Spam: A researcher reported that 71.8% of all email sent in a recent month was spam. A system manager at a large corporation believes that the percentage at his company may be 69%. He examines a random sample of 500 emails received at an email server, and finds that 365 of the messages are spam. Can you conclude that the percentage of emails that are spam is greater than 69% ? Use both =α0.01 and =α0.05 levels of significance and the P-value method with the TI-84 Plus calculator.

Slope-intercept form:  y = mx + b

(m is the slope, b is the y-intercept or the y value when x = 0 --> (0, y) or the point where the line crosses through the y-axis)

For lines to be parallel, they need to have the same slope.

[tex]y=-\frac{2}{3} x+12[/tex]    The slope is -2/3, so the parallel line's slope is also -2/3.

Now that you know the slope, substitute/plug it into the equation.

y = mx + b

[tex]y=-\frac{2}{3} x+b[/tex]   To find b, plug in the point (6, -11) into the equation, then isolate/get the variable "b" by itself

[tex]-11=-\frac{2}{3}(6)+b[/tex]

-11 = -4 + b     Add 4 on both sides to get "b" by itself

-7 = b

[tex]y=-\frac{2}{3} x-7[/tex]    

The slope-intercept form line that passes through the point (6, -11) and is parallel to the line y = -2/3x + 12 is y = -2/3x - 7.

The subject of this question is Mathematics, specifically algebra and geometry involving slope-intercept form. The question asks for the slope-intercept form of the line that passes through the given point (6, -11) and is parallel to the line y = -2/3x + 12.

In slope-intercept form, y = mx + b, m represents the slope of the line and b represents the y-intercept. We know that parallel lines have the same slope, so the slope of the line in question would be -2/3, same as the provided line.

To find the y-intercept (b), we use the point (6, -11) and the slope -2/3 in the slope-intercept equation: -11 = (-2/3) * 6 + b. Solving for b, we get b = -7. Hence, the equation of the line in slope-intercept form that passes through the given point and is parallel to the given line is y = -2/3x - 7.

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Answer:

2z^2+7z+6

because you multiply 2z with z and 2 then multiply 3 with z and 2 and combine like terms

Answer:

(2z + 3) (z +2)

combine like terms: (2z⋅z)+(2z⋅2)+3z+(3⋅2)

2z^2+7z+6

Answer:

-8/45

I hope this helped!

Step-by-step explanation:

-2/3 ÷ 3 3/4

Answer:

Step-by-step explanation:

Hello!

The mean life of 16 lithium batteries was estimated with a 95% CI:

(628.5, 661.5) hours

Assuming that the variable "X: Duration time (life) of a lithium battery(hours)" has a normal distribution and the statistic used to estimate the population mean was s Student's t, the formula for the interval is:

[X[bar]±[tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]]

The amplitude of the interval is calculated as:

a= Upper bond - Lower bond

a= [X[bar]+[tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]] -[X[bar]-[tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]]

and the semiamplitude (d) is half the amplitude

d=(Upper bond - Lower bond)/2

d=([X[bar]+[tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]] -[X[bar]-[tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]] )/2

d= [tex]t_{n-1;1-\alpha /2}* \frac{S}{\sqrt{n} }[/tex]

The sample mean marks where the center of the calculated interval will be. The terms of the formula that affect the width or amplitude of the interval is the value of the statistic, the sample standard deviation and the sample size.

Using the semiamplitude of the interval I'll analyze each one of the posibilities to see wich one will result in an increase of its amplitude.

Original interval:

Amplitude: a= 661.5 - 628.5= 33

semiamplitude d=a/2= 33/2= 16.5

1) Having a sample with a larger standard deviation.

