Suppose 100 stastisticians attended a conference of the American Statistical Society. At the dinner, among the menu options were a Caesar salad, roast beef, and apple pie. 35 had the Caesar salad, 28 had the roast beef, and 45 had the apple pie for dessert. Also, 15 had at least two of those three offerings, and 2 had all three. How many attendees had none of the three

Answer:

The probability that he makes one of the two free throws is 0.38

Step-by-step explanation:

Hello!

Considering the situation:

A pro basketball player shoots two free throws.

The following events are determined:

A: "He makes the first free throw"

Ac: "He doesn't make the first free throw"

B: "He makes the second free throw"

Bc: "He doesn't make the second free throw"

It is known that

P(A)= 0.48

P(B/A)= 0.62

P(B/Ac)= 0.38

You need to calculate the probability that he makes one of the two free throws.

There are two possibilities, that "he makes the first throw but fails the second" or that "he fails the first throw and makes the second"

Symbolically:

P(A∩Bc) + P(Ac∩B)

Step 1.

P(A)= 0.48

P(Ac)= 1 - P(A)= 1 - 0.48= 0.52

P(Ac∩B) = P(Ac) * P(B/Ac)= 0.52*0.38= 0.1976≅ 0.20

Step 2.

P(A∩B)= P(A)*P(B/A)= 0.48*0.62= 0.2976≅ 0.30

P(A)= P(A∩B) + P(A∩Bc)

P(A∩Bc)= P(A) - P(A∩B)= 0.48 - 0.30= 0.18

Step 3

P(Ac∩B) + P(A∩Bc) = 0.20 + 0.18= 0.38

I hope this helps!

The probability that the player makes one of the two free throws using the multiplicative rule is 0.4952 or 49.52%.

To find the probability that the player makes one of the two free throws using the multiplicative rule, we need to consider the two possible ways he can do this:

We can calculate the probability for each case and sum them up to find the total probability:

p(make 1st and miss 2nd) = (0.48) * (0.62) = 0.2976

p(miss 1st and make 2nd) = (0.52) * (0.38) = 0.1976

Total probability = p(make 1st and miss 2nd) + p(miss 1st and make 2nd) = 0.2976 + 0.1976 = 0.4952

Therefore, the probability that the player makes one of the two free throws is 0.4952, or 49.52%.

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Answer:

C.

Step-by-step explanation:

The amount of property tax that Ian pays is $10,890.

Tax is a compulsory sum of money levied on goods and services by the government. Property tax is the tax paid on property that is owned by an individual or group of individuals.

Property tax = tax rate x assessed value x market price

0.088 x 0.55 x 225,000 = $10,890

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Answer: the probability that during a randomly selected month, PCE's were between $775.00 and $990.00 is 0.9538

Step-by-step explanation:

Since the amount that the company spent on personal calls followed a normal distribution, then according to the central limit theorem,

z = (x - µ)/σ

Where

x = sample mean

µ = population mean

σ = standard deviation

From the information given,

µ = $900

σ = $50

the probability that during a randomly selected month PCE's were between $775.00 and $990.00 is expressed as

P(775 ≤ x ≤ 990)

For (775 ≤ x),

z = (775 - 900)/50 = - 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.0062

For (x ≤ 990),

z = (990 - 900)/50 = 1.8

Looking at the normal distribution table, the probability corresponding to the z score is 0.96

Therefore,

P(775 ≤ x ≤ 990) = 0.96 - 0.0062 = 0.9538

The probability that PCE's were between $775 and $990 is 0.9579 or 95.79%.

To find the probability that the PCE's were between $775 and $990 during a randomly selected month, we first need to standardize the values using the standard normal distribution. Formula for standardization is:

Z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

Using the formula, we calculate the standard scores for the given values:

Z1 = ($775 - $900) / $50 = -2.50

Z2 = ($990 - $900) / $50 = 1.80

Next, we use the standard normal distribution table or a calculator to find the corresponding probabilities for these z-scores. The probability between the z-scores -2.50 and 1.80 is the difference between their corresponding cumulative probabilities:

Prob(Z1 < Z < Z2) = Prob(Z < Z2) - Prob(Z < Z1)

Using the standard normal distribution table, we can find the probabilities:

Prob(Z < -2.50) = 0.0062

Prob(Z < 1.80) = 0.9641

Finally, we calculate the probability between the z-scores:

Prob(Z1 < Z < Z2) = 0.9641 - 0.0062 = 0.9579

Therefore, the probability that PCE's were between $775 and $990 during a randomly selected month is approximately 0.9579 or 95.79%.

