Show triangle XMZ is congruent to triangle YMZ. Support each part of your answer. Hint: mark the diagram.

Answer: 63 defective widgets

Step-by-step explanation:

Given that the proportion should not exceed 5%, that is:

p< or = 5%.

So we take p = 5% = 0.05

q = 1 - 0.05 = 0.095

Where q is the proportion of non-defective

We need to calculate the standard error (standard deviation)

S = √pq/n

Where n = 1024

S = √(0.05 × 0.095)/1024

S = 0.00681

Since production is to maximize profit(profitable), we need to minimize the number of defective items. So we find the limit of defective product to make this possible using the Upper Class Limit.

UCL = p + Za/2(n-1) × S

Where a is alpha of confidence interval = 100 -90 = 10%

a/2 = 5% = 0.05

UCL = p + Z (0.05) × 0.00681

Z(0.05) is read on the t-distribution table at (n-1) degree of freedom, which is at infinity since 1023 = n-1 is large.

Z a/2 = 1.64

UCL = 0.05 + 1.64 × 0.00681

UCL = 0.0612

Since the UCL in this case is a measure of proportion of defective widgets

Maximum defective widgets = 0.0612 ×1024 = 63

Alternatively

UCL = p + 3√pq/n

= 0.05 + 3(0.00681)

= 0.05 + 0.02043 = 0.07043

UCL =0.07043

Max. Number of widgets = 0.07043 × 1024

= 72

Answer:

(a)f(t)=1090-5t

(b)f(0)=1090metres

(c)g(t)=1600-9t

(d)g(0)=1600metres

(e)The Hare

Step-by-step explanation:

Total Distance =1600 metres

(a)The tortoise has a 510m headstart and a speed of 5m/s

Distance=Speed X Time

Distance at 5m/s = 5t

Total Distance covered by the tortoise at any time t= 510+5t

Therefore, The tortoise's distance from the finish line (in meters) in terms of the number of seconds t since the start of the race is given as:

f(t)=1600-(510+5t)

f(t)=1090-5t

(b)f(0)=1090-(5X0)

=1090metres

(c)The hare has a speed of 9m/s

Distance=Speed X Time

Distance at 9m/s = 9t

Total Distance covered by the hare at any time t= 9t

Therefore, The hare's distance from the finish line (in meters) in terms of the number of seconds t since the start of the race is given as:

g(t)=1600-9t

(d)

g(0)=1600-(9X0)=1600metres

(e)The race is finished when the distance from the finish line=0

For the Tortoise

f(t)=1090-5t=0

1090=5t

t=218seconds

For the Hare

g(t)=1600-9t=0

9t=1600

t=177.8seconds

The hare takes a shorter time to reach the finish line so he won the race.

Answer:

(B) 4 minutes after the rocket was launched, the rocket is 516 miles above the ground

Step-by-step explanation:

The function h(t) represents the height of the rocket above the launchpad after a time t minutes.

h(4) = 516 means that when t = 4 minutes, the height h is 516 miles above the launch pad. Note that the time t is measured from when the rocket is launched.

The first option indicates a rate of change, which is a velocity. This is not indicated in the question because velocity is a time derivative of the the height function.

Option C implies that the rocket is 516 miles currently but we do not know what time currently is from the time of launch.

The fourth option has reversed the roles of the variable by implying the time is 516 minutes while the height is 4 miles which is not what the function means.

Answer:

Below the classical model, economic growth is necessarily achieved because of stability in the wage level. For instance, one case of unemployment predominates at a real wage (W / P)1.

Currently, the excessive labor supply would lower the actual wage level before labor supply equals its demand. Eventually, real wage rates would decline to (W / P)F, whereby aggregate labor demand is perfectly matches by aggregate labor supply.

Only the supply side of the production market for products defines the quantity of output & jobs in the classical model.

As the classical method is supply-determined, it states that equiproportional increases (or declines) will not alter the supply of labor in both the rate of money wage and the price level.

Answer: the probability that a measurement exceeds 13 milliamperes is 0.067

Step-by-step explanation:

Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = current measurements in a strip.

µ = mean current

σ = standard deviation

From the information given,

µ = 10

σ = 2

We want to find the probability that a measurement exceeds 13 milliamperes. It is expressed as

P(x > 13) = 1 - P(x ≤ 13)

For x = 13,

z = (13 - 10)/2 = 1.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.933

P(x > 13) = 1 - 0.933 = 0.067

Final answer:

To find the probability that a current measurement exceeds 13 milliamperes in a normally distributed set with mean 10 mA and SD 2 mA, calculate the Z-score and use a normal distribution table or software.

Explanation:

The student's question seems to mistakenly mix units (millimeters instead of milliamperes), but assuming the intent was to refer to electrical current and not physical measurements of wire, we will proceed on the basis that the actual question is about the probability of a current measurement exceeding 13 milliamperes.

