MATH 1325 – EXAM 4 NAME: ______________________________ SHOW ALL WORK. ANSWERS WITHOUT WORK WILL RECEIVE NO CREDIT. YOU MUST USE A PENCIL. READ ALL DIRECTIONS. POINTS WILL BE DEDUCTED FOR FAILURE TO FOLLOW DIRECTIONS. TRUE/FALSE – WRITE THE WORD THAT BEST DESCRIBES THE GIVEN STATEMENT BY WRITING EITHER "TRUE" OR "FALSE" IN THE SPACE PROVIDED TO THE LEFT OF THE PROBLEM. __________ 1. THE ABSOLUTE MAXIMUM OF A FUNCTION ALWAYS OCCURS WHERE THE DERIVATIVE HAS A CRITICAL FUNCTION. __________ 2. IMPLICIT DIFFERENTIATION CAN BE USED TO FIND dy dx WHEN x IS DEFINED IN TERMS OF y . __________ 3. IN A RELATED RATES PROBLEM, THERE CAN BE MORE THAN TWO QUANTITIES THAT VARY WITH TIME. __________ 4. A CONTINUOUS FUNCTION ON AN OPEN INTERVAL DOES NOT HAVE AN ABSOLUTE MAXIMUM OR MINIMUM. __________ 5. IN A RELATED RATES PROBLEM, ALL DERIVATIVES ARE WITH RESPECT TO TIME. MULTIPLE CHOICE – CHOOSE THE ONE ALTERNATIVE THAT BEST COMPLETES THE STATEMENT OR ANSWERS THE QUESTION BY CIRCLING THE CORRECT LETTER. 6. FIND THE MAXIMUM ABSOLUTE EXTREMUM AS WELL AS ALL VALUES OF x WHERE IT OCCURS ON THE SPECIFIED DOMAIN

Answer:

a) The 95% of confidence intervals for the average spending

(23.872 , 32.128)

b) The calculated value t= 1< 1.711( single tailed test) at 0.05 level of significance with 24 degrees of freedom.

The null hypothesis is accepted

A survey of 25 young professionals statistically that the population mean is less than $30

Step-by-step explanation:

Step:-(i)

Given data a survey of 25 young professionals fond that they spend an average of $28 when dining out, with a standard deviation of $10

The sample size 'n' = 25

The mean of the sample   x⁻  = $28

The standard deviation of the sample (S) = $10.

Level of significance ∝=0.05

The degrees of freedom γ =n-1 =25-1=24

tabulated value t₀.₀₅ = 2.064

Step 2:-

The 95% of confidence intervals for the average spending

([tex](x^{-} - t_{\alpha } \frac{S}{\sqrt{n} } ,x^{-} + t_{\alpha }\frac{S}{\sqrt{n} } )[/tex]

[tex](28 - 2.064 \frac{10}{\sqrt{25} } ,28 + 2.064\frac{10}{\sqrt{25} } )[/tex]

( 28 - 4.128 , 28 + 4.128)

(23.872 , 32.128)

a) The 95% of confidence intervals for the average spending

(23.872 , 32.128)

b)

Null hypothesis: H₀:μ<30

Alternative Hypothesis: H₁: μ>30

level of significance ∝ = 0.05

The test statistic

[tex]t = \frac{x^{-}-mean }{\frac{S}{\sqrt{n} } }[/tex]

[tex]t = \frac{28-30 }{\frac{10}{\sqrt{25} } }[/tex]

t = |-1|

The calculated value t= 1< 1.711( single tailed test) at 0.05 level of significance with 24 degrees of freedom.

The null hypothesis is accepted

Conclusion:-

The null hypothesis is accepted

A survey of 25 young professionals statistically that the population mean is less than $30

If the conditions for statistical inference for testing a difference in population proportions been met then we can say that  -(E )All conditions for making statistical inference have been met.

