You wish to test the following claim ( H a ) at a significance level of α = 0.01 . H o : μ = 60.8 H a : μ ≠ 60.8 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 8 with mean M = 66.9 and a standard deviation of S D = 10.7 . What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = -.1228 Incorrect The p-value is... less than (or equal to) α greater than α Correct This p-value leads to a decision to... reject the null accept the null fail to reject the null Incorrect

Answer:

  all real numbers: -∞ < x < ∞

Step-by-step explanation:

Any polynomial function is defined for all real numbers. This one is, too.

The domain is ...

  -∞ < x < ∞

_____

Additional comment

Polynomial functions are often used to model things in the real world. One common application is the use of a quadratic function to model ballistic motion (height versus time). Such functions are defined for all values of the independent variable (time), but only have practical application over a smaller domain (time and height ≥ 0). Any question of domain might need to consider the practical domain, where the function is useful.

Answer:

0.3222 = 32.22% probability that the mean weight of the sample babies would differ from the population mean by more than 45 grams.

Step-by-step explanation:

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

Normal probability distribution

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

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

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

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

Central Limit Theorem

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

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

In this problem, we have that:

[tex]\mu = 3366, \sigma = \sqrt{244036} = 494, n = 118, s = \frac{494}{\sqrt{118}} = 45.48[/tex]

Probability if differs by more than 45 grams?

Less than 3366-45 = 3321 or more than 3366+45 = 3411. Since the normal distribution is symmetric, these probabilities are equal. So we find one of them, and multiply by them.

Less than 3321.

pvalue of Z when X = 3321. So

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

By the Central Limit Theorem

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

[tex]Z = \frac{3321 - 3366}{45.48}[/tex]

[tex]Z = -0.99[/tex]

[tex]Z = -0.99[/tex] has a pvalue of 0.1611

2*0.1611 = 0.3222

0.3222 = 32.22% probability that the mean weight of the sample babies would differ from the population mean by more than 45 grams.

Answer:

53/100?

Step-by-step explanation:

I mean you can't really simplify it any more. There's no LCD, GCD. Can't divide it evenly.

To solve it I just looked at the number of places, (3 which mean 100.)

Then I looked at the numbers, (53.)

So you just turn that into a fraction...

Answer:

well there are many ways to but id say 8/15

Step-by-step explanation:

Given:

Given that the quadrilateral ABCD is inscribed in the circle.

The measure of ∠A is (14z - 7)°

The measure of ∠C is (8z)°

The measure of ∠D is (10z)°

We need to determine the measures of ∠A, ∠B, ∠C and ∠D

Value of z:

We know the property that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

Thus, we have;

[tex]\angle A+ \angle C=180^{\circ}[/tex]

Substituting the values, we have;

[tex]14z-7+8z=180[/tex]

       [tex]22z-7=180[/tex]

             [tex]22z=187[/tex]

                 [tex]z=8.5[/tex]

Thus, the value of z is 8.5

Measure of ∠A:

The measure of ∠A can be determined by substituting the value of z.

Thus, we have;

[tex]\angle A=14(8.5)-7[/tex]

[tex]\angle A=119-7[/tex]

[tex]\angle A=112^{\circ}[/tex]

Thus, the measure of ∠A is 112°

Measure of ∠C:

The measure of ∠C can be determined by substituting the value of z.

Thus, we have;

[tex]\angle C=8(8.5)[/tex]

[tex]\angle C =68^{\circ}[/tex]

Thus, the measure of ∠C is 68°

Measure of ∠D:

The measure of ∠D can be determined by substituting the value of z.

Thus, we have;

[tex]\angle D=10(8.5)[/tex]

[tex]\angle D=85^{\circ}[/tex]

Thus, the measure of ∠D is 85°

Measure of ∠B:

The angles B and D are supplementary.