To calculate the probability that a measurement exceeds 13 milliamperes when the current measurements in a strip of wire are normally distributed with a mean of 10 milliamperes and standard deviation of 2 milliamperes, we need to use the Z-score formula:

Z = (X - μ) / σ

where X is the value in question (13 milliamperes), μ is the mean (10 milliamperes), and σ is the standard deviation (2 milliamperes).

Plugging in the values:

Z = (13 - 10) / 2 = 1.5

After finding the Z-score, we would look up this value in a standard normal distribution table or use a statistical software to find the probability that Z exceeds 1.5, which gives us the probability that a measurement exceeds 13 milliamperes. For this Z-score, the probability is approximately 6.68%.

Answer:

a) the probability is 0.12 (12%)

b) the probability is 0.61 (61%)

c) the probability is 0.476 (47.6%)

a) the probability is 0.538 (53.8%)

Step-by-step explanation:

a) denoting the event H= hits her target and G= a gust of wind appears hen

P(H∩G) = probability that a gust of wind appears * probability of hitting the target given that is windy = 0.3* 0.4 = 0.12 (12%)

b) for any given shot

P(H)= probability that a gust of wind appears*probability of hitting the target given that is windy + probability that a gust of wind does not appear*probability of hitting the target given that is not windy = 0.3*0.4+0.7*0.7 = 0.12+0.49 = 0.61 (61%)

c) denoting P₂  as the probability of hitting once in 2 shots  and since the archer can hit in the first shot or the second , then

P₂ = P(H)*(1-P(H))+ (1-P(H))*P(H) = 2*P(H) *(1-P(H)) = 2*0.61*0.39= 0.476 (47.6%)

d) for conditional probability we can use the theorem of Bayes , where

M= the archer misses the shot → P(M) = 1- P(H) = 0.39

S= it is not windy when the archer shots → P(S) = 1- P(G) = 0.7

then

P(S/M) = P(S∩M)/P(M) = 0.7*(1-0.7)/0.39 = 0.538 (53.8%)

where P(S/M)  is the probability that there was no wind when the archer missed the shot

Answer: 60 gallons of the 20% solution and 120 gallons of the 50% solution should be mixed.

Step-by-step explanation:

Let x represent the number of gallons of 20% solution that should be mixed.

Let y represent the number of gallons of 50% solution that should be mixed.

A 20% solution of fertilizer is to be mixed with a 50% solution of fertilizer in order to get 180 gallons of a 40% solution. This means that

0.2x + 0.5y = 0.4×180

0.2x + 0.5y = 72- - - - - - - - - - - -1

Since the total number of gallons is 180, it means that

x + y = 180

Substituting x = 180 - y into equation 1, it becomes

0.2(180 - y) + 0.5y = 72

36 - 0.2y + 0.5y = 72

- 0.2y + 0.5y = 72 - 36

0.3y = 36

y = 36/0.3

y = 120

x = 180 - y = 180 - 120

x = 60

To get a 40% solution by mixing a 20% and 50% solution, you need 60 gallons of the 20% solution and 120 gallons of the 50% solution.

To solve this problem, we can use the concept of mixtures. Let's represent the amount of 20% solution as x gallons and the amount of 50% solution as y gallons. From the given information, we can set up the following equations:

x + y = 180 (equation 1)

0.2x + 0.5y = 0.4 * 180 (equation 2)

Simplifying equation 2 gives us 0.2x + 0.5y = 72. To eliminate decimals, we can multiply both sides by 10 to get 2x + 5y = 720.

Now we have a system of equations. We can solve it by substitution or elimination. Let's use elimination:

Multiplying equation 1 by 2 gives us 2x + 2y = 360. Subtracting this from equation 2 gives us 3y = 360. Dividing both sides by 3 gives us y = 120. Substituting this value into equation 1 gives us x + 120 = 180. Subtracting 120 from both sides gives us x = 60.

Therefore, we need 60 gallons of the 20% solution and 120 gallons of the 50% solution.

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Answer:

a) Mean = 1.6, standard deviation = 1.21

b) 18.87% probability that zero online retail orders will turn out to be fraudulent.

c) 32.82% probability that one online retail order will turn out to be fraudulent.

d) 48.31% probability that two or more online retail orders will turn out to be fraudulent.

Step-by-step explanation:

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.

The mean of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

In this problem, we have that:

[tex]p = 0.08, n = 20[/tex]

a. What are the mean and standard deviation of the number of online retail orders that turn out to be fraudulent?

Mean

[tex]E(X) = np = 20*0.08 = 1.6[/tex]

Standard deviation

[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{20*0.08*0.92} = 1.21[/tex]

b. What is the probability that zero online retail orders will turn out to be fraudulent?

This is P(X = 0).

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

[tex]P(X = 0) = C_{20,0}.(0.08)^{0}.(0.92)^{20} = 0.1887[/tex]

18.87% probability that zero online retail orders will turn out to be fraudulent.

c. What is the probability that one online retail order will turn out to be fraudulent?