Step-by-step explanation:

We can test the claim and the assumptions  about the population proportion  under the following conditions

Thus we can say that by studying the question we can say that the above mentioned condition have been met.Hence

If the conditions for statistical inference for testing a difference in population proportions been met then we can say that  -(E )All conditions for making statistical inference have been met.

The conditions for statistical inference for testing a difference in population proportions have not been met.

In order to test for a difference in population proportions, certain conditions need to be met. Firstly, the condition for independence must be met, which means that random samples need to be selected. Secondly, the distribution of the difference in sample proportions needs to be approximately normal. This is determined by checking whether nO(pC) is greater than or equal to 10 and nO(1-pC) is greater than or equal to 10. In this case, the conditions for statistical inference have not been met because random samples were not selected (option A) and nN(pC) is not greater than or equal to 10 (option C).

Answer:

20 feet

Step-by-step explanation:

Answer:

0.0046 = 0.46% probability that the sample mean would differ from the population mean by more than 3.1 gallons

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

[tex]\mu = 86, \sigma = 7, n = 41, s = \frac{7}{\sqrt{41}} = 1.0932[/tex]

If 41 racing cars are randomly selected, what is the probability that the sample mean would differ from the population mean by more than 3.1 gallons?

Less than 86 - 3.1 = 82.9 or more than 86 + 3.1 = 89.1. Since the normal distribution is symmetric, these probabilities are equal, which means that we can find one of them and multiply by 2.

Less than 82.9

pvalue of Z when X = 82.9. So

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

By the Central Limit Theorem

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

[tex]Z = \frac{82.9 - 86}{1.0932}[/tex]

[tex]Z = -2.84[/tex]

[tex]Z = -2.84[/tex] has a pvalue of 0.0023

2*0.0023 = 0.0046

0.0046 = 0.46% probability that the sample mean would differ from the population mean by more than 3.1 gallons

Final answer:

The probability that the sample mean of 41 racing cars would differ from the population mean by more than 3.1 gallons is 0.0046 or 0.46%.

Explanation:

To calculate the probability that the sample mean of 41 racing cars would differ from the population mean by more than 3.1 gallons, you use the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough. In this case, 41 is usually considered large enough. The formula to calculate the z-score for a given sample mean difference is z = (X - µ) / (σ/√n), where X is the sample mean, µ is the population mean, σ is the standard deviation, and n is the sample size.

First, we calculate the standard error of the mean (SEM): SEM = σ/√n = 7/√41 ≈ 1.0934. Next, we calculate the z-score for a difference of 3.1 gallons: z = 3.1 / 1.0934 ≈ 2.8354.

Now, we look up the z-score in standard normal distribution tables or use a calculator to find the probability corresponding to a z-score of 2.8354, which gives us the probability of a sample mean being 3.1 gallons or more away from the population mean on one side. However, since the question asks for the probability of the sample mean differing by more than 3.1 gallons on either side, we have to multiply this probability by 2 to get the final answer.

To obtain the final probability, we use statistical software or tables, which typically provide the probability for the region beyond a certain z-score. If P(z > 2.8354) is the probability of being beyond 2.8354 standard deviations from the mean on one side, the probability of being more than 3.1 gallons away from the mean on either side is 2 * P(z > 2.8354). Assuming P(z > 2.8354) = 0.0023 (from a standard normal distribution table), the final probability would be 2 * 0.0023 = 0.0046 or rounded to four decimal places, 0.0046.

Answer:

We conclude that the average attention span for teenage boys is less than 5 minutes.

Step-by-step explanation:

We are given that the mean time to distraction for teenage boys working on an independent task was 4 minutes.

Suppose that this mean was based on a random sample of 50 teenage Australian boys and that the sample standard deviation was 1.4 minutes.

