1. Consider an athlete running a 40-m dash. The position of the athlete is given by , where d is the position in meters and t is the time elapsed, measured in seconds. Compute the average velocity of the runner over the given time intervals. a. b. c. d. e. Use the preceding answers to guess the instantaneous velocity of the runner at sec. ( ) 3 4 6 t dt t = + [1.95, 2.05] [1.995, 2.005] [1.9995, 2.0005] [2, 2.00001] t = 2

Answer: 4.85%

Step-by-step explanation: productivity = parcel delivered / number of drivers.

At when parcel delivered was 103, 000 the average number of drivers was 84 thus making the productivity to be

103,000/ 84 = 1226.19

At when the parcel delivered was 112, 800 the average number of drivers was 96 thus making the productivity to be

112, 000/ 96 = 1166.67

Thus the change in productivity = 1226.19 - 1166.67 = 59.52.

Initial productivity = 1226.19

final productivity = 1166.67

% change in productivity = change in productivity / initial productivity * 100

% change in productivity = 59.52 /1226 * 100

% change in productivity = 4.85%

Answer:

a)

Step-by-step explanation:

Given problem:

Which of the following best describes galena’s cleavage (see animation in placemark)?

Options:

a)three directions at ~90

b) three directions not at ~90°

c) two directions at ~90°

d) two directions not at ~90°

Solution:

- As per the animation where the place-mark is observed as a point. We can see that corner point is a point of intersection of three planes, left, right and bottom. We can also see that these three planes are at right angles @ 90 degrees to each other. Hence, option A.

Answer:

The correct option is (d) 11.070

Step-by-step explanation:

The test statistic for Goodness of fit test for k observations is:

[tex]\chi^{2}=\sum\frac{(Observed-Expected)^{2}}{Expected}[/tex]

This statistic follows a Chi-square distribution with (k - 1) degrees of freedom and α level of significance.

Here α = 0.05 and degrees of freedom is, k - 1 = 6 - 1 = 5 d.f.

Use the chi-square table for the critical value.

[tex]\chi^{2}_{(5)}=11.070[/tex]

Thus, the correct option is (d).

Answer:

a) 99.24% chance HLI will find a sample mean between 5.5 and 7.1 hours.

b) 81.64% probability that the sample mean will be between 5.9 and 6.7 hours.

Step-by-step explanation:

To solve this question, it is important to know the Normal probability distribution and the Central Limit Theorem

Normal probability distribution

Problems of normally distributed samples can be 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 random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].

In this problem, we have that:

[tex]\mu = 6.3, \sigma = 2.1, n = 49, s = \frac{2.1}{\sqrt{49}} = 0.3[/tex]

A) What is the chance HLI will find a sample mean between 5.5 and 7.1 hours?

This is the pvalue of Z when X = 7.1 subtracted by the pvalue of Z when X = 5.5.

By the Central Limit Theorem, the formula for Z is:

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

X = 7.1

[tex]Z = \frac{7.1 - 6.3}{0.3}[/tex]

[tex]Z = 2.67[/tex]

[tex]Z = 2.67[/tex] has a pvalue of 0.9962

X = 5.5

[tex]Z = \frac{5.5 - 6.3}{0.3}[/tex]

[tex]Z = -2.67[/tex]

[tex]Z = -2.67[/tex] has a pvalue of 0.0038

So there is a 0.9962 - 0.0038 = 0.9924 = 99.24% chance HLI will find a sample mean between 5.5 and 7.1 hours.

B) Calculate the probability that the sample mean will be between 5.9 and 6.7 hours.

This is the pvalue of Z when X = 6.7 subtracted by the pvalue of Z when X = 5.9

X = 6.7

[tex]Z = \frac{6.7 - 6.3}{0.3}[/tex]

[tex]Z = 1.33[/tex]

[tex]Z = 1.33[/tex] has a pvalue of 0.9082

X = 5.9

[tex]Z = \frac{5.9 - 6.3}{0.3}[/tex]

[tex]Z = -1.33[/tex]

[tex]Z = -1.33[/tex] has a pvalue of 0.0918.

So there is a 0.9082 - 0.0918 = 0.8164 = 81.64% probability that the sample mean will be between 5.9 and 6.7 hours.

Answer:

a. 74.63%

b. 33.63%

c. 87.67%

Step-by-step explanation:

If 59% (0.59) of the workers are married, then It means (100-59 = 41%) of the workers are not married.

If 43% (0.43) of the workers are College graduates, then it means (100-43= 57%) of the workers are not college graduates.

If 1/3 of college graduates are married, it means portion of graduate that are married = 1/3 * 43% = 1/3 * 0.43 = 0.1433.