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Answer:

0.24

Step-by-step explanation:

Answer: 6/25 as a decimal would be 0.24

Step-by-step explanation:0.240 and 0.24 are both the same thing, the 0 behind the 4, doesn't have value. But, most teachers would prefer you to put it as 0.24. Also, in order to turn a fraction into a decimal, you just divide the top number (numerator) by the bottom number (denominator). 6 divided by 25 is 0.24

Answer: The expected utility is 0.59.

Step-by-step explanation:

Since we have given that

[tex]U(c)=\sqrt{c}[/tex]

Probability that he will be injured = 0.1

Probability that he will not be injured = 0.9

If Billy is not injured this season, he will receive an income of 25 million dollars.

and

If he is injured, his income will be only 10,000 dollars.

According to question, the expected utility is given by

[tex]E[x]=0.9\times \sqrt{(0.01)}+0.1\times \sqrt{25}\\\\E[x]=0.9\times 0.1+0.1\times 5\\\\E[x]=0.09+0.5\\\\E[x]=0.59[/tex]

Hence, the expected utility is 0.59.

Final answer:

To calculate Billy's expected utility, we need to multiply his utility function by the probability of each outcome and sum the results. His expected utility is approximately 3333.33.

Explanation:

To calculate Billy's expected utility, we need to multiply his utility function by the probability of each outcome and sum the results. In this case, Billy's utility function is U(c) = c^(1/2), where c represents his income. If Billy is not injured, his income will be $25 million, and if he is injured, his income will be $10,000. The probability of being injured is 0.1, and the probability of not being injured is 0.9.

Expected utility = U(income if not injured) * P(not injured) + U(income if injured) * P(injured)

Expected utility = U($25 million) * 0.9 + U($10,000) * 0.1

Expected utility = (25 million)^(1/2) * 0.9 + (10,000)^(1/2) * 0.1

Solving this equation, we find that the expected utility for Billy is approximately 3333.33.

Answer:

1.  3/4 + 3/4 + 3/4 +3/4

2. 0.75 * 4

Step-by-step explanation:

1. add 3/4 four times

3/4 + 3/4 + 3/4 +3/4

2. You can turn 3/4 into a decimal.  3/4 =0.75

0.75 * 4

Final answer:

3/4 × 4 can be solved through repeated addition by adding 3/4 to itself four times to get 9/4 or 2 1/4. Alternatively, by simplifying before multiplying, recognizing that 4 is the reciprocal of 1/4, we easily find that the product is 3.

Explanation:

When solving 3/4 × 4 using repeated addition, we use the concept that multiplying a number by a whole number is the same as adding that number to itself that many times. In this case, 3/4 is added to itself 4 times:

We have four 3/4's, and when we add them up, we get:

When we add 3/2 (1 1/2) and 3/4, we can convert 1 1/2 into 6/4 to make it easier to add the fractions, obtaining:

Therefore, 3/4 × 4 equals 9/4 or 2 1/4 through repeated addition.

Another way to approach the problem is by simplifying before multiplying. Since we are multiplying by 4, which is the reciprocal of 1/4, we can simplify by understanding that:

Thus, by canceling out the common factors (4 in the numerator and 4 in the denominator), the multiplication becomes 3 × 1, which equals 3. This satisfies the condition that as long as we perform the same operation on both sides of the equals sign, the expression remains an equality.

Answer:

[tex]t=\frac{2.73-3.39}{\frac{1.22}{\sqrt{26}}}=-2.758[/tex]    

[tex]df=n-1=26-1=25[/tex]  

[tex]p_v =P(t_{(25)}<-2.758)=0.0054[/tex]  

Since the p value is lower than the significance level 0.1 we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significanlty lower than 3.39 personas at 10% of significance.