Thus, we have;

[tex]\angle B+ \angle D=180^{\circ}[/tex]

Substituting the values, we get;

[tex]\angle B+ 85^{\circ}=180^{\circ}[/tex]

        [tex]\angle B=95^{\circ}[/tex]

Thus, the measure of ∠B is 95°

Answer:

A: 112°

B: 95°

C: 68°

D: 85°

Step-by-step explanation:

Opposite angles add up to 180

8z + 14z - 7 = 180

22z = 187

z = 8.5

A = 14(8.5) - 7

A = 112

C = 8(8.5)

C = 68

D = 10(8.5)

D = 85

B = 180 - 85

B = 95

Answer:

The rectangle divided in 2 equal parts

Step-by-step explanation:

Answer:

The middle one is correct because, if you look at the question, you need to find the 1 with more than 1/3. so just think of them as fractions. It will make it easier.

Step-by-step explanation:

A rectangle divided into six equal parts. One of the parts is shaded. = 1/6

A rectangle divided into four equal parts. One of the parts is shaded. = 1/4

A rectangle divided into two equal parts. One of the parts is shaded. = 1/2

A rectangle divided into three equal parts. One of the parts is shaded. = 1/3

So 1/2 is more than 1/3.

Answer:

Benny's mom's age is 400% of Benny's age.

Step-by-step explanation:

Let p represent the percent of his mom's age to Benny's age.

48=P(12)

p=4=400%

Answer:

We need to find the unit rates for the fruit. To do this, divide the cost by the number of pounds

Grapefruit:

cost/pounds

$3/6 pounds

3/6=0.5

0.5 per pound

Apples:

cost/pounds

$2.65/5 pounds

2.65/5=0.53

0.53 per pound

Answer:

180

Step-by-step explanation:

The 3 interior angles of any triangle will always add to 180 degrees.

Answer:

(x+3) (x-6) = 0

Step-by-step explanation:

A quadratic equation is in the form of ax²+bx+c. The quadratic function whose zeros are 3 and -6 is (x² + 3x - 18).

A quadratic equation is an equation whose leading coefficient is of second degree also the equation has only one unknown while it has 3 unknown numbers. It is written in the form of ax²+bx+c.

Given that the quadratic function whose zeros are 3 and -6. Therefore, the  quadratic function can be written as,

(x-3)(x+6)

= x² + 6x - 3x -18

= x² + 3x - 18

Hence, the quadratic function whose zeros are 3 and -6 is (x² + 3x - 18).

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Answer:

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

[tex]X \sim N(60,13.5)[/tex]  

Where [tex]\mu=60[/tex] and [tex]\sigma=13.5[/tex]

And for this case we select a sample size of n= 81. Since the distribution for X is normal then we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard error of the mean would be:

[tex]\sigma_{\bar X} =\frac{13.5}{\sqrt{81}}= 1.5[/tex]

4.1.5

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 variable of interest of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(60,13.5)[/tex]  

Where [tex]\mu=60[/tex] and [tex]\sigma=13.5[/tex]

And for this case we select a sample size of n= 81. Since the distribution for X is normal then we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard error of the mean would be:

[tex]\sigma_{\bar X} =\frac{13.5}{\sqrt{81}}= 1.5[/tex]

4.1.5

The standard error of the mean for the sample of automobiles is 1.5 mph ,i.e, option 4, which is found by dividing the standard deviation, 13.5 mph, by the square root of the sample size, 9, because the sample size is 81 vehicles.

The standard error of the mean (SEM) is the standard deviation ()divided by the square root of the sample size (n). For the given data, with a sample size of 81 automobiles and a standard deviation of 13.5 mph, the standard error of the mean can be calculated as follows:

SEM = 13.5 /  81 = 13.5 / 9 = 1.5 mph

Therefore, the value for the standard error of the mean is 1.5.

