in this case is calculated using the sample mean and the standard deviation of the population. It is equal to (sample mean - population mean) / (standard deviation / sqrt(sample size)).

To determine if the results are significant at the 5% level, we compare the test statistic to the critical value for a one-tailed test with a significance level of 0.05. If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the mean weekly food budget for all households in the community is higher than the national average.

Answer:

-6-(-2) is equivalent to

-6 +2

and 2-6

a) and b) are correct options

Answer:

a and b.

Step-by-step explanation:

-6-(-2) = -6 + 2 = -4.

2 - 6 = -4.

the limits of the 99% confidence interval are approximately [tex]\( (230.068, 233.799) \).[/tex]

To find the limits of the 99% confidence interval (CI) for the true average natural frequency (Hz), we'll use the formula for confidence intervals for the mean:

[tex]\[ \text{CI} = \bar{x} \pm z \frac{s}{\sqrt{n}} \][/tex]

Where:

- [tex]\( \bar{x} \)[/tex] is the sample mean,

- [tex]\( z \)[/tex] is the z-score corresponding to the desired confidence level,

- [tex]\( s \)[/tex] is the sample standard deviation,

- [tex]\( n \)[/tex] is the sample size.

Given:

- Sample size (n ) = 5,

- 95% CI: Lower limit = 230.061, Upper limit = 233.807.

1. Calculate the sample mean [tex](\( \bar{x} \)):[/tex]

[tex]\[ \bar{x} = \frac{\text{Lower limit} + \text{Upper limit}}{2} \][/tex]

[tex]\[ \bar{x} = \frac{230.061 + 233.807}{2} \][/tex]

[tex]\[ \bar{x} = \frac{463.868}{2} \][/tex]

[tex]\[ \bar{x} = 231.934 \][/tex]

2. Calculate the sample standard deviation [tex](\( s \)):[/tex]

To calculate the sample standard deviation, we need to know the z-score corresponding to the desired confidence level.

3. Find the z-score for a 99% confidence level:

At 99% confidence level, the critical z-value is approximately 2.576 (you can find this value using a standard normal distribution table or a calculator).

4. Substitute the values into the confidence interval formula:

[tex]\[ 231.934 \pm 2.576 \frac{s}{\sqrt{5}} \][/tex]

We know that the width of the interval at 95% confidence level is[tex]\( 233.807 - 230.061 = 3.746 \).[/tex]

So, we can set up an equation to find \( s \):

[tex]\[ 2.576 \frac{s}{\sqrt{5}} = \frac{3.746}{2} \][/tex]

[tex]\[ \frac{s}{\sqrt{5}} = \frac{3.746}{2.576 \times 2} \][/tex]

[tex]\[ \frac{s}{\sqrt{5}} \approx \frac{3.746}{5.152} \][/tex]

[tex]\[ s \approx \frac{3.746 \times \sqrt{5}}{5.152} \][/tex]

[tex]\[ s \approx \frac{3.746 \times 2.236}{5.152} \][/tex]

[tex]\[ s \approx \frac{8.3753}{5.152} \][/tex]

[tex]\[ s \approx 1.6247 \][/tex]

Now, we can substitute [tex]\( s = 1.6247 \)[/tex] into the CI formula to find the limits of the 99% interval:

[tex]\[ 231.934 \pm 2.576 \frac{1.6247}{\sqrt{5}} \][/tex]

[tex]\[ 231.934 \pm 2.576 \times \frac{1.6247}{\sqrt{5}} \][/tex]

[tex]\[ 231.934 \pm 2.576 \times 0.7241 \][/tex]

[tex]\[ 231.934 \pm 1.8658 \][/tex]

Therefore, the limits of the 99% confidence interval are approximately [tex]\( (230.068, 233.799) \).[/tex]

Answer:

21 I believe

Step-by-step explanation:

3 x 5 = 15

15 + 1 = 16

because its one for every three blueberry then there are 5 strawberries and 16 blueberries so 5 + 16 = 21

Answer: there are 12 blueberries and 4 strawberries in the bowl

Step-by-step explanation:

The total number of blueberries and strawberries contained in the bowl is 16. The ratio of blueberries to strawberries is 3:1

Total ratio will be sum of the proportion the blueberries to the proportion of strawberries.

