what is the simplest form of this expression? m(m+4)+m(m-2)

Answer:

one mile per hour

Step-by-step explanation:

Answer:

It is 60 miles per hour

Step-by-step explanation: average speed = total distance / total time

Answer:

$ 63.75

Step-by-step explanation:

Final answer:

The sale price of a $75 iPod with a 15% discount is calculated by determining the discount amount ($11.25) and subtracting it from the original price, resulting in a sale price of $63.75. A similar calculation method is used to find the total cost of an $85 jacket with 7.5% sales tax, totaling $91.38.

Explanation:

To calculate the sale price of an iPod that originally costs $75 with a 15% discount, we first need to calculate the amount of the discount. To do this, multiply the original price of $75 by the discount rate of 15%.

$75 × 0.15 = $11.25

Now, subtract the discount from the original price to find the sale price:

$75 - $11.25 = $63.75

So, the sale price of the iPod is $63.75.

To illustrate using a similar example, let's imagine Emily purchased a jacket for $85 and needs to calculate the total cost including a 7.5% sales tax. First, find the amount of the sales tax by multiplying the cost of the jacket by the tax rate:

$85 × 0.075 = $6.38

Then, add the sales tax to the original price of the jacket to find the total cost:

$85 + $6.38 = $91.38

Therefore, the total cost of the jacket, including tax, is $91.38.

Answer:

Jade's statement is incorrect because "8 divided by ½" does not mean the same thing as "½ of 8".

Explanation:

No, I don't agree with Jade's statement.

When we say "8 divided by ½", it means we are dividing 8 into two equal parts, each representing one-half. Mathematically, [tex]\(8 \div \frac{1}{2}\)[/tex] means we are dividing 8 by [tex]\(\frac{1}{2}\)[/tex] , which is equivalent to multiplying 8 by the reciprocal of [tex]\(\frac{1}{2}\)[/tex], which is 2. So:

[tex]\[ 8 \div \frac{1}{2} = 8 \times 2 = 16 \][/tex]

On the other hand, "½ of 8" means finding half of 8, which is indeed 4. However, this is not the same as "8 divided by ½".

Answer: the answer is 2/3

Answer:

3 ounces

Step-by-step explanation:

1/3 of a pound is 6 ounces.

6 divided by 2 is 3.

Each pile of beans weighs 3 ounces.

Answer:

[tex]x = \sqrt{5}\\\\y = \frac{15}{ \sqrt{5} }[/tex]

Step-by-step explanation:

According to the information of the problem

[tex]xy = 15[/tex]

And

[tex]S = 3x+y[/tex]

If you solve for [tex]y[/tex] on the first equation you get that

[tex]y = {\displaystyle \frac{15}{x}}[/tex]

then you have that

[tex]S = {\displaystyle 3x + \frac{15}{x} }[/tex]

If you find the derivative of the function you get that

[tex]S' = {\displaystyle 3 - \frac{15}{x^2}} = 0\\[/tex]

The equation has two possible solutions but you are looking for POSITIVE numbers that make [tex]S[/tex] as small as possible.

Then

[tex]x = \sqrt{5}\\\\y = \frac{15}{ \sqrt{5} }[/tex]

Answer:

m\angle ABC=124^o

Step-by-step explanation:

Answer: b

Step-by-step explanation: ik

Answer:$27

Step-by-step explanation:

cost price(cp)=$36

Percentage off=25

sale price=sp

Percentage off=(cp-sp)/cp x 100

25=(36-sp)/36 x 100

Cross multiplying we get

25x36=100(36-sp)

900=100(36-sp)

Divide both sides by 100 we get

900/100=100(36-sp)/100

9=36-sp

Collect like terms

sp=36-9

sp=27

Answer: x^2+8x+15

Step-by-step explanation:

(x+5)(x+3)

x^2+3x+5x+15

x^2+8x+15

To expand the polynomial (x + 5)(x + 3), multiply each term from the first binomial by each term from the second binomial and combine like terms.

To expand the polynomial (x + 5)(x + 3), we need to use the distributive property. We multiply each term from the first binomial by each term from the second binomial. So, we have:

Combining like terms, we get x^2 + 3x + 5x + 15. Simplifying further, we have the expanded polynomial:

x^2 + 8x + 15.