Step-by-step explanation:

Data given and notation  

[tex]\bar X=2.73[/tex] represent the sample mean

[tex]s=1.22[/tex] represent the sample standard deviation

[tex]n=26[/tex] sample size  

[tex]\mu_o =3.39[/tex] represent the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true mean is less than 3.39 persons, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 3.39[/tex]  

Alternative hypothesis:[tex]\mu < 3.39[/tex]  

The statistic is given by:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

Calculate the statistic

We can replace in formula (1) the info given like this:  

[tex]t=\frac{2.73-3.39}{\frac{1.22}{\sqrt{26}}}=-2.758[/tex]    

P-value

The degreed of freedom are given by:

[tex]df=n-1=26-1=25[/tex]  

Since is a one sided lower test the p value would be:  

[tex]p_v =P(t_{(25)}<-2.758)=0.0054[/tex]  

Conclusion  

Since the p value is lower than the significance level 0.1 we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is significanlty lower than 3.39 personas at 10% of significance.

Answer:

a) The 95% CI for the mean surgery time is (133.05, 140.75).

b) The 99.5% CI for the mean surgery time is (131.37, 142.43).

c) The level of confidence of the interval (133.9, 139.9) is 69%.

d) The sample size should be 219 surgeries.

e) The sample size should be 377 surgeries.

Step-by-step explanation:

We have a sample, of size n=132, for which the mean time was 136.9 minutes with a standard deviation of 22.6 minutes.

a) We have to find a 95% CI for the mean surgery time.

The critical value of z for a 95% CI is z=1.96.

The margin of error of the CI can be calculated as:

[tex]E=z\cdot s/\sqrt{n}=1.96*22.6/\sqrt{132}=44.296/11.489=3.85[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=\bar x-E=136.9-3.85=133.05\\\\UL=\bar x+E=136.9+3.85=140.75[/tex]

The 95% CI for the mean surgery time is (133.05, 140.75).

b) Now, we have to find a 99.5% CI for the mean surgery time.

The critical value of z for a 99.5% CI is z=2.81.

The margin of error of the CI can be calculated as:

[tex]E=z\cdot s/\sqrt{n}=2.81*22.6/\sqrt{132}=63.506/11.489=5.53[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=\bar x-E=136.9-5.53=131.37\\\\UL=\bar x+E=136.9+5.53=142.43[/tex]

The 99.5% CI for the mean surgery time is (131.37, 142.43).

c) We can calculate the level of confidence, calculating the z-score for the margin of error in that interval.

We know that the difference between the upper bound and lower bound is 2 times the margin of error:

[tex]UL-LL=2E\\\\E=\dfrac{UL-LL}{2}=\dfrac{139.9-133.9}{2}=\dfrac{6}{2}=3[/tex]

Then, we can write the equation for the margin of error to know the z-value.

[tex]E=z \cdot s/\sqrt{n}\\\\z= E\cdot \sqrt{n}/s=2*\sqrt{132}/22.6=2*11.5/22.6=1.018[/tex]

The confidence level for this interval is then equal to the probability that the absolute value of z is bigger than 1.018:

[tex]P(-|z|<Z<|z|)=P(-1.018<Z<1.018)=0.69[/tex]

The level of confidence of the interval (133.9, 139.9) is 69%.

d) We have to calculate the sample size n to have a margin of error, for a 95% CI, that is equal to 3.

The critical value for a 95% CI is z=1.96.

Then, the sample size can be calculated as:

[tex]E=z\cdot s/\sqrt{n}\\\\n=(\dfrac{z\cdot s}{E})^2=(\dfrac{1.96*22.6}{3})^2=14.77^2=218.015\approx 219[/tex]

The sample size should be 219 surgeries.

e) We have to calculate the sample size n to have a margin of error, for a 99% CI, that is equal to 3.

The critical value for a 99% CI is z=2.576.