Total ratio = 3+1 = 4

To determine how many of each berry are in the bowl,

Number of blueberries in the bowl will be

(Proportion of blueberries / total ratio ) × 100

This becomes

3/4 × 16 = 12 blueberries

Number of strawberries in the bowl will be

(Proportion of strawberries / total ratio ) × 100

This becomes

1/4 × 16 = 4 strawberries

Answer:

-cos(6)+sin(3)+cos(5)

=-0.53538809312 (using calculator)

Step-by-step explanation:

here, F = sin(x)dx + cos(y)dy

then f = -cos(x) + sin(y)

( because, [tex]\nabla f[/tex] should be F. by applying [tex]\nabla[/tex] operator on f we must obtain F. so to satisfy this condition f must be -cosx + siny.

where, [tex]\nabla g(x,y) =[/tex] partial derive of g(x,y) with respect to x +  partial derive of g(x,y) with respect to y )

={-cos(-6)}+sin(3) -[{-cos(-5)}+sin(0)]

=-cos(6)+sin(3)-[-cos(5)]    [since, sin(0)=0, cos(-a)=cos(a) where, a>0]

= -cos(6)+sin(3)+cos(5)

=-0.53538809312 (using calculator)

The evaluation of the line integral is: -0.53538809312 (using a calculator) or -cos(6)+sin(3)+cos(5)

If F is a vector field and if it is defined on a domain D and F

= ∇f for any scalar function f on D

Then f is known as the potential function of F

where, F = sin(x)dx + cos(y)dy

then f = -cos(x) + sin(y)

( because ∇f should be F. by applying ∇ operator on f we must obtain F. so to satisfy this condition f must be -cosx + siny.

where, ∇g(x,y) partial derive of g(x,y) with respect to x +  partial derive of g(x,y) with respect to y )

Therefore, the line integral of F over the given curve C

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Answer:

a)  (50.30 , 52.50)

b) (50.85 , 51.95)

c) (50.68 , 52.12)

d)  (51.02 , 51.78)

e) 209

Step-by-step explanation:

(a)  Sample Mean = 51.4

σ = 2.8

Sample Size, n = 25

Standard Error, E = [tex]\frac{\sigma}{\sqrt{n}}[/tex] = 0.56

z critical value for 95% confidence interval

z = 1.96

Margin of Error (ME) = z × E = 1.0976

95% confidence interval is given as

⇒ Mean ± ME

= 51.4 ± 1.0976

or

= (50.30 , 52.50)

b) Sample Mean = 51.4

σ = 2.8

Sample Size, n = 100

Standard Error, E = [tex]\frac{2.8}{\sqrt{100}}[/tex] = 0.28

z critical value for 95% confidence interval

z = 1.96

Margin of Error (ME) = z × E = 0.5488

95% confidence interval is given as

⇒ Mean ± ME

= 51.4 ± 0.5488

or

= (50.85 , 51.95)

c) Sample Mean = 51.4

σ = 2.8

Sample Size, n = 100

Standard Error, E = [tex]\frac{2.8}{\sqrt{100}}[/tex] = 0.28

z critical value for 99% confidence interval

z = 2.5758

Margin of Error (ME) = z × E = 0.7212

99% confidence interval is given as

⇒ Mean ± ME

= 51.4 ± 0.7212

or

= (50.68 , 52.12)

d) Sample Mean = 51.4

σ = 2.8

Sample Size, n = 100

Standard Error, E = [tex]\frac{2.8}{\sqrt{100}}[/tex] = 0.28

z critical value for 82% confidence interval

z = 1.3408

Margin of Error (ME) = z × E = 0.3754

82% confidence interval is given as

⇒ Mean ± ME

= 51.4 ± 0.3754

or

= (51.02 , 51.78)

e) Margin of Error, ME = (width of interval) ÷ 2 = 0.5

Now,

σ = 2.8

as ME = z × Standard Error,

z = 2.5758  for 99% confidence level

For ME = 0.5,

i,e

[tex]\frac{z\times\sigma}{\sqrt{n}}[/tex] = 0.5

or

[tex]\frac{2.5758 \times2.8}{\sqrt{n}}[/tex] = 0.5

or

n = [tex](\frac{2.5758 \times2.8}{0.5})^2[/tex]

or

n = 208.06

or

n ≈ 209

The question involves calculating different confidence intervals for an unknown population parameter. The computations require using the given standard deviation, sample size, and sample mean, along with appropriate Z-scores. The calculated intervals range between about 49.896 and 52.904 Watts for a 95% CI with a sample size of 25, and as narrow as between 50.626 and 52.174 Watts for an 82% CI with a sample size of 100. Using a desire for a 99% CI width of 1.0, a necessary sample size of about 43 is computed.