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Answer:

y= 7.5

Step-by-step explanation:

25y = 15y +75

-15y -15y

10y = 75

÷10 ÷10

y = 7.5

Answer:

The height of tree house is 70.71 feet

Step-by-step explanation:

We are given that A 100-foot rope from the top of a tree house to the ground forms a 45∘ angle of elevation from the ground

Refer the attached figure

Length of rope AC = Hypotenuse =100 feet

The top of a tree house to the ground forms a 45∘ angle of elevation from the ground =[tex]\angle ACB = 45^{\circ}[/tex]

We are supposed to find the height of tree house i.e.AB = Perpendicular

So, Using trigonometric ratio

[tex]Sin \theta = \frac{perpendicular}{Hypotenuse}\\Sin 45= \frac{AB}{AC}\\\frac{1}{\sqrt{2}}=\frac{AB}{100}\\100 \times \frac{1}{\sqrt{2}}=AB\\70.71=AB[/tex]

Hence The height of tree house is 70.71 feet

Answer:

A. False

B. False

C. False

D. True

Step-by-step explanation:

A. False

Remember that E[aX+b] = aE[X] + b, therefore the equality would be true only if E[X] = 1.

B. False

Remember that [tex]V[aX+b] = a^2 V[X][/tex].

C. False

In other words they are telling you in words that [tex]V[X+b] = V[X]+b[/tex], that is false because [tex]V[X+b] = V[X][/tex]

D. True

That is a property of the expected value E[X+b] = E[X]+b

Option D is correct in stating that adding a constant to each value of a random variable shifts the expected value of that variable by that same constant. Adding a constant does not alter the variance.

The correct answer among the given options is D. This statement says that if a constant c is added to each possible value of a discrete random variable X, then the expected value of X will be shifted by that same constant amount. This is in accordance with the properties of expected value, where the expected value of (X + c) is E(X) + c. On the other hand, adding a constant does not affect the variance of a random variable. Variance measures the dispersion of a variable's possible values from its expected value, and adding a constant shifts every possible value, including the expected value, by the same amount, leaving dispersion unchanged.

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Answer:

Yes. There is enough evidence to support the claim that there are fewer errors on average when the yellow ball is used.

Step-by-step explanation:

The question is incomplete:

The sample data is:

Yellow 5 2 6 7 2 5 3 8 4 9

White 7 6 8 5 9 11 8 3 6 10

This is a hypothesis test for the difference between populations means.

The claim is that there are fewer errors on average when the yellow ball is used.

Then, the null and alternative hypothesis are:

[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0[/tex]

The significance level is α=0.05.

The sample 1 (yellow ball errors), of size n1=10 has a mean of 5.1 and a standard deviation of 2.42.

The sample 2 (white balls errors), of size n2=10 has a mean of 7.3 and a standard deviation of 2.41.

The difference between sample means is Md=-2.2.

[tex]M_d=M_1-M_2=5.1-7.3=-2.2[/tex]

The estimated standard error of the difference between means is computed using the formula:

[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2+\sigma_2^2}{n}}=\sqrt{\dfrac{2.42^2+2.41^2}{10}}\\\\\\s_{M_d}=\sqrt{\dfrac{11.665}{10}}=\sqrt{1.166}=1.08[/tex]

Then, we can calculate the t-statistic as:

[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{-2.2-0}{1.08}=\dfrac{-2.2}{1.08}=-2.037[/tex]

The degrees of freedom for this test are:

[tex]df=n_1+n_2-1=10+10-2=18[/tex]

This test is a left-tailed test, with 18 degrees of freedom and t=-2.037, so the P-value for this test is calculated as (using a t-table):

[tex]P-value=P(t<-2.037)=0.028[/tex]

As the P-value (0.028) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that there are fewer errors on average when the yellow ball is used.

Yes, there are fewer errors on average when the yellow ball is used and this can be determined by using the given data.

The Hypothesis test is carried out in which null and alternate hypothesis is given below:

[tex]\rm H_0 : \mu_1-\mu_2=0[/tex]

[tex]\rm H_a : \mu_1-\mu_2<0[/tex]

Now, determine the sample mean difference.

[tex]\rm M_d = M_1-M_2 = 5.1-7.3 = -2.2[/tex]

Now, determine the estimated standard error using the below formula:

[tex]\rm s =\sqrt{\dfrac{\sigma^2_1+\sigma^2_2}{n}}[/tex]

[tex]\rm s =\sqrt{\dfrac{(2.42)^2+(2.41)^2}{10}}[/tex]

s = 1.08

So, the t-statistics can be calculated as:

[tex]\rm t = \dfrac{M_d-(\mu_1-\mu_2)}{s}[/tex]

[tex]\rm t = \dfrac{-2.2-0}{1.08}=-2.037[/tex]

Now, determine the degree of freedom.