For question a, Probability that the worker is neither married nor a college graduate becomes:

= (probability of not married) + (probability of not a graduate) - (probability of not married * not a graduate)

= 0.41 + 0.57 - (0.41*0.57) = 0.98 - 0.2337

= 0.7463 = 74.63%

For question b, probability that the worker is married but not a college graduate becomes:

=(probability of married) * (probability of not a graduate.)

= 0.59 * 0.57

= 0.3363 = 33.63%

For question c, probability that the worker is either married or a college graduate becomes:

=probability of marriage + probability of graduate - (probability of married and graduate)

= 0.59 + 0.43 - (0.1433)

= 0.8767. = 87.67%

The probability that a randomly chosen worker is neither married nor a college graduate is 0%. The probability that a worker is married but not a college graduate is approximately 34%. The probability that a worker is married or a college graduate is approximately 75%.

To calculate the probabilities, we need to use the formulas for conditional probability. Let's solve each part:

a) To find the probability that a randomly chosen worker is neither married nor a college graduate, we can subtract the probability that the worker is married and the probability that the worker is a college graduate from 1. The probability of being married is 59%, and the probability of being a college graduate is 43%. Using the formula, 1 - 0.59 - 0.43 = 0.

b) To find the probability that a worker is married but not a college graduate, we can multiply the probability of being married and the probability of not being a college graduate. The probability of being married is 59%, and the probability of not being a college graduate is 57% (100% - 43%). Using the formula, 0.59 * 0.57 = 0.3363 (approximately 34%).

c) To find the probability that a worker is married or a college graduate, we can add the probabilities of being married and being a college graduate, and then subtract the probability of being both married and a college graduate to avoid double counting. The probability of being married is 59%, the probability of being a college graduate is 43%, and the probability of being both married and a college graduate is 1/3 of the college graduates (1/3 * 43%). Using the formula, 0.59 + 0.43 - (1/3 * 0.43) = 0.745 (approximately 75%).

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Answer: John must make at least 44 tips to reach his goal.

Step-by-step explanation:

Let x represent the number of tips that John must make this week to reach his goal.

John works at a restaurant and gets paid $80 per week plus $5 per tip. This means that the total amount that John would earn in a week if he makes x tips would be

80 + 5x

This week, John wants to earn at least $300. This is expressed as

80 + 5x ≥ 300

5x ≥ 300 - 80

5x ≥ 220

x ≥ 220/5

x ≥ 44

Answer:

[tex]z=-1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 -1.28*0.15=2.058[/tex]

So the value of height that separates the bottom 10% of data from the top 90% is 2.06.

For the upper limit since the distribution is symmetrical we can do this:

[tex]z=1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 +1.28*0.15=2.442[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 2.44.

And the best answer for this case would be:

C. [2.06, 2.44]

Step-by-step explanation:

Previous concepts

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".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the amount of juice of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(2.25,0.15)[/tex]  

Where [tex]\mu=2.25[/tex] and [tex]\sigma=0.15[/tex]

For this case we want the limits for the middle 80% values  of the distribution. so then we need 100-80= 20% of the area in the tails and 10% on each tail since the distribution is symmetrical.

We can use this condition for the lower limits

[tex]P(X>a)=0.9[/tex]   (a)

[tex]P(X<a)=0.1[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.1 of the area on the left and 0.9 of the area on the right it's z=-1.28. On this case P(Z<-1.28)=0.1 and P(z>-1.28)=0.9

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.1[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.1[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 -1.28*0.15=2.058[/tex]

So the value of height that separates the bottom 10% of data from the top 90% is 2.06.

For the upper limit since the distribution is symmetrical we can do this:

[tex]z=1.28<\frac{a-2.25}{0.15}[/tex]

And if we solve for a we got

[tex]a=2.25 +1.28*0.15=2.442[/tex]

So the value of height that separates the bottom 90% of data from the top 10% is 2.44.

And the best answer for this case would be:

C. [2.06, 2.44]

Answer:

a) 95% of the data falls between $135,000 and $175,000.

b) 81.5% of new homes priced between $135,000 and $165,000.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviations of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

We also have that:

50% of the measures are below the mean and 50% of the measures are above the mean.

34% of the measures are between 1 standard deviation below the mean and the mean, and 34% of the measures are between the mean and 1 standard deviations above the mean.

47.5% of the measures are between 2 standard deviations below the mean and the mean, and 47.5% of the measures are between the mean and 2 standard deviations above the mean.

49.85% of the measures are between 3 standard deviations below the mean and the mean, and 49.85% of the measures are between the mean and 3 standard deviations above the mean.

In this problem, we have that:

Mean = $155,000.

Standard deviation = $10,000.

(a) Between what two values do about 95% of the data fall?

By the Empirical Rule, 95% of the values fall within 2 standard deviations of the mean.