This is P(X = 1).

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

[tex]P(X = 1) = C_{20,1}.(0.08)^{1}.(0.92)^{19} = 0.3282[/tex]

32.82% probability that one online retail order will turn out to be fraudulent.

d. What is the probability that two or more online retail orders will turn out to be fraudulent?

Either one or less is fraudulent, or two or more are. The sum of the probabilities of these events is decimal 1. So

[tex]P(X \leq 1) + P(X \geq 2) = 1[/tex]

We want [tex]P(X \geq 2)[/tex]

So

[tex]P(X \geq 2) = 1 - P(X \leq 1)[/tex]

In which

[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]

From itens b and c

[tex]P(X \leq 1) = 0.1887 + 0.3282 = 0.5169[/tex]

[tex]P(X \geq 2) = 1 - P(X \leq 1) = 1 - 0.5169 = 0.4831[/tex]

48.31% probability that two or more online retail orders will turn out to be fraudulent.

The probability is an illustration of a binomial distribution.

The given parameters are:

n = 20

p = 0.08

The mean is calculated as:

[tex]\bar x = np[/tex]

So, we have:

[tex]\bar x = 20 * 0.08[/tex]

[tex]\bar x = 1.6[/tex]

The standard deviation is calculated as:

[tex]\sigma = \sqrt{\bar x * (1 - p)[/tex]

This gives

[tex]\sigma = \sqrt{1.6 * (1 - 0.08)[/tex]

[tex]\sigma = 1.21[/tex]

Hence, the mean is 1.6 and the standard deviation is 1.21

This is calculated as:

[tex]P(x) = ^nC_x * p^x * (1 - p)^{n-x}[/tex]

So, we have:

[tex]P(0) = ^{20}C_0 * 0.08^0 * (1 - 0.08)^{20 - 0}[/tex]

[tex]P(0) =0.1887[/tex]

The probability that zero online retail orders will turn out to be fraudulent is 0.1887

This is calculated as:

[tex]P(x) = ^nC_x * p^x * (1 - p)^{n-x}[/tex]

So, we have:

[tex]P(1) = ^{20}C_1 * 0.08^1 * (1 - 0.08)^{20 - 1}[/tex]

[tex]P(1) =0.3281[/tex]

The probability that one online retail orders will turn out to be fraudulent is 0.3281

This is calculated as:

[tex]P(x\ge 2) = 1 - P(0) - P(1)[/tex]

So, we have:

[tex]P(x\ge 2) = 1 - 0.1887 - 0.3281[/tex]

[tex]P(x\ge 2) = 0.4832[/tex]

The probability that two or more online retail orders will turn out to be fraudulent is 0.4832

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Answer:

a) 5.48% probability that a baby chosen at random weighs less than 5.5 pounds at birth

b) 0.28% probability that their average birth weight is less than 5.5 pounds

Step-by-step explanation:

To solve this question, the normal probability distribution and the central limit theorem are used.

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 sample means n of at least 30 can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

[tex]\mu = 7.5, \sigma = 1.25[/tex]

(a) What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth?

This is the pvalue of Z when X = 5.5. So

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

[tex]Z = \frac{5.5 - 7.5}{1.25}[/tex]

[tex]Z = -1.6[/tex]

[tex]Z = -1.6[/tex] has a pvalue of 0.0548

5.48% probability that a baby chosen at random weighs less than 5.5 pounds at birth

(b) You choose three babies at random. What is the probability that their average birth weight is less than 5.5 pounds?

[tex]n = 3, s = \frac{1.25}{\sqrt{3}} = 0.7217[/tex]

This is the pvalue of Z when X = 5.5. So

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

By the Central Limit Theorem

This is the pvalue of Z when X = 5.5. So

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

[tex]Z = \frac{5.5 - 7.5}{0.7217}[/tex]

[tex]Z = -2.77[/tex]

[tex]Z = -2.77[/tex] has a pvalue of 0.0028

0.28% probability that their average birth weight is less than 5.5 pounds

Answer:

(a) The mean is 52 and the median is 55.

(b) The first quartile is 44 and the third quartile is 60.

(c) The value of range is 33 and the inter-quartile range is 16.

(d) The variance is 100.143 and the standard deviation is 10.01.

(e) There are no outliers in the data set.

(f) Yes

Step-by-step explanation:

The data provided is:

S = {55, 56, 44, 43, 44, 56, 60, 62, 57, 45, 36, 38, 50, 69, 65}

(a)

Compute the mean of the data as follows:

[tex]\bar x=\frac{1}{n}\sum x\\=\frac{1}{15}[55+ 56+ 44+ 43+ 44+ 56+ 60+ 62+ 57+ 45 +36 +38 +50 +69+ 65]\\=\frac{780}{15}\\=52[/tex]

Thus, the mean is 52.