In statistics, confidence intervals (CI) provide an estimated range of values which is likely to include an unknown population parameter. Given the parameters of standard deviation (σ), sample size (n), and the sample mean (x), we can compute the confidence intervals. This involves finding the standard error of the mean (σ/√n), and using Z-scores depending on the percentage of the confidence interval. For 95%, 99%, and 82% CIs, the Z-scores are approximately 1.96, 2.58, and 1.34 respectively.

For part (e), we want the width of the 99% interval to be 1.0. This involves setting the equation for the interval to 1.0 and solving for n. This results in n being approximately 42.64, but since we can't have a fractional part of an individual, we round up to 43.

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Answer:

250

Step-by-step explanation:

36% are rabbits, and 28% are dogs.  That means the percent that are hamsters is:

100% − 36% − 28% = 36%

There are 160 dogs and hamsters, which are 28% + 36% = 64% of the pets.  Therefore, we can write and solve a proportion:

160 / 64% = x / 100%

x = 250

There are a total of 250 pets.

Answer: 250 pets

Step-by-step explanation:

Let the total number of pets in the pet shop be x

36% of them are rabbits. This means that the number of pets that are rabbit 36/100 × x = 0.36×x = 0.36x

28% of them are dogs. This means that the number of pets that are dogs are 28/100 × x = 0.28×x = 0.28x

The rest are hamsters. This means that total number of hamsters is total number of pets minus sum of rabbits and dogs.

Number of hamsters = x -(0.28x + 0.36x) = 0.36x

If there are 160 dogs and hamsters,it means that

0.28x + 0.36x = 160

0.64x =160

x = 160/0.64

x = 250

Answer:

y = -2(x - 4)^2 + 4.

Step-by-step explanation:

Vertex form:

y = a(x - 4)^2 + 4    (because the vertex is at (4, 4)).

To find the value of a we substitute the point (2, -4):

-4 = a(2-4)^2 + 4

4a = -8

a = -2.

Answer: -204.8

Step-by-step explanation: if there are and more questions like this use m   a   t    h   w    a  y

Answer:

The p-value is 0.175

Step-by-step explanation:

We have the null hypothesis [tex]H_{0}: \mu = 18[/tex] and the alternative hypothesis [tex]H_{1}: \mu \neq 18[/tex] (two-tailed alternative). Because the sample size is n = 17 and the test statistic is t=-1.421, we know that this last value comes from a t distribution with n-1=17-1=16 degrees of freedom. Therefore, the p-value is given by 2P(T < -1.421) because the p-value is the probability of getting a value as extreme as the observed value and because of the simmetry of the t distribution. Here, T has a t distribution with 16 df and we are using the t distribution because the sample size is small. So, 2P(T < -1.421) = 0.1745

The p-value is approximately 0.173, indicating that we do not have enough evidence to reject the null hypothesis. The mean data speed at airports is not significantly different from the claimed value of 18.00 Mbps.

Certainly! To find the p-value for the given hypothesis test, we can follow these steps:

1. **Set up Hypotheses:**

  - Null Hypothesis (H0): The mean [tex](\(\mu\))[/tex] data speed is equal to 18.00 Mbps.

  - Alternative Hypothesis (Ha): The mean [tex](\(\mu\))[/tex] data speed is different from 18.00 Mbps.

2. **Identify Test and Parameters:**

  - We are conducting a one-sample t-test.

  - Sample size (n) is 17.

  - Test statistic (t) is -1.421.

3. **Degrees of Freedom:**

  - Degrees of freedom (df) is calculated as n - 1, which is 16.

4. **Determine the Critical Region:**

  - Since it's a two-tailed test (not equal), we need to find the critical values for a significance level [tex](\(\alpha\))[/tex] of 0.05.

5. **Calculate the p-value:**

  - Using the t-distribution table or statistical software, we find the p-value associated with t = -1.421 and df = 16.

6. **Make a Decision:**

  - If the p-value is less than the significance level (commonly 0.05), we reject the null hypothesis.