So

155000 - 2*10000 = 135,000

155000 + 2*10000 = 175,000

95% of the data falls between $135,000 and $175,000.

(b) Estimate the percentage of new homes priced between $135,000 and $165,000?

We have to find how many fall between $135,000 and the mean($155,000) and how many fall between the mean and $165,000

$135,000 and the mean

$135,000 is two standard deviations below the mean.

By the empirical rule, 47.5% of the measures are between 2 standard deviations below the mean and the mean.

So 47.5% of the measures are between $135,000 and the mean

Mean and $165,000

$165,000 is one standard deviation above the mean.

By the empirical rule, 34% of the measures are between the mean and 1 standard deviations above the mean.

So 34% of the measures are between the mean and $165,000.

$135,000 and $165,000

47.5% + 34% = 81.5% of new homes priced between $135,000 and $165,000.

Final answer:

The answer explains how to determine the range of values where 95% of data falls based on mean and standard deviation and estimates the percentage of new homes within a specific price range.

Explanation:

The mean price for new homes from a sample of houses is $155,000 with a standard deviation of $10,000.

(a) Between what two values do about 95% of the data fall?

(b) Estimate the percentage of new homes priced between $135,000 and $165,000?

Answer:

a) W'(t) = 1.51 -0.0138t + 0.000762t² pounds/day

Step-by-step explanation:

The median weight of a boy with age between 0 and 36 months is given by:

[tex]W(t) = 8.38+1.51t-0.0069t^2 + 0.000254t^3[/tex]

To find the rate of change of weight with respect to time, that is, the change in weight measured in pounds caused by a unit change in time, measured in days, simply derivate the weight function over time:

[tex]\frac{d(W(t)}{dt}= W'(t) = 1.51-0.0138t + 0.000762t^2[/tex]

The rate of change is 1.51 -0.0138t + 0.000762t² pounds/day.

Answer:

1. ΔXYZ is a right Δ with altitude YU.

Given

2. ΔXYZ ~ ΔYUZ

Right Triangle Altitude Similarity Theorem

3. VW || XY

Given

4. ∠VWZ ≅ ∠XYZ

Corresponding angles

5. ∠Z ≅ ∠Z

Reflexive property of congruence

6. ΔXYZ ~ ΔVWZ

AA Similarity postulate

7. ΔYUZ ~ ΔVWZ

Transitive property of similar triangles

Step-by-step explanation:

The first statement is given in the problem.  Since we know the altitude of a right triangle, we can use the Right Triangle Altitude Similarity Theorem to say that the triangles formed by the altitude are similar to each other and the original triangle.

Next, we are given in the problem statement that the lines VW and XY are parallel.  Therefore, ∠VWZ and ∠XYZ are corresponding angles, which makes them congruent.  And since ∠Z is equal to itself (by reflexive property), we can use AA similarity to say ΔXYZ and ΔVWZ are similar.

Finally, combining statements 2 and 6, we can use transitive property to say that ΔYUZ and ΔVWZ are similar.

Answer:

49.34% probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

Step-by-step explanation:

For each television, there are only two possible outcomes. Either they have defective speakers, or they do not. The probabilities of each television having defective speakers are independent from each other. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

[tex]p = 0.01, n = 500[/tex]

Find an approximate probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

This is

[tex]P(5 \leq X \leq 8) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)[/tex]

So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 5) = C_{500,5}.(0.01)^{5}.(0.99)^{495} = 0.1764[/tex]

[tex]P(X = 6) = C_{500,6}.(0.01)^{6}.(0.99)^{494} = 0.1470[/tex]

[tex]P(X = 7) = C_{500,7}.(0.01)^{7}.(0.99)^{493} = 0.1048[/tex]

[tex]P(X = 8) = C_{500,8}.(0.01)^{8}.(0.99)^{492} = 0.0652[/tex]

[tex]P(5 \leq X \leq 8) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.1764 + 0.1470 + 0.1048 + 0.0652 = 0.4934[/tex]

49.34% probability that 5 to 8 televisions (inclusive) in the shipment have defective speakers

The approximate probability that 5 to 8 out of 500 televisions have defective speakers is 0.49. This was calculated using the binomial distribution with n = 500 and p = 0.01. We summed P(X = 5), P(X = 6), P(X = 7), and P(X = 8) to find the result.

To find the approximate probability that 5 to 8 televisions out of 500 have defective speakers, we can use the binomial distribution.

The probability mass function for a binomial distribution is given by:

[tex]P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)[/tex]

where C(n, k) is the binomial coefficient.

We need to calculate P(5 ≤ X ≤ 8), which is P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8).