The median for odd set of values is the computed using the formula:

[tex]Median=(\frac{n+1}{2})^{th}\ obs.[/tex]

Arrange the data set in ascending order as follows:

36, 38, 43, 44, 44, 45, 50, 55, 56, 56, 57, 60, 62, 65, 69

There are 15 values in the set.

Compute the median value as follows:

[tex]Median=(\frac{15+1}{2})^{th}\ obs.=(\frac{16}{2})^{th}\ obs.=8^{th}\ observation[/tex]

The 8th observation is, 55.

Thus, the median is 55.

(b)

The first quartile is the middle value of the upper-half of the data set.

The upper-half of the data set is:

36, 38, 43, 44, 44, 45, 50

The middle value of the data set is 44.

Thus, the first quartile is 44.

The third quartile is the middle value of the lower-half of the data set.

The upper-half of the data set is:

56, 56, 57, 60, 62, 65, 69

The middle value of the data set is 60.

Thus, the third quartile is 60.

(c)

The range of a data set is the difference between the maximum and minimum value.

Maximum = 69

Minimum = 36

Compute the value of Range as follows:

[tex]Range =Maximum-Minimum\\=69-36\\=33[/tex]

Thus, the value of range is 33.

The inter-quartile range is the difference between the first and third quartile value.

Compute the value of IQR as follows:

[tex]IQR=Q_{3}-Q_{1}\\=60-44\\=16[/tex]

Thus, the inter-quartile range is 16.

(d)

Compute the variance of the data set as follows:

[tex]s^{2}=\frac{1}{n-1}\sum (x_{i}-\bar x)^{2}\\=\frac{1}{15-1}[(55-52)^{2}+(56-52)^{2}+...+(65-52)^{2}]\\=100.143[/tex]

Thus, the variance is 100.143.

Compute the value of standard deviation as follows:

[tex]s=\sqrt{s^{2}}=\sqrt{100.143}=10.01[/tex]

Thus, the standard deviation is 10.01.

(e)

An outlier is a data value that is different from the remaining values.

An outlier is a value that lies below 1.5 IQR of the first quartile or above 1.5 IQR of the third quartile.

Compute the value of Q₁ - 1.5 IQR as follows:

[tex]Q_{1}-1.5QR=44-1.5\times 16=20[/tex]

Compute the value of Q₃ + 1.5 IQR as follows:

[tex]Q_{3}+1.5QR=60-1.5\times 16=80[/tex]

The minimum value is 36 and the maximum is 69.

None of the values is less than 20 or more than 80.

Thus, there are no outliers in the data set.

(f)

Yes, the data provided indicates that the Walmart is meeting its goal for reducing the number of hourly employees

Answer:

Step-by-step explanation:

Answer:

Part a

For this case after do the operations we got a value for the correlation coeffcient of:

[tex] r =0.506[/tex]

With this value we can find the determination coefficient:

[tex] r^2 = 0.506^2 = 0.256[/tex]

And with this value we can analyze the proportion of variance explained by one variable and the other. So we can conclude that 25.6% of the mother's BMI variation is explained by the daugther's BMI.

Part a

Since the BMI is a relation between height and weight, other possible variables that can explain the variability are (weight , height, age)  

Step-by-step explanation:

Previous concepts

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.

And in order to calculate the correlation coefficient we can use this formula:  

[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]  

Solution to the problem

Part a

For this case after do the operations we got a value for the correlation coeffcient of:

[tex] r =0.506[/tex]

With this value we can find the determination coefficient:

[tex] r^2 = 0.506^2 = 0.256[/tex]

And with this value we can analyze the proportion of variance explained by one variable and the other. So we can conclude that 25.6% of the mother's BMI variation is explained by the daugther's BMI.

Part a

Since the BMI is a relation between height and weight, other possible variables that can explain the variability are (weight , height, age)  

Answer:

solution attached below

Step-by-step explanation:

To determine the drag coefficient using the false-position method, start with initial guesses and iterate until the approximate relative error falls below 5%.

To determine the drag coefficient needed for a bungee jumper to have a velocity of 46 m/s after 9 s of free fall using the false-position method, we can follow these steps:

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Answer:

The answer is a population parameter.

Step-by-step explanation:

Population can include people, but other examples include objects, event, businesses, and so on. Population is the entire pool from which statistical sample is drawn.

A parameter is a value that describes a characteristics of an entire population, such as population mean, because you can almost never measure an entire population, you usually don't know the real value of a parameter.

Consider all possible sample of size N that can be drawn from a given population (either with or without replacement). For example, we can compute a statistics (such as the mean and the standard deviation ) that will vary from sample to sample. In this manner we obtain a distribution of statistics that is called Sampling distribution.

In statistics, a sampling distribution is the theoretical distribution of a sample statistic that arises from drawing all possible samples of a specific size from a population. It helps to quantify the variability and predictability of sample statistics when used as estimates for population parameters.