Answer:

[tex]Amplitude = /\frac{3}{2}/[/tex]

[tex]Range = [\frac{-3}{2} , \frac{3}{2}][/tex]

[tex]Period = \pi[/tex]

Step-by-step explanation:

Given: [tex]y = \frac{3}{2}cos\frac{t}{2}[/tex]

Comparing the equation with the standard form of cosine function :

[tex]y = A cos(Bx- C)[/tex]

where:

[tex]A = Amplitude[/tex]

Formula for calculating Amplitude is given as:

[tex]Amplitude = /A/[/tex]

The formula for calculating Period is given as :

[tex]Period =\frac{2\pi }{B}[/tex]

[tex]B = \frac{1}{2}[/tex]

Therefore , with the comparison

[tex]A = /\frac{3}{2}/[/tex]

which means that:

[tex]Range = \frac{-3}{2} , \frac{3}{2}[/tex]

[tex]Period = \frac{2\pi }{2}[/tex]

[tex]Period = \pi[/tex]

Answer:

3'2+3'2

Step-by-step explanation:

Answer:

3'2+3'2 is your answer

Answer:

Given:

Mean, x' = 94.32

s.d = 1.20

n = 16

a) For null hypothesis:

H0 : u= 95

For alternative hypothesis:

Ha : u≠95

Level of significance, a = 0.01

(Two tailed test)

We reject null hypothesis, h0, if P<0.01 level of significance.

Calculating the test statistic, we have:

[tex] Z = \frac{x' - u_0}{s.d / \sqrt{n}} = \frac{94.32-95}{1.20/ \sqrt{16}}[/tex]

= -2.266

= -2.27

Calculating the P value:

P value = 2P(Z < -2.27)

Using the standard normal table,

NORMSDIST(-2.27)

= 0.0116

P value= 2(0.0116)

= 0.0232

Since P value(0.0232) is greater than level of significance (0.01), we fail to reject the null hypothesis H0.

We can say that there is enough evidence to conclude the data does not support that average melting point differs from the level of 95 at the level of 0.01 significance level.

b) B(u = 94)

= P(when u=94, do not reject H0)

Using the standard nkrmal table, the z-score corresponding to

Z(0.01/2)= 0.005 will be

[tex]Z_a_/_2 = 2.58[/tex]

[tex]B(94) = ∅[Z_a_/_2+ \frac{u_0-u}{s.d- \sqrt{n}}] - ∅[-Z_a_/_2+ \frac{u_0-u}{s.d/ \frac{n}}] [/tex]

[tex]B(94) = ∅[2.58+ \frac{95-94}{1.20- \sqrt{16}}] - ∅[-2.58+ \frac{95-94}{1.2/ \frac{16}}] [/tex]

= ∅(5.91)-∅(0.75)

P(Z≤5.91)-P(Z≤0.75)

From standard normal table, we have:

P(Z≤0.75)=0.7734, P(Z≤5.91)=1

= 1 - 0.7734

= 0.2266

Probability of making type II error when u=94 is 0.2266

c) Probability of committing type II error when u= 94 at a significance level of 0.01 will be =0.10.

B(94) = 0.10

Finding sample size, n for a two tailed test:

[tex] n = [\frac{s.d(Z_a_/_2+Z_B)}{u_0-u}]^2 [/tex]

Using standard normal table, Z score corresponding to a/2 = 0.005 cummulative area(1-0.005 = 0.995) is Z= 2.58

Z score corresponding to 0.10 cummulative area(1-0.10 = 0.90) is Z = 1.28

Our sample size n, wil be=

[tex] n = [\frac{1.2(2.58+1.28)}{95-94}]^2 [/tex]

[tex] = [\frac{4.632}{1}]^2 [/tex]

= 21.46

= 22

Sample size = 22

The z-score for the hypothesis test can be calculated and compared with the critical values for a two-tailed level of .01. The probability of a Type II error can be calculated using the standard normal table, and the required sample size to ensure β(94)=0.1 when α=0.01 can be calculated using the appropriate formula. Remember, in the calculations, to use the values provided in the question.