Using a binomial calculator or statistical software, we get:

Summing these probabilities gives:

P(5 ≤ X ≤ 8) ≈ 0.1755 + 0.1447 + 0.1040 + 0.0681

≈ 0.4923

Therefore, the approximate probability that 5 to 8 televisions have defective speakers is 0.49 (rounded to two decimal places)

Answer:

a) 23%

Step-by-step explanation:

To find the price change as percentage :

Multiply 100 by 160 then divide it by 130

100 × 160 ÷ 130 = 123 approximately which means there's an additional 23% to the 100% price

Answer:

(a) The probability that a selected joint was judged to be defective by neither of the two inspectors is 0.906.

(b) The probability that a selected joint was judged to be defective by inspector B but not by inspector A is 0.0213.

Step-by-step explanation:

The sample of joints randomly selected is, n = 10,000.

Number of joints judged defective by inspector A is, n (A) = 727.

The probability that a joint is judged defective by inspector A is:

[tex]P(A)=\frac{n(A)}{n} =\frac{727}{10000} =0.0727[/tex]

Number of joints judged defective by inspector B is, n (B) = 756.

The probability that a joint is judged defective by inspector B is:

[tex]P(B)=\frac{n(B)}{n} =\frac{756}{10000} =0.0756[/tex]

Number of joints judged defective by at least one of the inspectors is,

n (At least 1) = 940.

The probability that a joint is judged defective by at least one of the inspectors is:

[tex]P(At\ least\ 1)=\frac{n(At\ least\ 1)}{n} =\frac{940}{10000}=0.094[/tex]

(a)

Compute the probability that a selected joint was judged to be defective by neither of the two inspectors as follows:

P (At least 1) = 1 - P (Less than 1)

                    = 1 - P (None)

     P (None) = 1 - P (At least 1)

                    [tex]=1-0.094\\=0.906[/tex]

Thus, the probability that a selected joint was judged to be defective by neither of the two inspectors is 0.906.

(b)

Compute the probability that a selected joint was judged to be defective by inspector B but not by inspector A as follows:

P (B but not A) = P (At least 1) - P (A)

                        [tex]=0.094-0.0727\\=0.0213[/tex]

Thus, the probability that a selected joint was judged to be defective by inspector B but not by inspector A is 0.0213.

Answer:

The probability that more than 8 and less than 12 customers pay in cash is 0.0931.

Step-by-step explanation:

Let X = a customer at David's gasoline station pay in cash.

The probability of a customer paying in cash is, P (X) = p = 0.40

The number of customers at the gasoline station during a 1-hour period is,

n = 15.

Then the random variable X follows a binomial distribution, Bin (15, 0.40).

The probability function for a Binomial distribution is:

[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x}[/tex]

Compute the probability that more than 8 and less than 12 customers pay in cash as follows:

[tex]P(8< X< 12)=P(X<12)-P(X<8)\\=P(X=9)+P(X=10)+P(X=11)\\=[{15\choose 9}(0.40)^{9}(1-0.40)^{15-9}]+[{15\choose 10}(0.40)^{10}(1-0.40)^{15-10}]\\+[{15\choose 11}(0.40)^{11}(1-0.40)^{15-11}]\\=0.0612+0.0245+0.0074\\=0.0931[/tex]

Thus, the probability that more than 8 and less than 12 customers pay in cash is 0.0931.

The problem comes to calculating binomial probabilities for when 9, 10, and 11 customers pay in cash and adding them together. This scenario applies to binomial distribution, where we have a success (paying in cash) happening with a probability of 40%.

This question is a problem of the binomial distribution. The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial (often referred to as success and failure). In this case, the success is the customer paying in cash, which happens 40% of the time according to past evidence.

The formula for the binomial distribution is:

P(X = k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where

Carry out this calculation for k=9, 10, 11, and then add these probabilities together to get the probability that more than 8 and less than 12 customers pay in cash.

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Answer:

Step-by-step explanation:

$97.50=15hours

$x=1hour

$15x= 97.50

X= 97.50/15

X= 6.5$

Therefore

1hour=6.5$

Therefore

130/6.5

=20hours

Answer:

a)

[tex]H_0: \pi\geq0.78\\\\H_a: \pi<0.78[/tex]

b) The Type I error occurs when we reject a null hypothesis that is actually true. In this case, it means we conclude that the arrival time have improved, when it didn't.

The Type II error occurs when we accept a null hypothesis that is actually false. In this case, although the arrival times have really improved, the evidence from the sample was not enough to show that improvement.

c) In this case, the Type I error is more serious, because it gives the wrong impression of improvement and no further actions will be taken to reduce the times.

Step-by-step explanation:

a) If you want to determine if the responders are arriving within 8 minutes of the call more often, you have to evaluate the proportion of accidents in which the arrival time is less than 8 minutes and compare it with the known proportion of π=0.78.