Let [tex]\mu[/tex] = average attention span for teenage boys

SO, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 5 minutes    {means that the average attention span for teenage boys is more than or equal to 5 minutes}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 5 minutes    {means that the average attention span for teenage boys is less than 5 minutes}

The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;

                    T.S.  = [tex]\frac{\bar X -\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean attention time span for teenage boys = 4 min

             s = sample standard deviation = 1.4 min

             n = sample of teenage boys = 50

So, the test statistics  =  [tex]\frac{4-5}{\frac{1.4}{\sqrt{50} } }[/tex]  ~  [tex]t_4_9[/tex]

                                     =  -5.051

Now at 0.01 significance level, the t table gives critical value of -2.405 at 49 degree of freedom for left-tailed test. Since our test statistics is less than the critical value of t as -2.405 > -5.051, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the average attention span for teenage boys is less than 5 minutes.

Answer:

  600

Step-by-step explanation:

As t gets very large, the exponential term goes to zero, and the expression nears the value ...

  P(∞) = 600/(1 +5·0) = 600

The maximum number of squirrels the park can support is modeled as being 600.

Answer:

E. The entire population is measured in both cases, so the actual difference in means can be computed and a confidence interval should not be used.

Step-by-step explanation:

The entire population of Greek and Egyptian mathematicians was already measured by Amara. She can measure the actual difference in the mean. Therefore, she needn't use a confidence interval. It is mostly common for a researcher to be more interested in the difference between means than in the specific values of the means. The difference in sample means is used to compute the difference in population means.

Amara's sample is a random selection of Greek and Egyptian mathematicians. It is a smaller group drawn from the population that has the characteristics of the entire population. The observations and conclusions made against the sample data are attributed to the population. In this case, the entire position is measured.

A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups (Greek and Egyptian mathematicians) which may be related in certain features. It is mostly used when the data sets, like the data set recorded as the outcome from rolling a die 50 times, would follow a normal distribution and may have unknown variances. A t-test is used as a hypothesis testing tool, which allows testing of an assumption applicable to a population. Once the actual difference is known, a confidence interval should not be used.

Answer:

257/2 degrees or 128.5 degrees

Step-by-step explanation:

radians to degrees is x radians * 180/π

[tex]\frac{257\pi }{360} * \frac{180}{\pi }= \frac{257}{2}[/tex]

it is 257/2 degrees or 128.5 degrees

Answer:

a) X ~ exp ( 10 )

b) E(X) = 0.1 , Var (X) = 0.01

c)  P ( X < 0.07 ) = 0.00698

Step-by-step explanation:

Solution:-

- The spike train, used to study neural activity, the given time in between consecutive spikes (ISI) where the firing rate = 10 neurons per seconds.

- Denote a random variable "X"represent a single interspike interval (ISI) having an exponential distribution.

- Where X follows exponential distribution defined by event rate parameter i.e λ.

                               X ~ Exp ( λ )

- The event rate (λ) is the number of times an event occurs per unit time. Since we are studying a single interspike interval (ISI) - which corresponds to the firing rate. So, event rate (λ) = firing rate = 10 neurons per second. Hence, the distribution is:

                               X ~ Exp ( 10 )

- The expected value E(X) denotes the amount of time in which a single an event occurs; hence, the time taken for a single neuron.

                               E(X) = 1 / λ

                               E(X) = 1 / 10

                               E(X) = 0.1 s per neuron.

- The variance is the variation in the time taken by a single neuron to be emitted. It is defined as:

                               Var (X) = 1 / λ^2

                               Var (X) = 1 / 10^2

                               Var (X) = 0.01 s^2

- The probability that ISI is less than t = 0.07 seconds: P ( X < t = 0.07 s):

- The cumulative distribution function for exponential variate "X" is:

                                P ( X < t ) = 1 - e^(-λ*t)

- Plug the values and the determine:

                                P ( X < 0.07 ) = 1 - e^(-0.1*0.07)

                                                      = 1 - 0.99302

                                                      = 0.00698

Answer:

Answer:

The first box plot.

Step-by-step explanation:

Step-by-step explanation:

In a box plot, the lower quartile is the right hand side of the box.  In the first plot, this is 3.

The upper quartile is the left hand side of the box.  In the first plot, this is 8.

The interquartile range is the difference between these two values:

8-3 = 5.