The null hypothesis will be H0: μ = 95 and the alternative hypothesis Ha: μ ≠ 95. Since we are given σ = 1.20 and the sample mean ¯x = 94.32, we can perform a z-test.

The z-score is calculated as: z = (¯x - μ) / (σ / √n), here n is the number of samples i.e., 16. Plug in the given values to calculate the z-score.

If the obtained z-score lies within the critical values of a two-tailed level of .01 (which for a normal distribution is approximately -2.576 and 2.576), we can't reject the null hypothesis.

Beta error, β(94), is the probability of a Type II error when actual mean (µ) is 94. To calculate this, you first need to find the z value for the α level .01, and then use the standard normal table to find the corresponding values for β(94).

To ensure that β(94)=0.1 when α=0.01, use the formula for sample size in hypothesis testing. You would need to find the z-scores associated with α and β, and then use these in the formula: n = [(Zα √2π + Zβ)^2 σ^2]/ (μ - μ0)^2.

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Final Answer:

The balance of the loan after the 5-year grace period, with no payments made and interest compounding weekly at an annual rate of 4.8%, would be approximately $2884.50.

Explanation:

To solve this problem, we want to calculate the future value of the loan after 5 years of weekly compounding interest. The formula for the future value of an investment compounded more frequently than once a year is

[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt}, \][/tex]

where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.

Given:
- P = $2250 (the initial loan amount)
- r = 4.8\% = 0.048 (the annual interest rate converted to a decimal)
- n = 52 (the loan is compounded weekly)
- t = 5 (the loan is for a period of 5 years)

Now we will plug these values into the formula to calculate the future value of the loan.

[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \\\\\\\[ A = 2250 \left(1 + \frac{0.048}{52}\right)^{52 \times 5} \][/tex]

Calculating the term inside the parentheses first:

[tex]\[ \frac{0.048}{52} = 0.00092307692 \] (approximately)[/tex]

Adding 1 to this result:

[tex]\[ 1 + 0.00092307692 = 1.00092307692 \] (approximately)[/tex]

Now we need to raise this to the power of 260 (which is [tex]\( 52 \times 5 \)[/tex]):

[tex]\[ \left(1.00092307692\right)^{260} \] (use a calculator for this step)[/tex]

After doing this calculation, you should get a value of approximately 1.282002. (Remember, the exact result might vary slightly depending on the precision of your calculator.)

Finally, we can calculate the future value A:

[tex]\[ A = 2250 \times 1.282002 \\\\\[ A \approx 2884.505 \][/tex]

Therefore, the balance of the loan after the 5-year grace period, with no payments made and interest compounding weekly at an annual rate of 4.8%, would be approximately $2884.50.

Answer:

is there any measurements included?

Step-by-step explanation:

The question mainly discusses the artistic attributes and historical aspects of a sculpture, including descriptions of paint used, rather than providing information needed to calculate the area painted.

The question provided does not directly relate to computing the area painted as it mostly describes the artistic characteristics and historical context of a sculpture. It talks about the sculpture's features, color schemes used for painting such as black hair, red and blue dress with a yellow belt, and the use of paint on the sculpture's body and face.

To calculate the area painted, one would ideally need the dimensions of each face of the model or sculpture to sum up their areas if it were a geometric problem in Mathematics. However, the question mainly focuses on the art critique, interpretation, and visual description of the sculpture, highlighting its historical significance and the artistic techniques employed.

Answer:

x = 14

Step-by-step explanation:

Let's solve your equation step-by-step.

6(x − 5) = 54

Step 1: Simplify both sides of the equation.

6(x − 5) = 54

(6)(x) + (6)(−5) = 54(Distribute)

6x + − 30 = 54

6x − 30 = 54

Step 2: Add 30 to both sides.

6x − 30 + 30 = 54 + 30

6x = 84

Step 3: Divide both sides by 6.

6x/6 = 84/6

x = 14

The ratio of the number of women to the total number of participants in the triathlon is 4:13.