A sampling distribution refers to the "distribution of a sample statistic". This is option C from your list. This term describes the probability distribution of a statistic based on a random sample. For example, if we study random samples of a certain size from any population, the mean score will form a distribution. This is the sampling distribution of the mean. Similarly, variance, standard deviations and other statistics also have sampling distributions. The purpose of a sampling distribution is to quantify the variation and uncertainty that arises when we use sample statistics (like the mean) to estimate population parameters (like the population mean).

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Answer:

[tex]r=\frac{19M}{14h}[/tex]

Step-by-step explanation:

The equation is given as:

[tex]M=\frac{7r2h}{19}[/tex]

Assuming all the unknown variables are positive, we can make [tex]r[/tex] the subject of the formula to obtain it in terms of M & h:

[tex]M=\frac{7r2h}{19}\\M\times19=7r2h\\\\\frac{19M}{2h}=7r\\\\r=\frac{19M}{2h\times7}\\\\r=\frac{19M}{14h}[/tex]

or [tex]r=1.3571M/h[/tex]

Hence, r as in terms of M& H is given as

[tex]r=\frac{19M}{14h} \ or \ 1.3571M/h[/tex]

Answer:

[tex]ydv = (v +2)dy\\[/tex]            

Step-by-step explanation:

We are given the following differential equation:

[tex]y dx = 2(x + y) dy[/tex]

We have to substitute

[tex]x = vy[/tex]

Differentiating we get,

[tex]\dfrac{dx}{dy} = v + y\dfrac{dv}{dy}[/tex]

Putting value in differential equation, we get,

[tex]y dx = 2(x + y) dy\\\\y\dfrac{dx}{dy}=2(x+y)\\\\y(v+y\dfrac{dv}{dy}) = 2(vy + y)\\\\vy + y^2\dfrac{dv}{dy} = 2vy +2y\\\\y^2\dfrac{dv}{dy}=vy +2y\\\\y^2dv = y(v+2)dy\\ydv = (v +2)dy\\[/tex]

is the differential equation after substitution.

The given homogeneous differential equation y dx = 2(x + y) dy can be rewritten in terms of y and v using the substitution x = vy. The result is the differential equation y dv/dy = v.

The given differential equation is y dx = 2(x + y) dy. To write his equation in terms of y and v using the substitution x = vy, we must first differentiate both sides of x = vy with respect to x to get 1 = v + y dv/dx. We rearrange this to get dx/dy = 1 / (v + y dv/dy). The original equation can now be rewritten after substituting these values, you will get y / (v + y dv/dy) = 2(v + y), simplifying, we get v = 2v + 2y, and after rearranging, we get y dv/dy = v. This is the differential equation in terms of v and y.

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Final answer:

The correct mean time for the first three finishers is 5 minutes, after subtracting the 45-second error from the initially recorded mean time of 5.75 minutes.

Explanation:

The correct mean time for the first three finishers, after adjusting for the stopwatch error, can be calculated by subtracting the 45 seconds from the initially recorded meantime. Since the mean time was calculated to be 5.75 minutes (or 345 seconds), we must correct each time by subtracting 45 seconds and then recalculating the mean.

To find the correct mean time, we do the following steps:

The correct mean time for the first three finishers is therefore 5 minutes.

Answer:

a) 0.9772 = 97.72% probability that the mean weight of the sample is less than 3.10 pounds

b) 2.88 pounds

Step-by-step explanation:

To solve this question, we have 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 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]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

[tex]\mu = 3, \sigma = 0.25, n = 25, s = \frac{0.25}{\sqrt{25}} = 0.05[/tex]

a) What is the probability that the mean weight of the sample is less than 3.10 pounds? Round your answer to four decimal places

This is the pvalue of Z when X = 3.10. So

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

By the Central Limit Theorem

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

[tex]Z = \frac{3.1 - 3}{0.05}[/tex]

[tex]Z = 2[/tex]

[tex]Z = 2[/tex] has a pvalue of 0.9772

0.9772 = 97.72% probability that the mean weight of the sample is less than 3.10 pounds

b) What value will the mean weight exceed with probability 0.99? Round your answer to two decimal places.

This is the value of X when Z has a pvalue of 1-0.99 = 0.01. So X when Z = -2.33.

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

[tex]-2.33 = \frac{X - 3}{0.05}[/tex]

[tex]X - 3 = -2.33*0.05[/tex]

[tex]X = 2.88[/tex]

Answer:

You earned an A.

Step-by-step explanation:

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

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

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

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

In this problem, we have that:

[tex]\mu = 79, \sigma = 6[/tex]

The top 15% of all scores have been designated As.

This means that if Z for the score has a pvalue of 1-0.15 = 0.85 or higher, the score is designated as A.

Your score is 89. Did you earn an A?

We have to find the pvalue of Z when X = 89. So

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

[tex]Z = \frac{89 - 79}{6}[/tex]

[tex]Z = 1.67[/tex]

[tex]Z = 1.67[/tex] has a pvalue of 0.9525. So yes, you earned an A.