Now, let's perform the calculation. Using statistical software or a t-distribution table, the p-value for t = -1.421 and df = 16 is approximately 0.173.

**Conclusion:**

Since the p-value (0.173) is greater than 0.05, we fail to reject the null hypothesis. There is not enough evidence to suggest that the mean data speed at airports is different from 18.00 Mbps.

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Answer: the system of equations are

x + y = 35

3x + 2y = 100

Step-by-step explanation:

Let x= the number of short answer questions.

Let y= the number of multiple choice questions.

Noah wants 35 questions on the exam. This means that

x + y = 35

He plans to mix short answer questions, worth 3 points, with multiple choice questions worth 2 points. This means that x short answer questions will give 3x points and y multiple choice questions will give 2y points

Since the exam is worth 100 points, then,

3x + 2y = 100 - - - - - - - -1

Substituting x = 35 - y into equation 1, it becomes

3(35 - y) + 2y = 100

105 - 3y + 2y = 100

y = 105 - 100 = 5

x = 35 - y = 35 - 5

x = 30

The system of equation telling the number of each question Noah can have on the test is [tex]x+y = 35\\3x + 2y = 100[/tex]. The value of variables evaluates to:

You can represent the unknown amounts by the use of variables. Follow whatever the description is and convert it one by one mathematically. For example if it is asked to increase some item by 4 , then you can add 4 in that item to increase it by 4. If something is for example, doubled, then you can multiply that thing by 2 and so on methods can be used to convert description to mathematical expressions.

For this case, we are given these facts:

Let there are:

Then, we get:

[tex]x+y = 35[/tex] (as total number of question is 35)

Thus, we get second equation as:

[tex]3x + 2y = 100[/tex] (as total maximum score = 100 points)

Therefore, the system of equation obtained for this condition is:

[tex]x+y = 35\\3x + 2y = 100[/tex]

From first equation, getting value of x in terms of  y, we get:

[tex]x = 35 -y[/tex]

Putting this in second equation, we get:

[tex]3x + 2y = 100\\3(35-y) + 2y = 100\\105 -3y + 2y = 100\\105-100 = y\\y = 5[/tex]

Putting this value of y in equation for x, we get;

[tex]x = 35 -y = 35 - 5 = 30[/tex]

Thus, the system of equation telling the number of each question Noah can have on the test is [tex]x+y = 35\\3x + 2y = 100[/tex]. The value of variables evaluates to:

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Answer:

The skitties are not evenly distributed by colour

Step-by-step explanation:

Given that Mr. T  aylor's 4th grade class uses Skittles to learn about probability. They open several randomly selected bags of Skittles and sort and count the different colors and want to determine if Skittles are evenly distributed by color.

[tex]H_0: Skitties are equally distributed\\H_a: atleast two are not equally distributed[/tex]

(Two tailed chi square test)

If all are equally distributed then expected values would be equal to 420/5 =104

Observed                       Red   Orange  Yellow   Purple   Green  Total

                                       107          101      87           115           10     420

Expected                         104          104     104          104         104    420

Chi square 0.0865 0.0865 2.7788 1.16345 84.9615 89.0769

Chi square is calculated as (obs-exp)^2/exp

Total chi square = 89.0769

df = 4

p value = <0.00001

Reject null hypothesis

The skitties are not evenly distributed by colour

Answer:

(B) Fail to reject the null hypothesis that the true population mean battery life of the smartphone is equal to 20 hours.

(C) There is not enough evidence at the α=0.05 level of significance to suggest that the true population mean battery life of the smartphone is less than 20 hours.

Step-by-step explanation:

1) Data given and notation    

[tex]\bar X=19.5[/tex] represent the battery life sample mean  

[tex]s=1.9[/tex] represent the sample standard deviation    

[tex]n=33[/tex] sample size    

[tex]\mu_o =20[/tex] represent the value that we want to test  

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)    

[tex]p_v[/tex] represent the p value for the test (variable of interest)    

2) State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean battery life is less than 20 :    

Null hypothesis:[tex]\mu \geq 20[/tex]    

Alternative hypothesis:[tex]\mu < 20[/tex]    

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

3) Calculate the statistic    

We can replace in formula (1) the info given like this:    

[tex]t=\frac{19.5-20}{\frac{1.9}{\sqrt{33}}}=-1.51[/tex]    

4) P-value    

First we need to calculate the degrees of freedom given by:

[tex]df=n-1=33-1=32[/tex]

Since is a one-side lower test the p value would be:    

[tex]p_v =P(t_{(32)}<-1.51)=0.0704[/tex]    

5) Conclusion    

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the average battery life it's not significantly different less than 20 hours at 5% of signficance. If we analyze the options given we have:

(A) Reject the null hypothesis that the true population mean battery life of the smartphone is equal to 20 hours. FALSE, we FAIL to reject the null hypothesis.