The standard deviation has a direct relationship with the semiamplitude of the interval, if you increase the standard deviation, it will increase the semiamplitude of the CI

↑d= [tex]t_{n-1;1-\alpha /2}[/tex] * ↑S/√n

2) Using a 99% confidence level instead of 95%.

d= [tex]t_{n_1;1-\alpha /2}[/tex] * S/√n

Increasing the confidence level increases the value of t you will use for the interval and therefore increases the semiamplitude:

95% ⇒ [tex]t_{15;0.975}= 2.131[/tex]

99% ⇒ [tex]t_{15;0.995}= 2.947[/tex]

The confidence level and the semiamplitude have a direct relationship:

↑d= ↑[tex]t_{n_1;1-\alpha /2}[/tex] * S/√n

3) Removing an outlier from the data.

Removing one outlier has two different effects:

1) the sample size is reduced in one (from 16 batteries to 15 batteries)

2) especially if the outlier is far away from the rest of the sample, the standard deviation will decrease when you take it out.

In this particular case, the modification of the standard deviation will have a higher impact in the semiamplitude of the interval than the modification of the sample size (just one unit change is negligible)

↓d= [tex]t_{n_1;1-\alpha /2}[/tex] * ↓S/√n

Since the standard deviation and the semiamplitude have a direct relationship, decreasing S will cause d to decrease.

4) Using a 90% confidence level instead of 95%.

↓d= ↓[tex]t_{n_1;1-\alpha /2}[/tex] * S/√n

Using a lower confidence level will decrease the value of t used to calculate the interval and thus decrease the semiamplitude.

5) Testing 10 batteries instead of 16. and 6) Testing 24 batteries instead of 16.

The sample size has an indirect relationship with the semiamplitude if the interval, meaning that if you increase n, the semiamplitude will decrease but if you decrease n then the semiamplitude will increase:

From 16 batteries to 10 batteries: ↑d= [tex]t_{n_1;1-\alpha /2}[/tex] * S/√↓n

From 16 batteries to 24 batteries: ↓d= [tex]t_{n_1;1-\alpha /2}[/tex] * S/√↑n

I hope this helps!

Answer:

0.22

Step-by-step explanation:

Sample given is 500, so use z-score for the critical value

Given 95% confidence interval;

∝=100-95 =5% =0.05

∝/2 = 0.05/2 =0.025 ----because you are interested with one tail area

1-0.025= 0.975 -----area to the left

proceed to z-table to read 0.975 = 1.96 as the critical value

standard deviation from the question is 2.5 but because this is a sample then;

standard error for the mean is= standard deviation/√sample= 2.5/√500

=0.1118

Margin of error=1.96*0.1118 =0.2191

The margin of error is approximately 0.219 inches. Therefore, the correct answer is approximately 0.22 inches. Option (c) is correct.

To find the margin of error for a 95% confidence interval, we first need to determine the critical value for a 95% confidence interval.

For a normal distribution, with a 95% confidence level, the critical value is approximately 1.96.

Then, we use the formula for the margin of error:

[tex]\[ \text{Margin of Error} = \text{Critical Value} \times \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \][/tex]

Given:

Let's calculate the margin of error:

[tex]\[ \text{Margin of Error} = 1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} = 1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} \approx1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} \approx 1.96 \times \frac{2.5}{22.36} \][/tex]

[tex]\[ \text{Margin of Error} \approx 1.96 \times 0.1118 \][/tex]

[tex]\[ \text{Margin of Error} \approx 0.219 \][/tex]

So, the margin of error is approximately 0.219 inches. Therefore, the correct answer is approximately 0.22 inches.

Answer:

:=?¿ZV16

Step-by-step explanation:

Answer:

[tex]\\\sum_{k=1}^{n+1}\frac{1}{k(k+1)}\\ \\ \\=\sum_{k=1}^n\frac{1}{k(k+1)}+\frac{1}{(n+1)(n+2)} \\ \\ =1-\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}\\ \\ \\ =1+\frac{1-(n+2)}{(n+1)(n+2)} \\ \\ \\ \\\sum_{k=1}^{n+1}\frac{1}{k(k+1)} =1-\frac{1}{n+2}[/tex]

Step-by-step explanation:

The question says; Proof that :

[tex]Let : P\left(n\right)=\sum_{k=1}^n\frac{1}{k(k+1)}=1-\frac{1}{n+1}[/tex]

Base case: P(1) = 1/2.