Answer:

a) 46.1

b) 46

c) Two modes: 46, 41          

Step-by-step explanation:

We are given the following sample of hours per week:

50, 53, 46, 46, 49, 43, 41, 41

a) mean of this data set

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{369}{8} = 46.1[/tex]

b) Median of data set

[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]

Sorted data:

41, 41, 43, 46, 46, 49, 50, 53

Median =

[tex]=\dfrac{4^{th}+5^{th}}{2} = \dfrac{46+46}{2} = 46[/tex]

The median of data is 46.

c) Mode of the data set

Mode is the most frequent observation in the data.

The mode of the data are 46 and 41 as they appeared two times.

Thus, there are two modes.

The mean of the data set is 46.1, the median is 46, and there are two modes: 41 and 46.

To answer the questions based on the provided data set of hours worked by eight adults, we need to perform some basic statistical calculations.

(a) Mean

The mean is calculated by summing all the data points and then dividing by the number of data points:

Mean = (50 + 53 + 46 + 46 + 49 + 43 + 41 + 41) / 8 = 369 / 8 = 46.1

(b) Median

First, we need to arrange the data in ascending order: 41, 41, 43, 46, 46, 49, 50, 53. As there are 8 data points (even number), the median is the average of the 4th and 5th values:

Median = (46 + 46) / 2 = 46

(c) Mode

The mode is the number that appears most frequently in the data set. Here, 41 and 46 both appear twice:

There are two modes: 41 and 46 and the mean of the data set is 46.1.

Answer:

0.0064 is the probability that Isaac will find his first defective can among the first 50 cans.    

Step-by-step explanation:

W are given the following in the question:

Probability of defective can =

[tex]P(A) = \dfrac{1}{90}[/tex]

We have to find the probability that Isaac will find his first defective can among the first 50 cans.

Then the number of adults follows a geometric distribution, where the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is

[tex]P(X=k) = (1-p)^{k-1}p[/tex]

We have to evaluate:

[tex]P(x = 50)\\= (1-\frac{1}{90})^{50-1}(\frac{1}{90})\\= 0.0064[/tex]

0.0064 is the probability that Isaac will find his first defective can among the first 50 cans.

Answer:

0.16

Step-by-step explanation:

For a normal distribution, the 68-95-99.7 rule says that 68% of the distribution lies within 1 standard deviation from the mean, 95% within two standard deviations and 99.7% within three standard deviations.

From the question,

Mean μ = 298 ml

Standard deviation σ = 3 ml

A value of 295 ml is within one standard deviation = 68%.

Since it's on the lower side, it's within 68% ÷ 2 = 34%.

The lower half below the mean is 50% of the distribution. Hence, for a selection less than 1 standard deviation, the probability is

50% - 34% = 16% = 0.16

Final answer:

The probability that a randomly selected bottle contains less than 295 ml of cola is calculated using the normal distribution. With a z-score of -1, ths probability is approximately 15.87%.

Explanation:

To determine the probability that a randomly selected bottle contains less than 295 ml of cola, we can use the properties of the normal distribution.

Given that the mean is 298 ml and the standard deviation is 3 ml, we must first find the z-score associated with 295 ml.

The z-score formula is z = (X - mu) / SD where X is the value for which we want to find the probability.

Substituting the given values into the formula, we get:

z = (295 - 298) / 3 = -1.

A z-score of -1 tells us that 295 ml is one standard deviation below the mean.

To find the probability corresponding to this z-score, we refer to a standard normal distribution table or use relevant statistical software. If we look up the probability for z = -1 in the z-table, we find that the area to the left of z is approximately 0.1587.

Therefore, the probability that a randomly selected bottle contains less than 295 ml of cola is about 15.87%.

Answer:

There are 21 comparisons to be made.

Step-by-step explanation:

The number of species of Odonata monitored every year is, n = 7.

It is provided that the density of each species is compared with each other.

The number of ways to compare the species (N) without repetition is:

[tex]N=\frac{n(n-1)}{2}\\=\frac{7(7-1)}{2}\\=\frac{7\times6}{2}\\=21[/tex]

Thus, there are 21 comparisons.

The 21 comparisons are as follows:

Specie 1 is compared with the remaining 6.

Specie 2 has already with he 1st so it is compared with the remaining 5.

Specie 3 has already with he 1st and 2nd so it is compared with the remaining 4.

Specie 4 has already with he 1st, 2nd and 3rd so it is compared with the remaining 3.

Specie 5 has already with he 1st, 2nd, 3rd and 4th so it is compared with the remaining 2.

Specie 6 has already with he 1st, 2nd, 3rd, 4th and 5th so it is compared with the remaining 1.

And the specie 7 has already been compared with the others.

Total number of comparisons = 6 + 5 + 4 + 3 + 2 + 1 = 21.