The sample parameter "p: proportion of accidents with arrival time of 8 minutes or less" will be used to test the hypothesis.

The null and alternative hypothesis will be:

[tex]H_0: \pi\geq0.78\\\\H_a: \pi<0.78[/tex]

Final answer:

The hypotheses for a significance test to determine whether first responders are arriving more often within 8 minutes are H0: p = 0.78 and HA: p > 0.78, with p representing the proportion of calls responded to within this timeframe. A Type I error involves mistakenly concluding an improvement, while a Type II error occurs by overlooking an actual improvement. The potential demotivating effect of a Type II error may render it more serious in this context.

Explanation:

The question seems to revolve around the concept of hypothesis testing in statistics and how it applies to the emergency response times in a city. The parameter of interest would be the proportion of emergency calls responded to within 8 minutes. So for part (a), the hypotheses could be stated as follows:

H0: p = 0.78 (The proportion of calls responded to within 8 minutes is 78% as it was last year.)

HA: p > 0.78 (The proportion of calls responded to within 8 minutes has increased from last year.)

In part (b), a Type I error would occur if the city concludes that the proportion of calls responded to within 8 minutes has increased when in reality, it has not. The consequence of a Type I error would be misallocating resources based on false success. A Type II error would occur if the city fails to recognize an actual improvement in response times. The consequence of this could lead to a lack of recognition and continued encouragement for first responders who have actually improved.

Part (c) asks which error is more serious. A Type II error may be considered more serious in this setting, as failing to acknowledge and react to an actual improvement could demotivate emergency personnel and affect future performances, possibly leading to life-threatening delays for accident victims.

Answer:

All options( option A, option B and option C)

Step-by-step explanation:

Mean, median and mode are used to summarize the data. Mode can be calculated for both quantitative and qualitative data but mean and median cannot be calculated for qualitative data.

Here outstanding balance on each account represents quantitative data and mode, median and mean all can summarize the quantitative data. So, mean, median and mode each can be used to summarize the data of outstanding balance on each account.

Answer:

the equation[tex]5(\frac{d^{2}x }{dt^{2} }) +4(\frac{dx}{dt})+9x=2cos3t[/tex] is a partial differential equation(PDE) because it contains unknown multi variables and their derivatives. This is a PDE of order 2.

The independent variable is x while the dependent variable is t.

The PDE is Linear.

Step-by-step explanation:

Partial Differential Equation (PDE): This is a differential equation that contains multi variables and their derivatives.

Ordinary Differential Equation (ODE): this is a differential equation containing a function of one independent variable and its derivatives.

Answer:

92.10% probability that a sheet selected at random from the population is between 29.75 and 30.5 inches long.

Step-by-step explanation:

Problems of normally distributed samples can be 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 = 30.05, \sigma = 0.2[/tex]

What is the probability that a sheet selected at random from the population is between 29.75 and 30.5 inches long?

This is the pvalue of Z when X = 30.5 subtracted by the pvalue of Z when X = 29.75

X = 30.5

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

[tex]Z = \frac{30.5 - 30.05}{0.2}[/tex]

[tex]Z = 2.25[/tex]

[tex]Z = 2.25[/tex] has a pvalue of 0.9878

X = 29.75

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

[tex]Z = \frac{29.75 - 30.05}{0.2}[/tex]

[tex]Z = -1.5[/tex]

[tex]Z = -1.5[/tex] has a pvalue of 0.0668

So there is a 0.9878 - 0.0668 = 0.9210 = 92.10% probability that a sheet selected at random from the population is between 29.75 and 30.5 inches long.

Answer:

  ai) 5.5

  aii) 22

  aiii) 4

  bi) 0

  bii) 0

  biii) undefined

Step-by-step explanation:

ai) The change in x is x2 -x1 = 7.5 -2 = 5.5 . . . . change in x

__

aii) The change in y is (4(7.5) +8) -(4(2) +8) = 4(7.5-2) = 22.0 . . . . change in y

__

aiii) The ratio of changes is (change in y)/(change in x) = 22/5.5 = 4

The change in y is 4 times the change in x.

___

bi) The difference is -5.1 -(-5.1) = 0 . . . . change in x

__

bii) The difference is (4(-5.1)+8) -(4(-5.1)+8) = 0 . . . . change in y

__

biii) The ratio of changes is (change in y)/(change in x) = 0/0 = undefined.

The multiplier of the change in x to get the change in y is undefined.

_____

Comment on part B

We know that the relative rates of change for x and y in this linear function are 1 : 4. However, we cannot compute that ratio directly when the change in x is 0. (The ratio holds for vanishingly small values of change in x, so is 4 in the limit as Δx → 0. That isn't what the problem asks.)