Inductive step: suppose P(n) has already been proven for some arbitrary n.  The statement P(n+1) is :

[tex]P\left(n+1\right)=\sum_{k=1}^{n+1}\frac{1}{k\left(k+1\right)}=1-\frac{1}{n+2}[/tex]

This concludes the proof by induction.

We Proof that:

The proof abuses the notation P(n) to make reference to the common values of the two sides of the equation to be proved. Moreover, it doesn't makes any sense to define P(n) as the common value of the two sides because it assumes the conclusion that the two sides are equal.

At the very least, the definition of P(n) in the first statement suppose to have be in quote or in parenthesis as shown below.

[tex]Let : P\left(n\right)=(\sum_{k=1}^n\frac{1}{k(k+1)}=1-\frac{1}{n+1})[/tex]

However , P(n) is a statement.

The proof writer confused stating P(n+1) with showing that it must be true; given that P(n) is true.

As such ; the correct proof for P(n+1) is:

[tex]\\\sum_{k=1}^{n+1}\frac{1}{k(k+1)}\\ \\ \\=\sum_{k=1}^n\frac{1}{k(k+1)}+\frac{1}{(n+1)(n+2)} \\ \\ =1-\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}\\ \\ \\ =1+\frac{1-(n+2)}{(n+1)(n+2)} \\ \\ \\ \\\sum_{k=1}^{n+1}\frac{1}{k(k+1)} =1-\frac{1}{n+2}[/tex]

Answer:

0.266

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 = 5.85, \sigma = 0.24[/tex]

Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than 6.0 cm.

This is 1 subtracted by the pvalue of Z when X = 6.

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

[tex]Z = \frac{6 - 5.85}{0.24}[/tex]

[tex]Z = 0.625[/tex]

[tex]Z = 0.625[/tex] has a pvalue of 0.734

1 - 0.734 = 0.266

Answer:

[tex]P(X>6)=P(\frac{X-\mu}{\sigma}>\frac{6-\mu}{\sigma})=P(Z>\frac{6-5.85}{0.24})=P(z>0.625)[/tex]

And we can find this probability using the complement rule and the normal standard table or excel:

[tex]P(z>0.625)=1-P(z<0.625)=1-0.734= 0.266[/tex]

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 diameters of mandarin oranges of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(5.85,0.24)[/tex]  

Where [tex]\mu=5.85[/tex] and [tex]\sigma=0.24[/tex]

We are interested on this probability

[tex]P(X>6)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X>6)=P(\frac{X-\mu}{\sigma}>\frac{6-\mu}{\sigma})=P(Z>\frac{6-5.85}{0.24})=P(z>0.625)[/tex]

And we can find this probability using the complement rule and the normal standard table or excel:

[tex]P(z>0.625)=1-P(z<0.625)=1-0.734= 0.266[/tex]

The correlation coefficient for the data is 0.0091.

This can be obtained by using the formula of correlation coefficient.

The following information is obtained from the table,

∑x  = 10

∑y = 10

∑xy = 42

∑x² = 42

∑y² = 42

The formula for finding the correlation coefficient,

r = n∑xy- ∑x∑y/(n∑x²-(∑x)²)(n∑y²-(∑y)²)

r = (5×42)-(10×10)/((5×42)-10²)((5×42)-10²)

 =[tex]\frac{110}{110.110}[/tex]

=[tex]\frac{1}{110}[/tex]

=0.0091

Thus, the correlation coefficient of the given data is 0.0091.

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Answer:

B) 1

Step-by-step explanation:

Got it right 2024 edg

This question is incomplete, in that the Excel File: data07-11.xlsx a. was not provided, but I was able to get the information on the Excel File: data07-11.xlsx a. from google as below:

57 61 86 74 72 73

20 57 80 79 83 74

The image of the Excel File: data07-11.xlsx a. is also attached below.