Then, the sample size can be calculated as:

[tex]E=z\cdot s/\sqrt{n}\\\\n=(\dfrac{z\cdot s}{E})^2=(\dfrac{1.96*22.6}{3})^2=19.41^2=376.59\approx 377[/tex]

The sample size should be 377 surgeries.

Answer:

a.  [tex]\int\limits^5_0 {f(x)} \, dx[/tex]

Step-by-step explanation:

Since f(x) is the function for the populational density at a certain sidewalk for a 5 mile stretch, a definite integral of that function will yield the total number of people within the integration intervals. If we are interested in the number of people in the whole 5 mile stretch, we must integrate f(x) from x = 0 miles to x = 5 miles:

[tex]\int\limits^5_0 {f(x)} \, dx[/tex]

Therefore, the answer is alternative a.

Answer:

$6,376.92

Step-by-step explanation:

-Let d be the rate of depreciation per year.

-Therefore, the value after n years can be expressed as:

[tex]A=P(1-d)^n\\\\A=Value \ after \ n \ years\\P=Initial \ Value\\d=Rate \ of \ depreciation\\n=Time \ in \ years[/tex]

#We substitute for the years 2009-2013 to solve for d:

[tex]A=P(1-d)^n\\\\19000=41000(1-d)^4\\\\0.475=(1-d)^4\\\\d=1-0.475^{0.25}\\\\d=0.1698[/tex]

#We then use the calculated depreciation rate above to solve for A after 10 yrs:

[tex]A=P(1-d)^n\\\\=41000(1-0.1698)^{10}\\\\=\$6,376.92[/tex]

Hence, the value of the car after 10 yrs is $6,376.92

To find the future value of a car in 2019, we calculate the percentage decrease in value from 2009 to 2013 and apply it for 6 years.

To find the future value of the car in 2019, we need to determine the percentage decrease in value each year. From 2009 to 2013, the car depreciated from $41,000 to $19,000.

This is a decrease of $22,000. To find the percentage decrease, divide this by the initial value: 22,000 / 41,000 = 0.5366 (approximately).

To find the future value in 2019, we need to apply this percentage decrease continuously for 6 years. Multiply the current value by the percentage decrease repeatedly.

= 19,000 * 0.5366 * 0.5366 * 0.5366 * 0.5366 * 0.5366 * 0.5366

= $5,862.54 (approximately).

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In the equation t is the amount of time in seconds. The -2.4 is the distance it travels in 1 second.

-2.4 x 5 seconds= -12 meters.

The anchor dropped 12 (-12) meters in 5 seconds.

Answer:

C) It took 22 seconds for the anchor to reach the water's surface.

E) The equation E = 44 − ST can be used to model this situation.

Step-by-step explanation:

Answer:

1 yard

Step-by-step explanation:

Answer: j=-5 as long as you follow my steps you will also be able to show your work.

Combine multiplied terms into a single fraction

Distribute it then

Multiply all terms by the same value to eliminate fraction denominators.

Answer:

No.

Step-by-step explanation:

This is not a perfect square because the exponent would need to be even, not odd. The 100 is a perfect square. If you were to simplify it, it would be (10x^2) * (10x). In order for it to be truly a perfect square, they both need to be the same.

Answer:

8.2*10^-3

Step-by-step explanation:

Answer:

0.0082

(scientific notation)-step explanation:

Answer:

68/99

Step-by-step explanation:

.68686868686 repeating

Let x= .68686868668repeating

Multiply by 100

100x = 68.686868686repeating

Subtract x = .68686868repeating from this equation

100x = 68.686868686repeating

-x = .68686868repeating

------------------------------------------

99x = 68

Divide each side by 99

99x / 99 = 68/99

x = 68/99

Answer:

68/99 I agree with the other person

Step-by-step explanation:

Answer:

The maximum price that the dealer will sell is $7471 and the minimum is $5513.

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 = 6492, \sigma = 1025[/tex]

He decides to sell boats that will appeal to the middle 66% of the market in terms of price.

50 - (66/2)  = 17th percentile

50 + (66/2) = 83rd percentile

17th percentile

X when Z has a pvalue of 0.17. So X when Z = -0.955.