Final answer:

Over the given interval, x changes by 5.5 and y changes by 38. The ratio of the change in y to the change in x is approximately 6.91. Over the given interval, x remains constant at -5.1 and y does not change. Therefore, the ratio of the change in y to the change in x is undefined.

Explanation:

i. To find the change in x, we subtract the initial value from the final value: 7.5 - 2 = 5.5.

ii. To find the change in y, we substitute the initial and final values of x into the equation y = 4x + 8 and subtract: y_final - y_initial = (4 * 7.5 + 8) - (4 * 2 + 8) = 38.

iii. To find the ratio of the change in y to the change in x, we divide the change in y by the change in x: 38 / 5.5 ≈ 6.91.

iv. To find the change in x, we subtract the initial value from the final value: -5.1 - -5.1 = 0.

v. To find the change in y, we substitute the initial and final values of x into the equation y = 4x + 8 and subtract: y_final - y_initial = (4 * -5.1 + 8) - (4 * -5.1 + 8) = 0.

vi. To find the ratio of the change in y to the change in x, we divide the change in y by the change in x: 0 / 0.

Answer:

You would expect 95% of your customers to stay in the store from 43.1 minutes to 57.5 minutes.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 50.3 minutes

Standard deviation = 3.6 minutes

Within what range would you expect 95% of your customers to stay in your store?

Within 2 standard deviations of the mean.

So from

50.3 - 2*3.6 = 43.1 minutes

To

50.3 + 2*3.6 = 57.5 minutes

You would expect 95% of your customers to stay in the store from 43.1 minutes to 57.5 minutes.

Answer:

Answers may vary but will most likely be close to 2.

Step-by-step explanation

                 first test:38%

                 second test:76%

    SIMULATION FIRST TEST

Randomly select a 2-digit number.

If the digit is between 00 and 35 then you passed the test,else you did not pass the test.

SIMULATION SECOND TEST

Randomly select a 2-digit number.

if the digit is between 00 and 75 then you passed the test,else you did not pass the test.

SIMULATION TRIAL

Perform the simulation of the first test.if you did not pass the first test then perform the simulation of the second test.

Record the number of trials needed to pass the first or second test.

Repeat 20 times and take the average of the 20 recorded number of trials

(what is the sum of recorded values divided by 20).

Note:you will most likely obtain a result of about two trials needed.

The estimated average number of tests drivers take to get a license is 1.9

To estimate the average number of tests drivers take to get a license, we can simulate the process using a simple random experiment. We will assume that each test attempt is independent of the others and that the probabilities of passing are 38% on the first attempt and 76% on subsequent attempts. Here's how we can perform the simulation:

1. For each driver, simulate the number of tests they take by generating a random number between 0 and 1. If the number is less than or equal to 0.38 (38%), the driver passes on the first attempt. Otherwise, they proceed to a second attempt.

2. For the second attempt and any subsequent attempts, generate another random number. If the number is less than or equal to 0.76 (76%), the driver passes.

3. Repeat this process until the driver passes the test. Record the number of attempts it took.

4. Perform this simulation for 20 drivers to get a sample size that will help estimate the average number of tests taken.

5. Calculate the average number of tests by summing the number of tests taken by all drivers and dividing by the number of drivers (20).

Let's perform the simulation:

1. For the first attempt, generate a random number for each of the 20 drivers and check if it's less than or equal to 0.38.

2. For any subsequent attempts, generate a random number for each driver who did not pass on the first attempt and check if it's less than or equal to 0.76.

3. Record the number of attempts for each driver.

4. Sum the total number of attempts and divide by 20 to find the average.

Now, let's calculate the average number of tests based on the simulation: Assuming we have performed the simulation and recorded the number of attempts for each driver, we would have a list of numbers. For example, it might look something like this:

 Driver 1: 1 attempt

Driver 2: 2 attempts

Driver 3: 1 attempt

Driver 20: 3 attempts

Let's say after performing the simulation, we have the following counts for each number of attempts:

1 attempt: 8 drivers

2 attempts: 7 drivers

3 attempts: 4 drivers

4 attempts: 1 driver

Now, we calculate the average number of tests taken by these 20 drivers:

Average = (8 * 1 + 7 * 2 + 4 * 3 + 1 * 4) / 20

Average = (8 + 14 + 12 + 4) / 20

Average = 38 / 20

Average = 1.9

a. PMF: [tex]\(P(X=k) = \frac{1}{2^k}\) for \(k=0,1,2,3\); \(P(X > 3) = \frac{1}{2^3}\)[/tex]

b. [tex]\(E(X) = \frac{11}{4}\)[/tex]

a. To calculate the probability mass function (PMF) of X, we need to consider the different scenarios leading to the player's success or removal from the game. Let P(X = k) be the probability that the player misses \(k\) shots before succeeding or getting removed.