Answer:

(a) P([tex]\bar X[/tex] [tex]\geq[/tex] 76) = 0.2327

(b) P(73 < [tex]\bar X[/tex] < 75) = 0.5035

(c) P([tex]\bar X[/tex] < 74.8) = 0.77035

Step-by-step explanation:

We are given that the mean of a population is 74 and the standard deviation is 16.

Assuming the data follows normal distribution.

Let [tex]\bar X[/tex] = sample mean

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

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

where, [tex]\mu[/tex] = population mean = 74

            [tex]\sigma[/tex] = standard deviation = 16

            n = sample size

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.

(a) Probability that a random sample of size 34 yielding a sample mean of 76 or more is given by = P([tex]\bar X[/tex] [tex]\geq[/tex] 76)

   P([tex]\bar X[/tex] [tex]\geq[/tex] 76) = P( [tex]\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] [tex]\geq[/tex] [tex]\frac{76-74}{\frac{16}{\sqrt{34} } }[/tex] ) = P(Z [tex]\geq[/tex] 0.73) = 1 - P(Z < 0.73)

                                                 = 1 - 0.7673 = 0.2327

The above probability is calculated by looking at the value of x = 0.73 in the z table which has an area of 0.7673.

(b) Probability that a random sample of size 120 yielding a sample mean of between 73 and 75 is given by = P(73 < [tex]\bar X[/tex] < 75) = P([tex]\bar X[/tex] < 75) - P([tex]\bar X[/tex] [tex]\leq[/tex] 73)

   P([tex]\bar X[/tex] < 75) = P( [tex]\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{75-74}{\frac{16}{\sqrt{120} } }[/tex] ) = P(Z < 0.68) = 0.75175

   P([tex]\bar X[/tex] [tex]\leq[/tex] 73) = P( [tex]\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] [tex]\leq[/tex] [tex]\frac{73-74}{\frac{16}{\sqrt{120} } }[/tex] ) = P(Z [tex]\leq[/tex] -0.68) =1 - P(Z < 0.68)

                                                 = 1 - 0.75175 = 0.24825

Therefore, P(73 < [tex]\bar X[/tex] < 75) = 0.75175 - 0.24825 = 0.5035

The above probability is calculated by looking at the value of x = 0.68 in the z table which has an area of 0.75175.

(c) Probability that a random sample of size 218 yielding a sample mean of less than 74.8 is given by = P([tex]\bar X[/tex] < 74.8)

   P([tex]\bar X[/tex] < 74.8) = P( [tex]\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{74.8-74}{\frac{16}{\sqrt{218} } }[/tex] ) = P(Z < 0.74) = 0.77035

The above probability is calculated by looking at the value of x = 0.74 in the z table which has an area of 0.77035.

Answer:

a) N(A∩B) = 0.9

b) N(A∩B) = 4.9

c) N(A or B) = 4.2

Step-by-step explanation:

Given that A and B each randomly, and independently, choose 3 of 10 objects;

P(A) = P(B) = 3/10 = 0.3

P(A') = P(B') = 1 - 0.3 = 0.7

a) chosen by both;

Probability of being chosen by both;

P(A∩B) = 0.3 × 0.3 = 0.09

Expected Number of objects being chosen by both;

N(A∩B) = P(A∩B) × N(total) = 0.09×10

N(A∩B) = 0.9

b) not chosen by either A or B;

Probability of not being chosen by either A or B;

P(A'∩B') = 0.7 × 0.7 = 0.49

Expected Number of objects being chosen by both;

N(A'∩B') = P(A'∩B') × N(total) = 0.49×10

N(A∩B) = 4.9

c) chosen by exactly one of A and B.

Probability of being chosen by exactly one of A and B

P(A∩B') + P(A'∩B) = 0.3×0.7 + 0.7 × 0.3 = 0.42

Expected Number of objects being chosen by both;

N(A or B) = 0.42 × 10

N(A or B) = 4.2

The expected number of objects chosen by both A and B is 0.9. The expected number of objects not chosen by either A or B is 9.1. The expected number of objects chosen by exactly one of A and B is 4.2.