(B) Fail to reject the null hypothesis that the true population mean battery life of the smartphone is equal to 20 hours. TRUE, we fail to reject the null hypothesis that the mean would be 20 or higher .

(C) There is not enough evidence at the α=0.05 level of significance to suggest that the true population mean battery life of the smartphone is less than 20 hours.  TRUE, we FAIL to reject the null hypothesis that the mean is greater or equal to 20 hours, so we reject the alternative hypothesis that the mean is less than 20 hours.

(D) There is enough evidence at the α=0.05 level of significance to support the claim that the true population mean battery life of the smartphone is not equal to 20 hours. FALSE the claim is not that the mean is different from 20. The real claim is: "Javier believes the mean battery life is less than 20 hours".

Javier's t-test with a test statistic of −1.51 and 32 degrees of freedom has a p-value greater than 0.05, hence we fail to reject the null hypothesis, indicating not enough evidence to suggest the true mean is less than the claimed 20 hours. Thus, the correct answer is option (C).

This problem involves conducting a one-sample t-test to determine whether the true population mean battery life is different from the claimed 20 hours. The null hypothesis here is H0: [tex](mu = 20)[/tex] hours, and the alternative hypothesis is Ha: [tex](mu < 20)[/tex] hours. With 32 degrees of freedom, Javier's calculated test statistic is t ≈ −1.51. To decide whether to reject or fail to reject the null hypothesis, we must compare the p-value to the significance level, [tex]\(\alpha = 0.05\)[/tex].

If the p-value is less than [tex]\(\alpha = 0.05\)[/tex], then we reject the null hypothesis. If the p-value is higher, we fail to reject. In this exercise, the p-value associated with Javier's test statistic of t ≈ −1.51 for a one-tailed test is greater than 0.05, thus we should fail to reject the null hypothesis (Option B). This implies that there is not enough evidence at the 0.05 level of significance to support the claim that the true population mean battery life of the smartphone is less than 20 hours (Option C).

Answer:

50

Step-by-step explanation:

Please see attachment .

Answer:

[tex]$ \sqrt{-1} = i $[/tex]

Step-by-step explanation:

(1) [tex]$ \sqrt{-100} $[/tex]

[tex]$ = \sqrt{100}.i = 10i $[/tex]

(1) - 10i

(2) [tex]$ -2\sqrt{50} $[/tex]

[tex]$ = -2\sqrt{50}  = -2(5) \sqrt{2} = -10\sqrt{2} $[/tex]

(2) - -10√2

(3) [tex]$ 5 + \sqrt{-13} $[/tex]

[tex]$ \implies 5 + \sqrt{13}i $[/tex]

(3) - 5 + √13i

(4) [tex]$ 5 + \sqrt{-120} $[/tex]

[tex]$ = 5 + \sqrt{120}i = 5 + 2\sqrt{30}i$[/tex]

(4) - 5 + 2√30i

(5) [tex]$ -10 +  \sqrt{-50} $[/tex]

[tex]$ = -10 + 5\sqrt{2} i $[/tex]

(5) - -10 + 5√2i

(6) - [tex]$ \sqrt{-45} $[/tex]

[tex]$ 3\sqrt{5}i $[/tex]

(6) - 3√5i

Answer:

C. Fail to reject Upper H 0. There is insufficient evidence to warrant rejection of the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150

Step-by-step explanation:

Hello!

You have two different independent samples and are asked to test if the population variances of both variables are the same.

Sample 1 (4000 B.C)

X₁: Breadth of a skull from 4000 B.C. (mm)

X₁~N(μ₁;σ₁²)

n₁= 30 skulls

X[bar]₁= 131.62 mm

S₁= 5.19 mm

Sample 2 (A.D. 150)

X₂: Breadth of a skull from 150 A.D. (mm)

X₂~N(μ₂;σ₂²)

n₂= 30 skulls

X[bar]₂= 136.07 mm

S₂= 5.35 mm

Since you want to test the variances, the proper test to do is an F-test for the population variance ratio. The hypothesis can be established as equality between variances or as a quotient between them.