[tex]\rm df = n_1+n_2-1[/tex]

df = 10 + 10 - 2

df = 18

Now, for this test, the p-value is 0.028 which is less than the significance level. Therefore, the null hypothesis is rejected.

For more information, refer to the link given below:

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Answer:

-P(paid) = 10/26 and P(paid|game) = 5/14.

-The events "paid" and "game" are not independent.

Step-by-step explanation:

Number of paid apps downloaded = 10

Number of free apps downloaded = 16

Total number of apps = 10 + 16 = 26

Thus;

P(paid) = 10/26

Now, it says she paid for 5 out if 14 which were game apps. Thus;

P(paid|game) = 5/14

Now, Two events are independent if the result of the second event is not affected by the result of the first event. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events.

In this question, paid and game are not affected by each other and the probability of of P(paid) and P(paid|game) occurring are not products of each individual event paid and game. Thus, they are not independent.

Answer:

 10/26 , 5/14 , not independent

Step-by-step explanation:

Answer:

(2). 26 (4). 547 (6). 3,132 (8). 46.1 (10). 10.6

Step-by-step explanation:

Answer:

Step-by-step explanation:

the volume of the prism is the amount of space inside the figure.

120 cubes fit inside the prism, so the figure has a volume of 120 square unite

Answer:

1. B 2. C

Step-by-step explanation:

Answer:

[tex]\hat p = \frac{0.244+0.326}{2}=0.285[/tex]

[tex] ME = \frac{0.326-0.244}{2}=0.041[/tex]

[tex] 0.285 \pm 0.041[/tex]

Step-by-step explanation:

For this case we have a confidence interval given as a percent:

[tex] 24.4\% \leq p \leq 32.6\%[/tex]

If we express this in terms of fraction we have this:

[tex] 0.244 \leq p \leq 0.326 [/tex]

We know that the confidence interval for the true proportion is given by:

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

And thats equivalent to:

[tex]\hat p \pm ME[/tex]

We can estimate the estimated proportion like this:

[tex]\hat p = \frac{0.244+0.326}{2}=0.285[/tex]

And the margin of error can be estimaed using the fact that the confidence interval is symmetrical

[tex] ME = \frac{0.326-0.244}{2}=0.041[/tex]

And then the confidence interval in the form desired is:

[tex] 0.285 \pm 0.041[/tex]

Answer:

(a) The total number of ways to select 4 officers from from 25 students is 12,650.

(b) The total number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is 5,148.

Step-by-step explanation:

(a)

It is provided that there are a total of n = 25 students.

Officers need to be elected for four positions:

President, Vice-President, Secretary, and Treasurer.

k = 4

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

[tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]

Compute the number of ways to select 4 students from from 25 as follows:

[tex]{25\choose 4}=\frac{25!}{4!(25-4)!}[/tex]

      [tex]=\frac{25!}{4!\times 21!}\\\\=\frac{25\times 24\times 23\times 22\times 21!}{4!\times 21!}\\\\=\frac{25\times 24\times 23\times 22}{4\times 3\times 2\times 1}\\\\=12650[/tex]

Thus, the total number of ways to select 4 officers from from 25 students is 12,650.

(b)

It is provided that of the 25 students, there are 13 girls and 12 boys in the class.

For the post of President and Treasurer only girls are selected.

For the post of Vice-President and Secretary only boys are selected.

Compute the number of ways to select 2 girls for the post of President and Treasurer as follows:

[tex]{13\choose 2}=\frac{13!}{2!(13-2)!}[/tex]

      [tex]=\frac{13!}{2!\times 11!}\\\\=\frac{13\times 12\times 11!}{2!\times 11!}\\\\=\frac{13\times 12}{ 2\times 1}\\\\=78[/tex]

Compute the number of ways to select 2 boys for the post of Vice-President and Secretary as follows:

[tex]{12\choose 2}=\frac{12!}{2!(12-2)!}[/tex]

      [tex]=\frac{12!}{2!\times 10!}\\\\=\frac{12\times 11\times 10!}{2!\times 10!}\\\\=\frac{12\times 11}{ 2\times 1}\\\\=66[/tex]

The number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is:

[tex]{13\choose 2}\times {12\choose 2}=78\times 66=5148[/tex]

Thus, the total number of ways the four officers are selected such that the President and Treasurer are girls and the Vice-President and Secretary are boys is 5,148.