The probability that a randomly selected Canadian baby is a large baby (weighing more than 4,000 grams) is approximately 0.187 or 18.7%.

To find the probability that a randomly selected Canadian baby is a large baby, we need to calculate the area under the normal distribution curve to the right of 4,000 grams. First, we calculate the z-score using the formula: z = (x - mean) / standard deviation. Plugging in the values, we get z = (4000 - 3500) / 560 = 0.8929.

Next, we need to find the area under the curve to the right of this z-score using a standard normal distribution table or a calculator. The cumulative probability from the table or calculator is approximately 0.187. This means that the probability of a randomly selected Canadian baby being a large baby (weighing more than 4,000 grams) is approximately 0.187 or 18.7%.

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Answer:

The 99% confidence interval for the population mean hours spent watching television per month is between 143.07 hours and 158.93 hours.

Step-by-step explanation:

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

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

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

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

Now, find M as such

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

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

[tex]M = 2.575*\frac{32}{\sqrt{108}} = 7.93[/tex]

The lower end of the interval is the mean subtracted by M. So it is 151 - 7.93 = 143.07 hours

The upper end of the interval is the mean added to M. So it is 151 + 7.93 = 158.93 hours

The 99% confidence interval for the population mean hours spent watching television per month is between 143.07 hours and 158.93 hours.

Answer and Step-by-step explanation:

The answer is attached below

In this exercise we have to use the knowledge of parameterization and calculate the direction and direction of the equation, so we have to:

A) Clockwise: [tex]t \in [ -\infty, 1/6][/tex]

B) Counter-clockwise: [tex]t \in [ 1/6, \infty][/tex]

C) [tex]\theta \in [ 0, 2 \pi][/tex]

D) [tex]t= 0 \ or \ t=1/3[/tex]

For this exercise, the following equations were informed:

[tex]x= cos(3t^2-t)\\y= sin(3t^2-t)\\t \in [ -\infty, \infty][/tex]

taking the parameterization we have that:

[tex]\phi = 3t^2 - t= t(3t-1)[/tex]

As t increases from [tex][ -\infty, \infty][/tex]  [tex]\phi[/tex] decreases, after 0 it becomes negative and after 1/3, goes on increasing. Also:

[tex]\frac{d\phi}{dt} = (6t-1)\\t= 1/6[/tex]

a) For clockwise begin [tex]\phi[/tex] must be decreasing, so:

[tex]t \in [ -\infty, 1/6][/tex]

b) For counter-clockwise  [tex]\phi[/tex] must be increasing, so:

[tex]t \in [ 1/6, \infty][/tex]

c) Entise circle gets traced out. For we know:

[tex]x= cos\theta\\y= sin\theta[/tex]

Circle gets traced out once for:

[tex]\theta \in [ 0, 2 \pi][/tex]

d) When point (1, 0) so:

[tex]1= cos(3t^2-t)\\0= sin(3t^2-t)\\t= 0 \or \ t=1/3[/tex]

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Answer:

a) P ( 3 ≤X≤ 5 ) = 0.02619

b) E(X) = 1

Step-by-step explanation:

Given:

- The CDF of a random variable X = { 0 , 1 , 2 , 3 , .... } is given as:

                    [tex]F(X) = P ( X =< x) = 1 - \frac{1}{(x+1)*(x+2)}[/tex]

Find:

a.Calculate the probability that 3 ≤X≤ 5

b) Find the expected value of X, E(X), using the fact that. (Hint: You will have to evaluate an infinite sum, but that will be easy to do if you notice that

Solution:

- The CDF gives the probability of (X < x) for any value of x. So to compute the P (  3 ≤X≤ 5 ) we will set the limits.

                   [tex]F(X) = P ( 3=<X =< 5) = [1 - \frac{1}{(x+1)*(x+2)}]\limits^5_3\\\\F(X) = P ( 3=<X =< 5) = [-\frac{1}{(5+1)*(5+2)} + \frac{1}{(3+1)*(3+2)}}\\\\F(X) = P ( 3=<X =< 5) = [-\frac{1}{(42)} + \frac{1}{(20)}}]\\\\F(X) = P ( 3=<X =< 5) = 0.02619[/tex]

- The Expected Value can be determined by sum to infinity of CDF:

                   E(X) = Σ ( 1 - F(X) )

                   [tex]E(X) = \frac{1}{(x+1)*(x+2)} = \frac{1}{(x+1)} - \frac{1}{(x+2)} \\\\= \frac{1}{(1)} - \frac{1}{(2)}\\\\= \frac{1}{(2)} - \frac{1}{(3)} \\\\= \frac{1}{(3)} - \frac{1}{(4)}\\\\= ............................................\\\\= \frac{1}{(n)} - \frac{1}{(n+1)}\\\\= \frac{1}{(n+1)} - \frac{1}{(n+ 2)}[/tex]

                   E(X) = Limit n->∞ [1 - 1 / ( n + 2 ) ]  

                   E(X) = 1

Answer:

Option a)  H0 : μ = 61 , Hα : μ > 61 .