The question is asking for the ratio of the number of women to the total number of participants in a triathlon. We know that there are 13 total participants and 9 of them are men. We can determine that there are 4 ladies by taking 9 out of 13. Therefore, the ratio of the number of women to the total number of participants is 4:13. This indicates that 4 women make up each of the 13 participants, according to our calculations.

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Given:

Given that the equation of the line is [tex]y=2x-4[/tex]

We need to determine the slope of the line.

Slope:

The general form of the equation of the line is [tex]y=mx+c[/tex] where m is the slope of the line.

Comparing the equation [tex]y=2x-4[/tex] with the general form of the equation of the line, we get;

Slope = m = 2.

Hence, the slope is 2.

Since, we need to determine the slope of the parallel lines and we know that the slope of the parallel lines are equal.

Thus, the slope of the parallel line is 2.

Hence, Option B is the correct answer.

Answer:

Option D is the correct answer.

Step-by-step explanation:

The possibility that a relationship between two or more variables is caused by something other than chance is known as statistical significance. Statistical hypothesis testing is used to ascertain whether the result of a data set is statistically significant. This test provides a P-value, which represents the probability that random chance could explain the result.

Generally, a P-value of 5% or lower is regarded to be statistically significant.

Statistical significance relates to whether an effect exists, whereas practical significance indicates the magnitude of the effect.

Thus, by the definitions of Statistical significance and practical significance provided, only option  D is correct, the other options are wrong.

The correct option is D- “Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application”.

The difference between statistical and practical significance involves whether findings are unlikely due to chance (statistical significance) or if the differences observed are large enough to matter in real applications (practical significance), correctly identified as option D.

The question asks to explain the difference between statistical significance and practical significance. The correct answer is option D. Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. This involves probability estimates and is often indicated by p-values like p < 0.05, which means the data is very unlikely to have been caused by chance factors alone, and therefore, there might be a real relationship. On the other hand, practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application. This considers if the findings have a real-world impact or importance beyond just being statistically unlikely due to chance.

Answer:

90

Step-by-step explanation:

work backwards, (18+12)3= 90

Plz brainliest :)

Answer:

Step-by-step explanation:

hope it helps to you

please mark it as brainlist

Final answer:

By creating equations representing the cost relationship between the washer and dryer, and solving for the unknown, we find that the cost of the dryer is $325.

Explanation:

To calculate the cost of the dryer when the combined cost of a washer and dryer is $584 and the washer costs $66 less than the dryer, we can set up an equation. Let's denote the cost of the dryer as D and the cost of the washer as W.

1. The first step is to express the washer's cost in terms of the dryer's cost:

W = D - 66

2. Then, use the combined cost to create an equation:

W + D = 584

3. Substitute the expression for W into the second equation:

(D - 66) + D = 584

4. Solve for D:

2D - 66 = 584

2D = 584 + 66

2D = 650

D = 650 / 2

D = 325

Therefore, the cost of the dryer is $325.

To cover a rectangle gift box with dimensions of 8 inches by 10 inches by 7 inches, you need to calculate the surface area by adding the area of all six faces. After calculation, you would need 412 square inches of wrapping paper.

To calculate how much wrapping paper is needed to cover a rectangle gift box exactly, we need to find the surface area of the box. The box dimensions are 8 inches wide, 10 inches long, and 7 inches tall.

Surface area of a rectangle box is calculated by adding the areas of all six faces. The areas of the faces are:

Calculating each area:

Adding all of these areas together gives us:

160 in² + 112 in² + 140 in² = 412 in².