Answer:

1 9/10 or 1.9

Answer:

1.9

Step-by-step explanation:

Answer:

The treatment should be stated by the four companies,since it more interested in the quality among each of the companies to be compared.

Step-by-step explanation:

From the example given, Since an electronic company wants to differentiate  their cell phones quality to the cell phones from their three main competitors.

If ANOVA is used to determine the average number of defects, then the treatment  should be defined for the four companies because it is more interested in comparing the quality among the different companies.

In ANOVA analysis for the given question, the treatments would be defined as the cell phones from the electronics company itself and three of its competitors. ANOVA evaluates if there is any significant difference in the average number of defects amongst these four groups.

In the situation described, an electronics company testing the quality of their cell phones against three competitors, the treatments in the context of ANOVA (Analysis of Variance) would be the four different companies' cell phones. Specifically, a one-way ANOVA is being used here since there is only one factor or variable (the company) affecting the outcome variable (the number of defects).

Each sample of phones (10 from each company) represents a level within the treatment. Therefore, the treatments here are the cell phones from the electronics company, and Cell Phone Company 1, Cell Phone Company 2, and Cell Phone Company 3.

This comparison is made by analyzing the variance within each group's data and between the groups. The essential goal of ANOVA is to test if there is any significant difference between these means. In this case, it is used to ascertain whether the mean number of defects is significantly different across the four companies.

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Answer:

20%

Step-by-step explanation:

(54/45) - 1 = 0.2 or 20%

Answer:

1/5

Step-by-step explanation:

4 of the 20 marbles are red, so the probability is 4/20 = 1/5.

Answer:

is there a pic i need more info...

Step-by-step explanation:

Answer:

864 square inches

Step-by-step explanation:

To solve this problem, first we need to know the area of each side of the cube(since the shrink wrap will cover these sides) :

The area of each side is A=[tex]L^{2}[/tex] where L is the length of the side. In this case, L= 4 inches.

Thus, the area of each side is [tex]A=L^{2}=4^{2} =16[/tex] square inches.

However, the cube has 6 sides so we have to multiply the area of each side by 6, this gives us [tex](16)(6)= 96[/tex] square inches. Thus, we need 96 square inches of shrink wrap for each cube.

Now, we have nine cubes, so we have to multiply those 96 square inches by 9, [tex](96)(9)= 864[/tex].

Thus, we need 864 square inches of shrink wrap to wrap 9 cubes.

Answer:

34.86% probability that it will be huge​ success

Step-by-step explanation:

Bayes Theorem:

Two events, A and B.

[tex]P(B|A) = \frac{P(B)*P(A|B)}{P(A)}[/tex]

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Receiving a favorable review.

Event B: Being a huge success.

Information on previous textbooks published show that 20 % are huge​ successes

This means that [tex]P(B) = 0.2[/tex]

99 % of the huge successes received favorable​ reviews

This means that [tex]P(A|B) = 0.99[/tex]

Probability of receiving a favorable review:

20% are huge​ successes. Of those, 99% receive favorable reviews.

30% are modest​ successes. Of those, 70% receive favorable reviews.

30% break​ even. Of those, 40% receive favorable reviews.

20% are losers. Of those, 20% receive favorable reviews.

Then

[tex]P(A) = 0.2*0.99 + 0.3*0.7 + 0.3*0.4 + 0.2*0.2 = 0.568[/tex]

Finally

[tex]P(B|A) = \frac{P(B)*P(A|B)}{P(A)} = \frac{0.2*0.99}{0.568} = 0.3486[/tex]

34.86% probability that it will be huge​ success

Answer:

9

Step-by-step explanation:

Plug in 3 for x, and the equation is 6 + 3, which is equal to 9.

Answer:

6 + x = 9

Step-by-step explanation:

If x equals 3

6 plus x (when x equals 3)

Thus, 6 plus 3

= 9

9 will be your answer

Answer:

a) D. The group of all of the babies whose recorded weights were examined

b) A. The weights of the babies at birth

c) i) Ux' = u = 3370;

s.d = 43.95

ii) Ux' = u = 3370

s.d = 35.109

Step-by-step explanation:

Given:

Mean, u = 3370

Standard deviation = 582

a) Population in statistics involves every member of the group in study.