Answer:

a) Point estimate of sample mean  = 68

b) Point estimate of standard deviation (4 decimals) = 17.8122

Step-by-step explanation:

a) Point estimate of sample mean, \bar{x}  =  ∑Xi / n = (57 + 61 + 85 + 74 + 73 + 72 + 20 + 58 + 81 + 78 + 84 + 73)/12 = 68

b) Point estimate of standard deviation = sqrt ∑ Xi² - n\bar{x}² / n-1)

= sqrt(((57 - 68)^2 + (61 - 68)^2 + (85 - 68)^2 + (74 - 68)^2 + (73 - 68)^2 + (72 - 68)^2 + (20 - 68)^2 + (58 - 68)^2 + (81 - 68)^2 + (78 - 68)^2 + (84 - 68)^2 + (73 - 68)^2)/11) = 17.8122

Answer:

Step-by-step explanation:

Recall that we say that d | a if there exists an integer k for which a = dk. So, let d = gcd(a,b) and let x, y be integers. Let t = ax+by.

We know that [tex]d | a, d | b[/tex] so there exists integers k,m such that a = kd and b = md. Then,

[tex] t = ax+by = (kd)x+(md)y = d(kx+my)[/tex]. Recall that since k,  x, m, y are integers, then (kx+my) is also an integer. This proves that d | t.

8 (y - 7) = - 16

Divide by 8 on both sides

y - 7 = - 2

Add 7 to both sides

y = 5

Answer:

y = 5

Step-by-step explanation:

8(y-7) = -16

Divide each side by 8

8(y-7)/8 = -16/8

y-7 = -2

Add 7 to each side

y -7+7 = -2+7

y = 5

Answer:

x>-8

Step-by-step explanation:

multiply both sides with -1

Answer:

8

Step-by-step explanation:

8

Answer:

The pairs are matched

Step-by-step explanation:

A. A line through the origin that makes an angle of [tex]\pi/6[/tex] with the positive x-axis.

Given a line through the origin that makes an angle of [tex]\pi/6[/tex] with the positive x-axis. The angle which the line makes with the x-axis is [tex]\pi/6[/tex].

B. A vertical line through the point (3, 3).

If a line passes through the point (3,3), x=3 and y=3. The vertical line through the point (3,3) is x=3

C. Given a circle center (h,k) and a center r, the standard form of the equation of the circle is given as:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

Therefore, for a circle with radius 5 and center (2, 3), the standard form equation is:

D. A circle centered at the origin with radius.

For a circle centered at the origin with radius r=4.

The radius of the circle is 4 units.

Answer:

[tex]\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

[tex]\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

Step-by-step explanation:

The critical numbers are found by the First Derivative Test, which consists in differentiating the function, equalizing it to zero and solving it:

[tex]g'(\theta) = 16 - 4\cdot \sec^{2} \theta[/tex]

Following equation needs to be solved:

[tex]16 - 4\cdot \sec^{2}\theta = 0[/tex]

[tex]\sec^{2}\theta = 4[/tex]

[tex]\cos^{2}\theta = \frac{1}{4}[/tex]

[tex]\cos \theta = \frac{1}{2}[/tex]

The solution is:

[tex]\theta = \cos^{-1} \frac{1}{2}[/tex]

Given that cosine is a periodical function, there are two subsets of solution:

[tex]\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

[tex]\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

The critical numbers of the function g(θ) = 16θ − 4 tan(θ) can be found by determining where the derivative of the function is zero or undefined. However, specific critical numbers can't be determined from the provided prompt due to complexity of the derivative.

The critical numbers of a function occur when the derivative of the function is equal to zero or undefined. Given the function g(θ) = 16θ − 4 tan(θ), the first step to be done is finding the derivative (g'(θ)) of the function.

This involves applying the rules of differentiation to each term: the coefficient rule (b), and chain rule for the tangent part. After calculating the derivative, we set g'(θ) equal to zero and solve for θ to determine the critical numbers. For the term involving tan(θ), it is undefined at θ = π/2 + πn, where n is an integer. Therefore, these are also considered as critical numbers.