The hypothesis is:

H₀: σ₁²/σ₂² = 1

H₁: σ₁²/σ₂² ≠ 1

Remember, when you express the hypothesis as a quotient of variances, if it's true that they are the same, the result will be 1, this is the number you'll use to replace in the F-statistic.

α: 0.05

F= (S₁²/S₂²) * (σ₁²/σ₂²) ~F[tex]_{n1-1;n2-1}[/tex]

F= (5.19/5.35)*1 = 0.97

The p-value = 0.5324

Since the p-value is greater than the level of significance, the decision is to not reject the null hypothesis.

Using critical values:

Left: F[tex]F_{n1-1;n2-1;\alpha /2} = \frac{1}{F_{n2-1;n1-1;1-\alpha /2} } = \frac{1}{F_{29;29;0.95} } = \frac{1}{2.10} } =0.47[/tex]

Right: [tex]F_{n1-1; n2-1; 1-\alpha /2} = F_{29; 29; 0.975} = 2.10[/tex]

The calculated F-value (0.97) is in the not rejection zone (0.47<F<2.10) ⇒ Don't reject the null hypothesis.

I hope this helps!

The hypotheses are that there is no significant difference (null) or that there is a significant difference (alternative) in skull breadths from 4000 B.C. and A.D. 150. An F-test is used to test these via the comparison of sample variances. The conclusion depends on the P-value: if it is greater than 5%, the null is accepted (option C), and if less, rejected (option D).

The null and alternative hypotheses for this question can be stated as follows:

There is no significant difference in the variation of maximal skull breadths in 4000 B.C. and A.D. 150.

There is a significant difference in the variation of maximal skull breadths in 4000 B.C. and A.D. 150.

To test these hypotheses, we use the F-test for the equality of two variances. The test statistic (F) is calculated by taking the ratio of the sample variances, which in this case would be (5.19^2) / (5.35^2).

The P-value associated with the F statistic is then used to determine the significance of the evidence against the null hypothesis. If the P-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

The conclusion of the hypothesis test depends on the calculated P-value. If P-value is less than 0.05, we conclude that there is a significant difference in the variation of skull breadths, thereby rejecting the null hypothesis (option D). If the P-value is greater than 0.05, we fail to reject the null hypothesis, concluding that the skull variations are not significantly different (option C).

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Answer:

a) (0.72, 0.78)

Step-by-step explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Description in words of the parameter p

[tex]p[/tex] represent the real population proportion of all residents in that county who think their local government did a good job

[tex]\hat p[/tex] represent the estimated proportion of all residents in that county who think their local government did a good job

n=1000 is the sample size required  

[tex]z_{\alpha/2}[/tex] represent the critical value for the margin of error  

The population proportion have the following distribution  

[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]

Numerical estimate for p

In order to estimate a proportion we use this formula:

[tex]\hat p =\frac{X}{n}[/tex] where X represent the number of people with a characteristic and n the total sample size selected.

[tex]\hat p=\frac{750}{1000}=0.75[/tex] represent the estimated proportion of all residents in that county who think their local government did a good job

Confidence interval

The confidence interval for a proportion is given by this formula  

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]  

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.  

[tex]z_{\alpha/2}=1.96[/tex]  

And replacing into the confidence interval formula we got:  

[tex]0.750 - 1.96 \sqrt{\frac{0.75(1-0.75)}{1000}}=0.72[/tex]  

[tex]0.750 + 1.96 \sqrt{\frac{0.75(1-0.75)}{1000}}=0.78[/tex]  

And the 95% confidence interval would be given (0.72;0.78).  

We are confident at 95% that the true proportion of people who think their local government did a good job is between (0.72;0.78).

a) (0.72, 0.78)

Answer:

35$

Step-by-step explanation:

Let the shirt be = X

And the tie =Y

X + Y= 48$

X = 22 + Y (The shirt costs $22 more than the tie)

22 + 2y = 48

2y = 26

y = 13

X= 48 – 13

X = 35

therefore, the cost of the shirt is $35

and the cost of the tie is $13

The cost of the shirt is 35$ such the shirt and tie together cost $48.

Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.