To find the expected number of objects chosen by both A and B, we can use the multiplication rule. Since A and B each independently choose 3 objects from a set of 10, the probability of choosing a specific object is 3/10 for both A and B. Therefore, the expected number of objects chosen by both A and B is (3/10)(3/10)(10) = 0.9.

To find the expected number of objects not chosen by either A or B, we can subtract the expected number of objects chosen by both A and B from the total number of objects. So, the expected number of objects not chosen by either A or B is 10 - 0.9 = 9.1.

To find the expected number of objects chosen by exactly one of A and B, we can use the addition rule. Since A and B each independently choose 3 objects, the probability of choosing a specific object is 3/10 for both A and B. Therefore, the expected number of objects chosen by exactly one of A and B is 2(3/10)(7/10)(10) = 4.2.

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Answer:

Correct options are A,B,D:

Step-by-step explanation:

Concert marketing: The college’s performing arts center wanted to investigate why ticket sales for the upcoming season significantly decreased from last year’s sales. The marketing staff collected data from a survey of community residents. Out of the 110 people surveyed, only 7 received the concert brochure in the mail.

The college's performing arts center should not calculate a confidence interval for the proportion of community residents who received the concert brochure by mail because there is no information on the randomness of the sample, and both n(p-hat) and n(1 minus p-hat) are less than 10.

The college's performing arts center should not calculate a confidence interval for the proportion of all community residents who received the concert brochure by mail based on the collected survey data for a few reasons. First, the sample needs to be random to provide a representative subset of the population, and there is no information provided to confirm that the sample used was random. Secondly, n(p-hat) is not greater than 10, which refers to the number of successes (7 residents received the brochure) not meeting the minimum requirement for the normal approximation to be valid. Lastly, n(1 minus p-hat) is also not greater than 10, indicating that the number of 'failures' (103 residents did not receive the brochure) also does not meet the condition for the central limit theorem to apply. Both the latter points touch upon the requirements for constructing a confidence interval for a proportion which is that np-hat as well as n(1-p-hat) should both be greater than 10 to approximate the binomial distribution with the normal distribution.

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Answer:

The length of side AB is 9 units.

Step-by-step explanation:

In order to find this answer, you must use the distance formula.

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

x2 = 0

x1 = 0

y2 = -3

y1 = 6

Since both x values are equal to zero, we do not need to plug these into our formula. It will just cancel out. Therefore, just plug in the y values.

[tex]\sqrt{(-3-6)^2}[/tex]

-3 - 6 = -9

(-9)^2 = 81

sqrt(81) = 9 units

Answer:

C)[tex]64 feet^{2}[/tex]

Step-by-step explanation:

The formula for the area of a trapezium is :

[tex]\frac{a+b}{2} *height[/tex]

A is the top line

B is the bottom line

To work this out you would first add 6 to 10, which is 16. Then you would divide 16 by 2, which is 8. Then you would multiply 8 by 8, which is 64 [tex]feet^{2}[/tex]

1) Add 6 to 10.

[tex]6+10=16[/tex]

2) Divide 16 by 2.

[tex]16/2=8[/tex]

3) Multiply 8 by 8.

[tex]8*8=64 feet^{2}[/tex]

Answer:

12

Step-by-step explanation:

let y be the value of the gift card and x be the cost of a pair of jeans

The equation for Pablo

y-2x=36    Add 2x to both sides

y=36+2x

The equation for Greg

y-5x=0      Add 5x to both sides

y=5x

Now that we have isolated y in both equations, lets set them equal to each other

36+2x=5x      Subtract 2x from both sides

36=3x            Divide both sides by 3

12=x                x=12, which means that 1 pair of jeans costs $12

Final answer:

The cost of one pair of jeans is $12.

Explanation:

To find the cost of one pair of jeans, we can set up an equation using the information given.

Let's say the cost of one pair of jeans is x. Pablo bought 2 pairs of jeans, so he spent 2x.