However, due to the complexity of the derivative, finding critical numbers usually requires using algebraic and trigonometric techniques or potentially numerical methods if the equation cannot be solved analytically. As the specific steps for this complex derivative calculation are not provided in the prompt, we can't provide the specific critical numbers.

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Answer:

using the calculator

Step-by-step explanation:

The mathematical value to 54-200 divided by 4 is -36.5

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

Given that we should find the value of 54-200 divided by 4, which can be expressed mathematically as ;

[tex]\frac{ 54-200}{4}[/tex]

Then we can find the value of the numerator as ;

[tex]54-200 = -146[/tex]

Then we have

[tex]\frac{-146}{4} \\\\= -36.5[/tex]

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Answer:

D. 12.5

Step-by-step explanation:

You can notice that ∠JMK and ∠JML are similar triangles.

That means that JM (from triangle on the left) is similar to KM (from triangle of the right)- the ratio between these must be:

[tex]\frac{6}{8}[/tex], that means that we can multiple KM (from triangle on the right) by this ratio to get ML (from triangle on the right).

[tex]6 * \frac{6}{8}=\frac{36}{8}=4.5[/tex]

Now we add JM to ML to to get JL - so 8 + 4.5 = 12.5

Answer:

12.5

Step-by-step explanation:

Answer:

565.2 in²

Step-by-step explanation:

2pi × r × (r + h)

2 × 3.14 × 5 × (5 + 13)

31.4 × 18

565.2

Answer: after driving 330 miles, the remaining fuel in each tank be the same.

Step-by-step explanation:

Let x represent the number of miles it will take for the the remaining fuel in each tank to be the same.

Jerry has a large car which holds 22 gallons of fuel and gets 20 miles per gallon. It means that the number of gallons needed to drive 1 mile is 1/20. Then the number of gallons needed to drive x miles is

1/20 × x = x/20

If the tank is full, then the number of gallons of fuel left after driving x miles is

22 - x/20

Kate has a smaller car which holds 16.5 gallons of fuel and gets 30 miles per gallon. It means that the number of gallons needed to drive 1 mile is 1/30. Then the number of gallons needed to drive x miles is

1/30 × x = x/30

If the tank is full, then the number of gallons of fuel left after driving x miles is

16.5 - x/30

For the remaining fuel in each tank to be the same, it means that

22 - x/20 = 16.5 - x/30

Multiplying both sides of the equation by 60(LCM), it becomes

1320 - 3x = 990 - 2x

- 2x + 3x = 1320 - 990

x = 330 miles

Answer:

5x-3

Step-by-step explanation:

x+4+7x-1-4x

5x-3

Answer:

7 points total

Step-by-step explanation:

So basically the x represents the total amount of times you landed on that color so since its one of each you just replace the x with one and solve normally. 1+4=5 points for green, 7(1)-1= 6 points for blue -4(1)= -4 points for red  

Answer:

Probability is the chance of you getting something

for example the chances (or probability) of the dice landing on 2 or 3 is 3 out of 6 (3/6)

hope this helps!

Answer:no

Step-by-step explanation:

expressions don't have equal sides

so it's an equation

please like and Mark as brainliest

Answer:

[tex]n=\frac{0.45(1-0.45)}{(\frac{0.05}{1.96})^2}=380.32[/tex]  

And rounded up we have that n=381

Step-by-step explanation:

Previous concepts

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

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

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

Solution to the problem

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]

The margin of error for the proportion interval is given by this formula:  

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

And on this case we have that [tex]ME =\pm 0.05[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

And replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.45(1-0.45)}{(\frac{0.05}{1.96})^2}=380.32[/tex]  

And rounded up we have that n=381

Final answer:

To estimate the proportion of out-of-state hotel guests with a 95% confidence level and a margin of error of 5%, a sample size of at least 385 guests is needed.