Here, the correct population is the group of all of the babies whose recorded weights were examined.

b) In statistics, variable involves data collected.

The correct variable is the weights of the babies at birth

c)

i) at X' = 175

Sample mean:

Ux' = u = 3370

Standard deviation:

[tex] \sigma x' = \frac{\sigma}{\sqrt{n}} = \frac{582}{\sqrt{175}} = 43.995 [/tex]

ii) at X' = 175

Sample mean:

Ux' = u = 3370

Stanard deviation:

[tex] \sigma x' = \frac{\sigma}{\sqrt{n}} = \frac{582}{\sqrt{350}} = 35.109 [/tex]

a. The population includes all babies whose recorded weights were examined, and the variable is the birth weights of the babies.

b. For sample sizes of 175, the mean birth weight remains 3370 grams, with a standard deviation of approximately 43.96 grams.

c.  For sample sizes of 350, the mean birth weight remains 3370 grams, with a standard deviation of approximately 31.15 grams.

Let's address each part of the question step-by-step to ensure clarity and accuracy.

Part (a): Identifying Population and Variable

To identify the correct population:

To identify the correct variable:

Hence, the correct answers are:

Part (b): Calculating Mean ([tex]\mu_\bar{x}[/tex]) and Standard Deviation ([tex]\sigma_\bar{x}[/tex]) for Sample Size 175

For samples of size 175, we need to find the mean and standard deviation of all possible sample means:

Part (c): Calculating Mean ([tex]\mu_\bar{x}[/tex]) and Standard Deviation ([tex]\sigma_\bar{x}[/tex]) for Sample Size 350

For samples of size 350, we perform similar calculations:

In summary, we have identified the population and variable, and calculated the mean and standard deviation for different sample sizes of 175 and 350.

Answer:

[tex] (0.79-0.72) -1.96 \sqrt{\frac{0.79(1-0.79)}{1200} +\frac{0.72(1-0.72)}{1500}} =0.0376[/tex]

[tex] (0.79-0.72) +1.96 \sqrt{\frac{0.79(1-0.79)}{1200} +\frac{0.72(1-0.72)}{1500}} =0.1024[/tex]

And the 95% confidence interval for the difference of the two proportions is given by:

[tex] 0.0376 \leq p_1 -p_2 \leq 0.1024[/tex]

Step-by-step explanation:

For this case we have the following info given:

[tex] X_1 = 948[/tex] number of people that they read at least one book in the last 3 months in 2011

[tex]n_1 = 1200[/tex] the sample size selected for 2011

[tex] X_2 = 1080[/tex] number of people that they read at least one book in the last 3 months in 2015

[tex]n_2 = 1500[/tex] the sample size selected for 2015

The estimated proportions people that they read at least one book in the last 3 months for each year are given by:

[tex]\hat p_1 = \frac{948}{1200}= 0.79[/tex]

[tex]\hat p_2 = \frac{1080}{1500}= 0.72[/tex]

And the confidence interval for the true difference of proportions is given by:

[tex](\hat p_1 -\hat p_2) \pm z_{\alpha/2} \sqrt{\frac{\hat p_1 (1-\hat p_1)}{n_1} +\frac{\hat p_2 (1-\hat p_2)}{n_2}}[/tex]

The confidence level is 95% so then the significance is 0.05 or 5% and [tex]\alpha/2 =0.025[/tex] and the critical value for this case using the normal standard distribution is:

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

And replacing into the confidence interval formula we got:

[tex] (0.79-0.72) -1.96 \sqrt{\frac{0.79(1-0.79)}{1200} +\frac{0.72(1-0.72)}{1500}} =0.0376[/tex]

[tex] (0.79-0.72) +1.96 \sqrt{\frac{0.79(1-0.79)}{1200} +\frac{0.72(1-0.72)}{1500}} =0.1024[/tex]

And the 95% confidence interval for the difference of the two proportions is given by:

[tex] 0.0376 \leq p_1 -p_2 \leq 0.1024[/tex]

Answer:

Alex can line eight 1-meter wide with the paper.

Step-by-step explanation:

- Alex has five rolls of shelf paper that is 800cm.

- She wants to use the paper to line the 1-meter wide shelves in her pantry.

- We want to determine how many 1-meter wide she can line with the paper.

- First, we know that

100cm = 1m

- we need to determine how many meters are in 800cm.

100cm = 1m

800cm = xm

100x = 800

x = 800/100

= 8

Therefore, 800cm is equivalent to 8m

Alex can line eight 1-meter wide with the paper.