In other words, an equation is a set of variables that are constrained through a situation or case.

Let's say the shirt cost is S while the tie is T

Together;

S + T = 48

And,

S = 22 + T

By substituting

22 + T + T = 48

2T = 26

T = 13

So,

S = 48 - 13 = 35

Hence "The cost of the shirt is 35$ such the shirt and tie together cost $48".

For more about the equation,

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Answer:

(a) 4.148 x 10^(12) ways

(b) 5,827,360 ways

Step-by-step explanation:

Number of Demonstrators (D)  = 44

Number of Repudiators (R) = 56

(a)

5 senate members must be Repudiators and 5 must be demonstrators, assuming that the order at which they are selected is irrelevant:

[tex]N= C^{D}_{5} * C^{R}_{5}\\N=\frac{56!}{5!(56-5)!} *\frac{44!}{5!(44-5)!} \\N=3,819,816*1,086,008\\N=4.148 *10^{12}[/tex]

(b)

Since there are two different positions, (speaker and vice speaker), order is important in this situation, and the total number of ways to select two senators from each party is:

[tex]N= P^{D}_{2} * P^{R}_{2}\\N=\frac{56!}{(56-2)!} *\frac{44!}{(44-2)!} \\N=3,080*1,892\\N=5,827,360[/tex]

The question involves applying the concept of combinations in mathematics to determine the number of ways committee members, speakers, and vice speakers can be selected from two different political parties.

The subject of the question involves two main components of mathematics: combinatorics and probability. This involves calculating the number of ways certain events can occur given a certain number of possibilities.

Let's first solve part (a) of your question. We are asked how many ways there are to select a committee of 10 senate members with the same number of Demonstrators and Repudiators. We want five senators from each party. Given there are 44 Demonstrators and 56 Repudiators, the number of ways we can pick a committee is the product of comb(44,5) and comb(56,5) which are the combinations of picking 5 out of 44 Demonstrators and 5 out of 56 Repudiators, respectively.

In part (b) of your question, we are asked how many ways there are for two speakers and two vice speakers to be selected, one from each party. This is simply comb(44,1) multiplied by comb(43,1) multiplied by comb(56,1) multiplied by comb(55,1). This is because we first choose 1 out of 44 Demonstrators for a speaker position, then 1 out of the remaining 43 Demonstrators for a vice speaker position, then 1 out of 56 Repudiators for a speaker position, and finally 1 out of the remaining 55 Repudiators for a vice speaker position.

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Answer:

The total length can be calculated by doing thesum of all sides

Step-by-step explanation:

The total length of planting around each stepping stone is 15x - 2.

To find the total length of planting around each stepping stone, we need to add up the lengths of all the sides of the irregular polygon.

The sides of the polygon are (3x-1), (2x+1), (4x-2), (4x-4), and (2x+4).

To find the total length, we can add up the lengths of all the sides:

(3x-1) + (2x+1) + (4x-2) + (4x-4) + (2x+4)

Simplifying the expression, we get:

15x - 2

Therefore, The total length of planting around each stepping stone is 15x - 2.

The probable question may be:

Belinda placed stepping stones in the shape of the irregular polygon shown. She will plant thyme around the edge of each stepping stone. What is the total length of planting around each stepping stone ?​

The sides of the irregular polygon are (3x-1), (2x+1), (4x-2), (4x-4), (2x+4)

Answer:

65%

Step-by-step explanation:

No, the events "drank coffee" and "drank tea" are not mutually exclusive, as there are 10% of employees drank both coffee and tea.

If there are 50% drank coffee and 10% of them enjoy both, then there are 40% of the employees enjoy only coffee.

Similarly, there are 15% of employees who only enjoy tea.

Then the probability of selecting a person who only enjoy tea or coffee is

40% + 15% = 65%

Answer:

No.

0.65 or 65%

The proportion of Greenville County adult residents who are 65 years or older is approximately 13%.

To find the proportion of Greenville County adult residents who are 65 years or older, we need to divide the number of residents who are 65 years or older by the total number of adult residents in the county.

The proportion can be found using the formula: Proportion = Number of residents who are 65 years or older / Total number of adult residents.

Substituting the given values, we have:

Proportion = 59,969 / 461,299 = 0.1299

Therefore, the proportion of Greenville County adult residents who are 65 years or older is approximately 0.13, or 13%.

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