Explanation:

To estimate the proportion of out-of-state guests at a local hotel with a margin of error of no more than 5% and a 95% level of confidence, we can use the formula for determining sample size for a proportion:

n = (Z^2 * p * (1 - p)) / E^2

Where:
- n is the sample size
- Z is the Z-score corresponding to the confidence level (1.96 for 95% confidence)
- p is the preliminary estimate of the proportion (0.45, or 45%, in this case)
- E is the desired margin of error (0.05, or 5%, here)

Substituting the known values, we get:

n = (1.96^2 * 0.45 * (1 - 0.45)) / 0.05^2

n = 384.16

Since we cannot have a fraction of a person, we would round up to the nearest whole number, which gives us a sample size of 385. Therefore, the hotel should sample at least 385 guests to meet their requirements.

Answer:

Pi/3, 2pi/3, 4pi/3, 5pi/3

Step-by-step explanation:

Edge 2021 :)

Answer:

63 intervals would be expected to contain the true population average.

Step-by-step explanation:

90% confidence interval:

Mean that we are 90% sure that the interval contains the true population mean. That is, 90% of the intervals are expected to contain the true population average.

Of these 70 intervals, how many of these intervals would you expect to contain the true population average?

Following the explained logic

0.9*70 = 63

63 intervals would be expected to contain the true population average.

Answer:

A) σ_x' = 1.4142

B) σ_x' = 1.4135

C) σ_x' = 1.4073

D) σ_x' = 1.343

Step-by-step explanation:

We are given;

σ = 10

n = 50

A) when size is infinite, the standard deviation of the sample mean is given by the formula;

σ_x' = σ/√n

Thus,

σ_x' = 10/√50

σ_x' = 1.4142

B) size is given, thus, the standard deviation of the sample mean is given by the formula;

σ_x' = (σ/√n)√((N - n)/(N - 1))

Thus, with size of N = 50,000, we have;

σ_x' = 1.4142 x √((50000 - 50)/(50000 - 1))

σ_x' = 1.4142 x 0.9995

σ_x' = 1.4135

C) at N = 5000;

σ_x' = 1.4142 x √((5000 - 50)/(5000 - 1))

σ_x' = 1.4073

D) at N = 500;

σ_x' = 1.4142 x √((500 - 50)/(500 - 1))

σ_x' = 1.343

The value of the standard error in each of the following are:

A) σ_x' = 1.4142

B) σ_x' = 1.4135

C) σ_x' = 1.4073

D) σ_x' = 1.343

Given:

σ = 10

n = 50

A) when size is infinite, the standard deviation of the sample mean is given by the formula;

σ_x' = σ/√n

Thus,

σ_x' = 10/√50

σ_x' = 1.4142

B) size is given, thus, the standard deviation of the sample mean is given by the formula;

σ_x' = (σ/√n)√((N - n)/(N - 1))

Thus, with size of N = 50,000, we have;

σ_x' = 1.4142 x √((50000 - 50)/(50000 - 1))

σ_x' = 1.4142 x 0.9995

σ_x' = 1.4135

C) at N = 5000;

σ_x' = 1.4142 x √((5000 - 50)/(5000 - 1))

σ_x' = 1.4073

D) at N = 500;

σ_x' = 1.4142 x √((500 - 50)/(500 - 1))

σ_x' = 1.343

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The capability of the process is determined using the process capability index (Cp). In this case, the process has a capability of 0.833, indicating that it is not meeting the specifications well.

To calculate the capability of the process, we need to use the process capability index (Cp). Cp is calculated by dividing the tolerance range by six times the standard deviation. The tolerance range in this case is 65 - 55 = 10 psi. So, Cp = 10 / (6 * 2) = 10 / 12 = 0.833. Since Cp is a measure of how well the process meets the specifications, a value closer to 1 indicates a better capability.

In this case, the process has a capability of 0.833, which means it is not meeting the specifications very well.

Keywords: capability, process capability index, tolerance range, standard deviation, specifications

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The process capability index (Cpk) for the grease aerosol can pressurization process is 0.67, indicating that the process is not capable of meeting the design specifications as it is less than the acceptable limit of 1.33.