To determine the number of 1-meter wide shelves Alex can line with the 800 cm long shelf paper, convert the total length to meters and divide by the shelf width. Alex can line 8 shelves with the paper.

To find out how many 1-meter wide shelves Alex can line with the 800 cm long shelf paper, we need to convert the total length of the paper to meters to match the shelf width.

Convert 800 cm to meters: 800 cm = 8 meters

Divide the total length of the paper by the width of each shelf: 8 meters / 1 meter = 8 shelves

Alex can line 8 shelves with the 1-meter wide shelf paper she has.

Answer:

a) 0.5.

b) 0.8413

c) 0.8413

d) 0.6826

e) 0.9332

f) 1

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 = 6, \sigma = 0.2[/tex]

(a) P(x > 6) =

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

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

[tex]Z = \frac{6-6}{0.2}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5.

1 - 0.5 = 0.5.

(b) P(x < 6.2)=

This is the pvalue of Z when X = 6.2. So

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

[tex]Z = \frac{6.2-6}{0.2}[/tex]

[tex]Z = 1[/tex]

[tex]Z = 1[/tex] has a pvalue of 0.8413

(c) P(x ≤ 6.2) =

In the normal distribution, the probability of an exact value, for example, P(X = 6.2), is always zero, which means that P(x ≤ 6.2) = P(x < 6.2) = 0.8413.

(d) P(5.8 < x < 6.2) =

This is the pvalue of Z when X = 6.2 subtracted by the pvalue of Z when X  5.8.

X = 6.2

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

[tex]Z = \frac{6.2-6}{0.2}[/tex]

[tex]Z = 1[/tex]

[tex]Z = 1[/tex] has a pvalue of 0.8413

X = 5.8

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

[tex]Z = \frac{5.8-6}{0.2}[/tex]

[tex]Z = -1[/tex]

[tex]Z = -1[/tex] has a pvalue of 0.1587

0.8413 - 0.1587 = 0.6826

(e) P(x > 5.7) =

This is 1 subtracted by the pvalue of Z when X = 5.7.

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

[tex]Z = \frac{5.8-6}{0.2}[/tex]

[tex]Z = -1.5[/tex]

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

1 - 0.0668 = 0.9332

(f) P(x > 5) =

This is 1 subtracted by the pvalue of Z when X = 5.

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

[tex]Z = \frac{5-6}{0.2}[/tex]

[tex]Z = -5[/tex]

[tex]Z = -5[/tex] has a pvalue of 0.

1 - 0 = 1

Summary of probabilities

[tex](a)\ P(x > 6) = 0.5\\(b)\ P(x < 6.2) = 0.8413\\(c)\ P(x \leq 6.2) = 0.8413\\(d)\ P(5.8 < x < 6.2) = 0.6826\\(e)\ P(x > 5.7) = 0.9332\\(f)\ P(x > 5) \approx 1\\[/tex]

(a) [tex]\(P(x > 6)\)\\[/tex]

1. Calculate the z-score for [tex]\(x = 6\)[/tex] :

2. Find [tex]\(P(Z > 0)\)[/tex] :

(b) [tex]\(P(x < 6.2)\)[/tex]

1. Calculate the z-score for [tex]\(x = 6.2\)[/tex] :

2. Find [tex]\(P(Z < 1)\)[/tex]:

(c) [tex]\(P(x \leq 6.2)\)[/tex]

(d) [tex]\(P(5.8 < x < 6.2)\)[/tex]

1. Calculate the [tex]\(z\)[/tex]-score for [tex]\(x = 5.8\):[/tex]

2. Calculate the [tex]\(z\)[/tex]-score for [tex]\(x = 6.2\):[/tex]

3. Find [tex]\(P(Z < 1)\)[/tex] and [tex]\(P(Z < -1)\):[/tex]

4. Calculate [tex]\(P(-1 < Z < 1)\):[/tex]

(e) [tex]\(P(x > 5.7)\)[/tex]

1. Calculate the [tex]\(z\)[/tex]-score for [tex]\(x = 5.7\):[/tex]

2. Find [tex]\(P(Z > -1.5)\):[/tex]

(f) [tex]\(P(x > 5)\)[/tex]

1. Calculate the [tex]\(z\)[/tex]-score for [tex]\(x = 5\):[/tex]

2. Find [tex]\(P(Z > -5)\):[/tex]