Therefore, you would need 412 square inches of wrapping paper to cover the box exactly.

a. Chocolate need to cover a rectangle with a length of 9 in. and a perimeter of 22 inches is 18 pieces of 1 inch square piece chocolates

b.Chocolate need to cover a rectangle with a length of 5 in. and a perimeter of 20 inches is 20 pieces of 1 inch square piece chocolates

Step-by-step explanation

a. To cover a rectangle with a length of 9 in. and a perimeter of 22 in.

perimeter = 2(length+breadth)

Breadth = 22 -9×2  = 22-18 = 4 ÷ 2 = 2 inches

Area of rectangle = 9 × 2 = 18 inches

18 pieces of  1 inch square piece  Chocolate is needed to cover the rectangle

b. To cover a rectangle with a length of 5 in. and a perimeter of     20 in.

perimeter = 2(length+breadth)

Breadth = 20 - 5 ×2  = 10 = 10 ÷ 2 = 5 inches

Area of rectangle = 5 × 5 = 20 inches

20 pieces of  1 inch square piece  Chocolate is needed to cover the rectangle

Step-by-step explanation:

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Answer:

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

Step-by-step explanation:

For each voter, there are only two possible outcomes. Either the voter is a Democrat, or he is not. The probability of the voter being a Democrat is independent of other voters. So we use the binomial probability distribution to solve this question.

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.

62% of the voters are Democrats

This means that [tex]p = 0.62[/tex]

(a) What is the probability that two independently surveyed voters would both be Democrats?

This is P(X = 2) when n = 2. So

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

[tex]P(X = 2) = C_{2,2}.(0.62)^{2}.(0.38)^{0} = 0.3844[/tex]

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

Answer:

4.3%

Step-by-step explanation:

[tex]selling \: percent \\ = \frac{6}{140} \times 100 \\ \\ = \frac{60}{14} \\ \\ = 4.28 \\ = 4.3\% \\ [/tex]

As per linear equation, 4.3% of tickets are sold of the goal.

"A linear equation is an equation in which the highest power of the variable is always 1."

Given, a local charity has sold 6 raffle tickets.

The goal of selling tickets is 140.

Therefore, the percentage of sold tickets is

[tex]= \frac{6}{140}[/tex] ×100%

[tex]= 4.3[/tex]%

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Answer:

x + 3y = 18  is the equation in standard form.

In  slope-intercept form it is  y = -1/3 x + 6.

Step-by-step explanation:

The given line y = 3x + 2 has a slope of  3 so the line perpendicular to it has slope -1/3.

y - y1 = m(x - x1)  where m = slope and (x1, y1) is a point on the line.

Using the point (3, 5) our equation is:

y - 5 = -1/3(x - 3)     Multiply through by -3:

-3y + 15  = x - 3

x  + 3y =  15 + 3

x + 3y = 18 is the required equation.

Answer:

(a) Proportion of Defectives = 0.01

Expected Number Defective = 0.5

Probability of Rejecting the Shipment = 0.0138

(b) Proportion of Defectives = 0.05

Expected Number Defective = 2.5

Probability of Rejecting the Shipment = 0.459

(c) Proportion of Defectives = 0.1

Expected Number Defective = 5

Probability of Rejecting the Shipment = 0.888

Step-by-step explanation:

This is a binomial distribution problem due to the unchanging probability of getting a defective board, no matter the number of trials ran.

The expected number for binomial distribution is given as E(X) = np

The probability mass funaction for binomial distribution is given as

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = 50

x = Number of successes required = ≥3

p = probability of success = changing from question to question

q = probability of failure = 1 - p

Total number of boards tested = 50

Note that probability of rejecting the shipment for each of the sub-question is the probability that 3 or more boards are defective. That is, P(X ≥ 3)

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)]

a) Proportion of Defectives=0.01

Expected Number Defective = np = 0.01 × 50 = 0.5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.01

q = probability of failure = 1 - 0.01 = 0.99

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.01381727083 = 0.0138

b) Proportion of Defectives = 0.05

Expected Number Defective = np = 0.05 × 50 = 2.5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.05

q = probability of failure = 1 - 0.05 = 0.95

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.45946687728 = 0.459

c) Proportion of Defectives = 0.1

Expected Number Defective = np = 0.1 × 50 = 5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.1

q = probability of failure = 1 - 0.1 = 0.90

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.88827124366 = 0.888

Hope this Helps!!!