He has $36 remaining on his gift card, which means the total amount he spent is the original amount on the gift card minus the remaining balance, which is 2x = original amount - $36.

Greg bought 5 pairs of jeans, so he spent 5x.

He has no money left on his gift card, so the total amount he spent is 5x = original amount.

We can write these two equations:

2x = original amount - $36 ... Equation 1

5x = original amount ... Equation 2

To solve this system of equations, we can substitute Equation 2 into Equation 1:

2x = 5x - $36

3x = $36

x = $12

Therefore, the cost of one pair of jeans is $12.

Step-by-step explanation:

you are going to have to be more specific... please let us know which ones are the legs

Answer:

109/10 or 10 9/10

Step-by-step explanation:

Dividing something by 10 puts it one place after the decimal point.

so 9/10 becomes 0.9.

Answer:

10 9/10

Step-by-step explanation:

Answer:

Step-by-step explanation:

1) The sample and the population of this study is the friends who replied his email which includes in his contact list. then, the number of the replied to his email are 9 friends.

population: the whole friends include in his contact list.

3) Type I error occurs when one incorrectly rejects the null hypothesis

Here there is possibility of type I error

The entire peach pie contains 2240 calories.

To find the number of calories in the entire peach pie, we need to multiply the number of calories in each slice by the total number of slices. In this case, there are 8 equal slices in the peach pie, and each slice has 280 calories. Therefore, the total number of calories in the entire peach pie is 8 multiplied by 280, which equals 2240 calories.

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Answer:

We accept H₀ we don´t have evidence of differences between the information from the sample and the population mean

Step-by-step explanation:

From data and excel (or any statistics calculator) we get:

X = 249,6 ml         and       s  1,26 ml

Sample mean and sample standard deviation respectively.

Population mean  μ₀  = 250 ml

We have a normal distribution but we dont know the standard deviation of the population. Furthermore we have a two tails test since we are finding if the sample give us evidence of differences ( in both senses ) when we compare them with the amount of water spec ( 250 ml )

Our test hypothesis are: null hypothesis       H₀    X = μ₀

Alternative Hypothesis                                 Hₐ     X ≠ μ₀

We also know that sample size is 8  therefore df  =  8 - 1   df = 7 , with this value and the fact that we are required to test at α = 0,05 ( two tails test)

t = 2,365

Then we evaluate our interval:

X ± t* (s/√n)   ⇒   249,6 ±  2,365 * ( 1,26/√8 )

 249,6 ±  2,365 * (1,26/2,83)  ⇒ 249,6 ±  2,365 *0,45

249,6 ± 1,052

P [ 250,652 ; 248,548]

Then the population mean 250 is inside the interval, therefore we must accept that the bottles have being  fill withing the spec. We accept H₀

Answer:

Because the p-value of 0.4304 is greater than the significance level of 0.05, we fail to reject the null hypothesis. We conclude the data provide convincing evidence that the mean amount of water in all the bottles filled that day does not differ from the target value of 250 milliliters.

Step-by-step explanation:

Answer:

a) (9/2)

b) (9/4)

Step-by-step explanation:

I = ∫¹₀ (x³ - 5x) dx = -(9/4)

a) ∫¹₀ (10x - 2x³) dx = -2 ∫¹₀ (x³ - 5x) dx

∫¹₀ (x³ - 5x) dx = -(9/4) from the given value for I

-2 ∫¹₀ (x³ - 5x) dx = -2 × (-9/4) = (9/2)

b) ∫¹₀ (5x - x³) dx = -1 ∫¹₀ (x³ - 5x) dx

∫¹₀ (x³ - 5x) dx = -(9/4) from the given value for I

-1 ∫¹₀ (x³ - 5x) dx = -1 × (-9/4) = (9/4)

Hope this Helps!!!

Answer:

a.

[tex]\int\limits_{0}^{1} 10x - 2x^3 \,dx = -2(\int\limits_{0}^{1} x^3 -5x\,dx) = -2*(-9/4) = 9/2[/tex]

b.