[tex]\[P(X = 0) = \frac{1}{2}\][/tex]

The player succeeds on the first shot.

[tex]\[P(X = 1) = \frac{1}{2} \cdot \frac{1}{3}\][/tex]

(The player misses the first shot, then succeeds on the second.)

[tex]\[P(X = 2) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{1}{4}\][/tex]

(The player misses the first two shots, then succeeds on the third.)

[tex]\[P(X = 3) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{1}{5}\][/tex]

(The player misses the first three shots, then succeeds on the fourth.)

[tex]\[P(X > 3) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5}\][/tex]

(The player misses the first four shots and is removed.)

b. To compute the expected value of X, we use the formula:

[tex]\[E(X) = \sum_{k=0}^{\infty} k \cdot P(X = k)\][/tex]

[tex]\[E(X) = 0 \cdot P(X = 0) + 1 \cdot P(X = 1) + 2 \cdot P(X = 2) + 3 \cdot P(X = 3) + \sum_{k=4}^{\infty} k \cdot P(X > 3)\][/tex]

Now plug in the values from part (a) into this formula and compute the sum. The result will give you the expected value of X.

Answer:

1.8413 is the radian measure of angle at the center of circle.

Step-by-step explanation:

We are given the following in the question:

Radius,r = 70.0 cm

Arc length,s = 127 cm

Formula:

[tex]\theta = \dfrac{s}{r}[/tex]

where [tex]\theta[/tex] is the angle measure in radians, s is the intercepted arc and r is the radius of the circle.

Putting the values, we get,

[tex]\theta = \dfrac{127}{70} = 1.8143[/tex]

Thus, 1.8413 is the radian measure of angle at the center of circle.

The radian measure of an angle in a circle, given an arc length of 127 cm and radius of 70 cm, is approximately 1.814. This is computed by dividing the arc length by the radius.

In mathematics, the measure of an angle in radians in a circle is given by the ratio of the length of the arc that it subtends and the radius of the circle in which this occurs. This is represented by the formula: θ = s / r where s is the arc length and r is the radius.

Given the radius r = 70 cm and arc length s = 127 cm, you can substitute these values into the formula to get the radian measure of the angle. Therefore, by substitution we get: θ = 127 cm / 70 cm which simplifies to approximately 1.814 radians.

To note, a radian is the standard unit of angular measure, used in many areas of mathematics. An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle, hence why this method works.

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Answer:

14

Step-by-step explanation:

The inter-quartile range (IQR) is the difference between the third and first quartile. The data gathered is:

57, 59, 60, 62, 62, 63, 64, 68, 70, 70, 71, 71, 73, 79, 87, 89, 90

The data set has 17 values, the first quartile is the average between the 4th and 5th values, while the third quartile is the average between the 13th and 14th values:

[tex]Q_1 = \frac{62+62}{2}=62\\Q_3 = \frac{73+79}{2}=76[/tex]

The IQR is:

[tex]IQR = Q_3 - Q_1 = 76 -62\\IQR = 14[/tex]

The Interquartile Range (IQR) of the given data for resting heart rate among college students is 11, calculated by finding the difference between the upper (third) and lower (first) quartiles.

Given the following data for resting heartrate among college students: 57, 59, 60, 62, 62, 63, 64, 68, 70, 70, 71, 71, 73, 79, 87, 89, 90, we are to find the Interquartile Range (IQR). The IQR is a statistical term that measures the statistical spread, or variability, of data points. It is calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1).

First, we need to arrange the data points in ascending order, which has already been done for us in this case. Next, let's divide the data into two halves: the lower half and the upper half. For our data, the lower half is 57 to 68 and the upper half is 70 to 90. Find the median (middle value) of each half. The lower half median (Q1) is 62, and upper half median (Q3) is 73.

Now, subtract Q1 from Q3 to find the IQR: 73 - 62 = 11. So, the IQR of the resting heart rates of the given college students data is 11.

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Answer:

m arc DC=48°

Step-by-step explanation:

Angles in a Circle

We know [tex]\overline{BE}[/tex] and [tex]\overline{AC}[/tex] are diameters, so

[tex]m\angle BFA+m\angle AFE=180^o[/tex].