The capability of the process in question relates to its ability to meet design specifications, which can be quantified using statistical measures like the process capability index (Cpk). To compute the Cpk, you need to determine the worst-case process capability scenario by comparing the distance of the process mean to the nearest specification limit in terms of standard deviations. The Cpk can be calculated using the formula:

Cpk = minimum [(USL - x) / (3σ), (x - LSL) / (3σ)]

Where USL is the upper specification limit, LSL is the lower specification limit, x is the process mean, and σ is the standard deviation.

For this process with an average of 61 psi and a standard deviation of 2 psi:

You would calculate two separate indices:

Thus, the Cpk would be the smaller of these two indices which is 0.67. A Cpk of 0.67 indicates that the process is not capable of meeting the design specifications since it is less than the acceptable limit of 1.33 for most industries.

Answer:

(a) The estimated proportion of defective in the population is 0.05.

(b) The 95% confidence interval for the proportion defective cups is (2.5%, 7.5%).

(c) Zack does not needs to return the lots.

Step-by-step explanation:

Let X = number of defective cups.

The random sample of cups selected is of size, n = 300.

The number of defective cps in the sample is, X = 15.

(a)

The proportion of the defective cups in the population can be estimated by the sample proportion because the sample selected is quite large.

The sample proportion of defective cups is:

[tex]\hat p=\frac{X}{n}=\frac{15}{300}=0.05[/tex]

Thus, the estimated proportion of defective in the population is 0.05.

(b)

The (1 - α)% confidence interval for population proportion is:

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

Compute the critical value of z for 95% confidence level as follows:

[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]

Compute the 95% confidence interval for p as follows:

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

     [tex]=0.05 \pm 1.96\times\sqrt{\frac{0.05(1-0.05)}{300}}[/tex]

     [tex]=0.05\pm 0.025\\=(0.025, 0.075)\\[/tex]

Thus, the 95% confidence interval for the proportion defective cups is (2.5%, 7.5%).

(c)

It is provided that Zack has an agreement with his supplier that he is to return lots that are 10% or more defective.

The 95% confidence interval for the proportion defective is (2.5%, 7.5%). This implies that 95% of the lots have 2.5% to 7.5% defective items.

Thus, Zack does not needs to return the lots.

The estimated proportion of defective in the population is 0.05 and the 95 percent confidence interval for the proportion defective is (0.025,0.075).

Given :

a) The formula given below is used in order to determine the estimated proportion of defective in the population.

[tex]\hat{p} = \dfrac{X}{n}[/tex]

[tex]\hat{p} = \dfrac{15}{300}[/tex]

[tex]\hat{p} = 0.05[/tex]

So, the estimated proportion of defective in the population is 0.05.

b) The below formula is used in order to determine the 95 percent confidence interval for the proportion defective.

[tex]CI =\hat{p}\pm z_{\alpha /2}\times \sqrt{ \dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

Now, substitute the known terms in the above expression.

[tex]CI =0.05 \pm 1.96\times \sqrt{ \dfrac{0.05(1-0.05))}{300}}[/tex]

[tex]CI = 0.05\pm 0.025[/tex]

So, the 95 percent confidence interval for the proportion defective is (0.025,0.075).

c) According to the given data, Zack has an agreement with his supplier that he is to return lots that are 10 percent or more defective.

So, from the above calculation, it can be concluded that he did not have to return the lots.

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Answer:

Step-by-step explanation:

The null hypothesis is the hypothesis that is assumed to be true. It is an expression that is the opposite of what the researcher predicts.

The alternative hypothesis what the researcher expects or predicts. It is the statement that is believed to be true if the null hypothesis is rejected.

From the given situation,

The mean number of pounds of fruit on this plot of land with the old fertilizer was 416 pounds. This is the null hypothesis.

H0 : µ = 416

Agriculture scientists believe that the new fertilizer may decrease the yield. This is the alternative hypothesis.

H0 : µ < 416

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