[tex]\int\limits_{0}^{1} 5x - x^3 \,dx = -1*(\int\limits_{0}^{1} x^3 -5x\,dx) = -1*(-9/4) = 9/4[/tex]

Step-by-step explanation:

According to the information given.

[tex]\int\limits_{0}^{1} x^3 - 5x \,dx = -9/4\\[/tex]

Now.

a.

[tex]\int\limits_{0}^{1} 10x - 2x^3 \,dx = -2(\int\limits_{0}^{1} x^3 -5x\,dx) = -2*(-9/4) = 9/2[/tex]

b.

[tex]\int\limits_{0}^{1} 5x - x^3 \,dx = -1*(\int\limits_{0}^{1} x^3 -5x\,dx) = -1*(-9/4) = 9/4[/tex]

Final answer:

The probability of more than 192 confidence intervals containing the true proportion when constructing 95% confidence intervals across 200 towns, which follows a binomial distribution.

Explanation:

When constructing confidence intervals for a proportion, if we state that we have 95% confidence, this means that if we were to take many samples and build a confidence interval from each sample, we would expect 95% of those intervals to contain the population proportion. In the case with 200 towns, 95% confidence suggests that approximately 190 out of 200 intervals would contain the true population proportion, as 95% of 200 is 190.

To find the probability that more than 192 of these intervals contain the true proportion, we would use the binomial distribution, where each interval has a 0.95 probability of containing the true proportion (success), and we are looking for the sum of the probabilities of 193, 194, ..., 200 successes out of 200 trials.

Answer:

(2x100,000) 200,000,00

Step-by-step explanation:

200,000

Answer: 2 hundred thousands

Explanation: To determine what the digit 2 means in 7,239,103, if we put 7,239,103 into the place value chart, we can recognize that the digit 2 is in the hundreds column of the thousands period.

So in this problem, 2 refers to 2 hundred thousands.

Place value chart is attached

in the image provided.

Answer:

16:27

Step-by-step explanation:

16 white squares and 27 shaded squares

The ratio of the area of the unshaded parts to the area of the shaded parts is 4/5.

The numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.

To find the ratio:

Required ratio =  number of unshaded columns / total number of shaded columns

In the given figure 1,

Total no. of columns =3 x 3 = 9

Number of columns shaded = 9 -1 = 8

In the given figure 2,

Total no. of columns = 4 x 4 = 16

Number of columns shaded = 16 - 4 = 12

In the given figure 3,

Total no. of columns = 5 x 5 = 25

Number of columns shaded = 25 - 9 = 16

Similarly;

In the figure 4,

Total no. of columns = 6 x 6 = 36

Number of columns shaded = 36 - 16 = 20

Required ratio =  number of unshaded columns / total number of shaded columns

​Required ratio = 16 / 20

​Required ratio = 4/5

Therefore, the ratio is 4/5.

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Answer:

D - Can't be determined

Step-by-step explanation:

10 * 2 *4 * 16 = 1280

1280/2 = 640

Answer: Angle Bisector

Step-by-step explanation: You divided into two equal angles, you bisected. You turned F, I, G and I, G, H  into two separate triangles.

Answer:

infinite solutions

Step-by-step explanation:

5(- 3x - 2) - (x - 3) = -4(4x + 5) + 13

Ditribute

-15x -10 -x +3 = -16x - 20+13

Combine like term

-16x -7 = -16x -7

Add 16x to each side

-16x+16x -7 = -16x+16x-7

-7 =-7

This is always true so we have infinite solutions.

answers

All real numbers are solutions

step by step

5(−3x−2)−(x−3)=−4(4x+5)+13

Step 1: Simplify both sides of the equation.