Since [tex]m\angle AFE=3x+12[/tex] and [tex]m\angle BFA=4x[/tex]

We set the equation

[tex]3x+12+4x=180^o[/tex]

Solving

[tex]7x=180-12=168[/tex]

[tex]x=24^o[/tex]

Thus

[tex]m\angle BFA=4(24)=96^o[/tex]

We also know

[tex]m\angle CFD\cong m\angle DFE[/tex]

Being [tex]\angle CFE[/tex] opposite to [tex]\angle BFA[/tex]

Then [tex]m\angle CFE=96^o[/tex]

It's divided in two equal angles [tex]\angle CFD\ and\ \angle DFE[/tex], thus m arc DC is half of 96°:

m arc DC=48°

First option

Answer:

∑E(x[tex]_{i}[/tex]) = 13.49167 floors

Step-by-step explanation:

The expected number of floors no one get off = ∑E(x[tex]_{i}[/tex]) where i is from 0 to 23

and E(x[tex]_{i}[/tex]) = ∑x[tex]_{i}[/tex]P(x[tex]_{i}[/tex])

here x[tex]_{i}[/tex] is the indicator of floor where no one gets off, its value is 0 when  atleast one person get off on its floor and 1 when when no one gets off.

Now,

P(x[tex]_{i}[/tex]=1) = (22/23)¹²

P(x[tex]_{i}[/tex]=0) = [1-(22/23)¹²]

Now,

E(x[tex]_{i}[/tex]) = ∑x[tex]_{i}[/tex]P(x[tex]_{i}[/tex]) = 0* [1-(22/23)¹²] + 1*(22/23)¹² =0.586594704

For total number of floors where no one gets off

∑E(x[tex]_{i}[/tex]) = E(x₁)+E(x₂)+E(x₃)........................+E(x₂₃)

∑E(x[tex]_{i}[/tex]) = 23*0.586594704

∑E(x[tex]_{i}[/tex]) = 13.49167 floors

Final answer:

To answer, we calculate the expectation by considering the probability of a floor being skipped by all 12 people in a 24-floor building. The expected number of floors not chosen by anyone is around 8.

Explanation:

To find this, we can use the concept of probability. For any given floor (except the first), the probability that a single person does not choose it is (23/23) for the first choice, and the probability that they do choose another floor is (22/23), considering there are 23 floors above the first. Since each of the 12 people chooses independently, the probability that a given floor is not chosen by any of the 12 people is ((22/23)¹²).

With 23 possible floors (excluding the first) for people to exit on, the expected number of floors not chosen by anyone is 23 * ((22/23)¹²). A calculation reveals this to be approximately 7.9. Therefore, there's an expectation that around 8 floors will have no one exiting on them, under the given conditions.

Answer: the selling price of the boat is $57.5

Step-by-step explanation:

The original price of a toy boat was $50. The boat is marked up 15% before it’s sold. This means that the amount by which the original price of the toy boat was increased would be

15/100 × 50 = 0.15 × 50 = $7.5

The selling price of the toy boat would be the sum of its original price and the amount by which it was marked up. It becomes.

50 + 7.5 = $57.5

Answer:

The parenthesis need to be kept intact while applying the DeMorgan's theorem on the original equation to find the compliment because otherwise it will introduce an error in the answer.

Step-by-step explanation:

According to DeMorgan's Theorem:

(W.X + Y.Z)'

(W.X)' . (Y.Z)'

(W'+X') . (Y' + Z')

Note that it is important to keep the parenthesis intact while applying the DeMorgan's theorem.

For the original function:

(W . X + Y . Z)'

= (1 . 1 + 1 . 0)

= (1 + 0) = 1

For the compliment:

(W' + X') . (Y' + Z')

=(1' + 1') . (1' + 0')

=(0 + 0) . (0 + 1)

=0 . 1 = 0

Both functions are not 1 for the same input if we solve while keeping the parenthesis intact because that allows us to solve the operation inside the parenthesis first and then move on to the operator outside it.

Without the parenthesis the compliment equation looks like this:

W' + X' . Y' + Z'

1' + 1' . 1' + 0'

0 + 0 . 0 + 1

Here, the 'AND' operation will be considered first before the 'OR', resulting in 1 as the final answer.

Therefore, it is important to keep the parenthesis intact while applying DeMorgan's Theorem on the original equation or else it would produce an erroneous result.

The error originates from an incorrect application of DeMorgan's theorem. The correct complement of (W · X) + (Y · Z) is (W' + X') · (Y' + Z'). This corrects the discrepancy seen for WXYZ = 1110.

The confusion here likely originates from a mistake in the DeMorgan's theorem application. According to DeMorgan's Theorem, the complement of (W · X) + (Y · Z) is given by (W' + X') · (Y' + Z'), not W' + X' · Y' + Z'.

So, if we have WXYZ = 1110, the given function (W · X) + (Y · Z) equals 1 because we have (1 · 1) + (1 · 0) = 1 + 0 = 1. Whereas, the correct complement function, (W' + X') · (Y' + Z') equals 0 because we have (0 + 0) · (0 + 1) = 0 · 1 = 0.

This explains why we were seeing both original function and the incorrectly applied complement function evaluating to 1 for the same input combination.

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