Step-by-step explanation:

We are given that the time, in number of days, until maturity of a certain variety of tomato plant is Normally distributed, with mean μ and standard deviation s = 2.4.

You select a simple random sample of four plants of this variety and measure the time until maturity. The sample yields x bar = 65.

You read on the package of seeds that these tomatoes reach maturity, on average, in 61 days.

Now, since we know about;

sample size, n = 4

sample standard deviation, s = 2.4

sample, x bar = 65

The only line which represent information about the population mean is reading on the package of seeds that these tomatoes reach maturity, on average, in 61 days.

This shows that population mean , [tex]\mu[/tex] = 61 and in null and alternate hypothesis only population mean is tested.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] = 61

Alternate Hypothesis, [tex]H_1[/tex] : [tex]\mu[/tex] > 61 { we are asked for later than expected}

Answer:

m=1.7

C=68 gr

Step-by-step explanation:

Function Modeling

We are given a relationship between the carbohydrates used by a professional tennis player during a strenuous workout and the time in minutes as 1.7 grams per minute. Being C the carbohydrates in grams and t the time in minutes, the model is

[tex]C=1.7t[/tex]

The slope m of the line is the coefficient of the independent variable, thus m=1.7

The graph of C vs t is shown in the image below.

To find how many carbohydrates the athlete would use in t=40 min, we plug in the value into the equation

[tex]C=1.7\cdot 40=68\ gr[/tex]

The slope of the line representing the rate of carbohydrate usage is 1.7. Multiply this rate (1.7 grams per minute) by the time (40 minutes) to find the total carbohydrates used, which is 68 grams.

The rate at which the tennis player uses carbohydrates is 1.7 grams per minute. In the context of a graph, this rate would represent the slope of the line. So, the slope of the line would be 1.7. Slope, in mathematics, is defined as the change in the y-value (vertical axis) divided by the change in the x-value (horizontal axis). Here, the rate of carbohydrate usage (1.7 grams per minute) is the change in the y-value (carbohydrates used) per change in x-value (time).

Now, you also want to know how many carbohydrates the athlete would use in 40 minutes. We know that the rate of carbohydrate usage is 1.7 grams per minute. So, to find the total amount of carbohydrates used in 40 minutes, you'd simply multiply the rate by the time:

1.7 grams/minute * 40 minutes = 68 grams

So, the athlete would use 68 grams of carbohydrates in 40 minutes.

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Answer:

Correct option: B. Less between group variability

Step-by-step explanation:

The Analysis of Variance (ANOVA) test is performed to determine whether there is a significant difference between the different group mean.

The hypothesis is defined as:

H₀: There is no difference between the group means, i.e. μ₁ = μ₂ = ... = μₙ

Hₐ: At least one of the mean is different from the others, i.e. μ[tex]_{i}[/tex] ≠ 0.

The test statistic is defined as:

[tex]F=\frac{SS_{between}}{SS_{within}}[/tex]

If the null hypothesis is true then the test statistic will be small and if it is false then the test statistic will be large.

In this case it is provided that the null hypothesis is true.

This implies that:

[tex]SS_{between}<SS_{within}[/tex]

Implying that the sum of squares for between group variability is less than within group variability.

Thus, if the null hypothesis is true there will be less between group variability.

Answer:

1/48 % or 6/48 % chance

Step-by-step explanation:

The probability of winning with the very first Lotto ticket you purchase of the Lotto game consisting of picking 6 of 48 numbers is 1/12271512 or approximately 0.0000000815.

A permutation is a process of calculating the number of ways to choose a set from a larger set in a particular order.

If we want to choose a set of r items from a set of n items in a particular order, we find the permutation nPr = n!/(n-r)!.

A combination is a process of calculating the number of ways to choose a set from a larger set in no particular order.

If we want to choose a set of r items from a set of n items in no particular order, we find the combination nCr = n!/{(r!)(n-r)!}.

In the question, we are asked to determine the probability of winning a lottery by picking 6 numbers from 48 numbers with the first ticket we purchase.

First, we need to calculate the number of combinations of choosing 6 numbers from 48 numbers. As we need to consider no particular order, we will use combinations,

48C6 = 48!/{(6!)(48-6)!} = 48!/(6!*42!) = (43*44*45*46*47*48)/(1*2*3*4*5*6*) (As 48! = 42!*43*44*45*46*47*48, and 42! cancels itself from the numerator and the denominator).

or, 48C6 = 12,271,512.

So, we get the number of combinations = 12,271,512.

We know that we will choose only one particular set of 6 numbers.

∴ The probability of winning on the very first ticket = 1/12,271,512 ≈ 0.0000000815

∴ The probability of winning with the very first Lotto ticket you purchase of the Lotto game consisting of picking 6 of 48 numbers is 1/12271512 or approximately 0.0000000815.

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