5(−3x−2)−(x−3)=−4(4x+5)+13

5(−3x−2)+−1(x−3)=−4(4x+5)+13(Distribute the Negative Sign)

5(−3x−2)+−1x+(−1)(−3)=−4(4x+5)+13

5(−3x−2)+−x+3=−4(4x+5)+13

(5)(−3x)+(5)(−2)+−x+3=(−4)(4x)+(−4)(5)+13(Distribute)

−15x+−10+−x+3=−16x+−20+13

(−15x+−x)+(−10+3)=(−16x)+(−20+13)(Combine Like Terms)

−16x+−7=−16x+−7

−16x−7=−16x−7

Step 2: Add 16x to both sides.

−16x−7+16x=−16x−7+16x

−7=−7

Step 3: Add 7 to both sides.

−7+7=−7+7

0=0

Answer:

[tex]z=\frac{0.843-0.653}{\sqrt{0.748(1-0.748)(\frac{1}{300}+\frac{1}{300})}}=5.358[/tex]    

[tex]p_v =2*P(Z>5.358) = 4.2x10^{-8}[/tex]  

Comparing the p value with 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 reject the null hypothesis, and we can say that we have singificantly differences between the two proportions.  

Step-by-step explanation:

Data given and notation  

[tex]X_{1}=253[/tex] represent the number with no defects in sample 1

[tex]X_{2}=196[/tex] represent the number with no defects in sample 1

[tex]n_{1}=300[/tex] sample 1

[tex]n_{2}=300[/tex] sample 2

[tex]p_{1}=\frac{253}{300}=0.843[/tex] represent the proportion of number with no defects in sample 1

[tex]p_{2}=\frac{196}{300}=0.653[/tex] represent the proportion of number with no defects in sample 2

z would represent the statistic (variable of interest)  

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

[tex]\alpha=0.05[/tex] significance level given

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if is there is a difference in the the two proportions, the system of hypothesis would be:  

Null hypothesis:[tex]p_{1} - p_2}=0[/tex]  

Alternative hypothesis:[tex]p_{1} - p_{2} \neq 0[/tex]  

We need to apply a z test to compare proportions, and the statistic is given by:  

[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)  

Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{253+196}{300+300}=0.748[/tex]  

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.  

Calculate the statistic  

Replacing in formula (1) the values obtained we got this:  

[tex]z=\frac{0.843-0.653}{\sqrt{0.748(1-0.748)(\frac{1}{300}+\frac{1}{300})}}=5.358[/tex]    

Statistical decision

Since is a two sided test the p value would be:  

[tex]p_v =2*P(Z>5.358) = 4.2x10^{-8}[/tex]  

Comparing the p value with 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 reject the null hypothesis, and we can say that we have singificantly differences between the two proportions.  

Answer:

The 95% confidence interval for the proportion is (0.2824, 0.3672). The upper limit of the interval is higher than 32% = 0.32, which means that the confidence interval does not support this claim.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 468, \pi = \frac{152}{468} = 0.3248[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3248 - 1.96\sqrt{\frac{0.3248*0.6752}{468}} = 0.2824[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3248 + 1.96\sqrt{\frac{0.3248*0.6752}{468}} = 0.3672[/tex]

The 95% confidence interval for the proportion is (0.2824, 0.3672). The upper limit of the interval is higher than 32% = 0.32, which means that the confidence interval does not support this claim.

In this problem, we apply mathematical concepts related to statistics to determine whether a campaign claim is supported by a survey. The calculation of a 95% confidence interval supports Mayor Waffleskate's claim that no more than 32% of voters want her defeated.

The subject in question refers to confidence intervals and population proportions, both of which are topics in statistics. To answer it, we need to construct a 95% confidence interval for the proportion of voters who wish to see Mayor Waffleskate defeated in the election, based on a sample size of 468 voters, of which 152 express this sentiment.

The first step is to calculate the sample proportion (p), which is the number of 'successes' over the sample size, or in this case 152/468, which equals approximately 0.325.

Next, we apply the formula for the confidence interval for a population proportion, which i

Because the confidence interval (0.278, 0.372) includes the value of 0.32 stated by the Waffleskate campaign, it can be concluded that the claim that no more than 32% of registered voters wish to see her defeated is plausible or supported by this survey at the 95% confidence level.

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