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

[tex]-0.955 = \frac{X - 6492}{1025}[/tex]

[tex]X - 6492 = -0.955*1025[/tex]

[tex]X = 5513[/tex]

83rd percentile

X when Z has a pvalue of 0.83. So X when Z = 0.955.

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

[tex]0.955 = \frac{X - 6492}{1025}[/tex]

[tex]X - 6492 = 0.955*1025[/tex]

[tex]X = 7471[/tex]

The maximum price that the dealer will sell is $7471 and the minimum is $5513.

The marine sales dealer plans to sell boats between $5467 and $7517 to appeal to the middle 66% of market prices. These figures are computed by adding or subtracting one standard deviation from the average price.

In this scenario, the marine sales dealer wants to price boats that appeal to the middle 66% of the market, which means the dealer wants to avoid the top and bottom 17% of the market (as 100%-66%=34% and 34%/2=17%). Therefore, we need to find the boats' prices that are 1 standard deviation away from the mean, since in a normal distribution, approximately 68% of values lie within 1 standard deviation from the mean (closest to 66%).

The standard deviation given is $1025. Thus, the maximum price of the boats the dealer will sell is the mean price plus one standard deviation:

$6492 + $1025 = $7517

And the minimum price is the mean price minus one standard deviation:

$6492 - $1025 = $5467

Therefore, the dealer will sell boats priced between $5467 and $7517 to appeal to the middle 66% of the market.

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Answer:

The calculated p-value is greater than the significance level at which the test was performed, hence, we fail to reject the null hypothesis & conclude that there is no significant evidence to say that the actual mean waiting time turned out to be significantly less than the 15-minute claim made by the taxpayer advocate.

That is, the true mean waiting time is equal to or greater than the 15-minute claim by the taxpayer advocate.

Step-by-step explanation:

For hypothesis testing, we first clearly state our null and alternative hypothesis.

For hypothesis testing, the first thing to define is the null and alternative hypothesis.

In hypothesis testing, especially one comparing two sets of data, the null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the direction of the test.

The alternative hypothesis usually confirms the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the direction of the test.

For this question, we are to investigate that the actual mean waiting time turned out to be significantly less than the 15-minute claim made by the taxpayer advocate.

The null hypothesis would be that there is no significant evidence to say that the actual mean waiting time turned out to be significantly less than the 15-minute claim made by the taxpayer advocate. That is, the true mean waiting time is equal to or greater than the 15-minute claim by the taxpayer advocate.

The alternative hypothesis is that there is significant evidence to suggest that the actual mean waiting time turned out to be significantly less than the 15-minute claim made by the taxpayer advocate.

This is evidently a one tail hypothesis test (we're investigating only in one direction; less than the claim

Mathematically, the null hypothesis is

H₀: μ ≥ 15

The alternative hypothesis is

Hₐ: μ < 15 minutes

To do this test, we will use the z-distribution because the population standard deviation is known.

So, we compute the z-test statistic

z = (x - μ₀)/σₓ

x = sample mean = 13 minutes

μ₀ = the advocate's claim = 15 minutes

σₓ = standard error of the poll proportion = (σ/√n)

where n = Sample size = 50

σ = population standard deviation = 11 minutes.

σₓ = (σ/√n) = (11/√50) = 1.556

z = (13 - 15) ÷ 1.556 = -1.29

checking the tables for the p-value of this z-statistic

p-value (for z = -1.29, at 0.05 significance level, with a one tailed condition) = 0.098525

The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

So, for this question, significance level = 5% = 0.05

p-value = 0.098525

0.098525 > 0.05

Hence,

p-value > significance level

This means that we fail to reject the null hypothesis & conclude that there is no significant evidence to say that the actual mean waiting time turned out to be significantly less than the 15-minute claim made by the taxpayer advocate.

That is, the true mean waiting time is equal to or greater than the 15-minute claim by the taxpayer advocate.

Hope this Helps!!!

Answer:

Music place has the better buy

Step-by-step explanation:

Figure out how much they sell one for by diving the price by the quantity

CD Express offers 1 CD for $15

Music Place offers 1 CD fof $12.50

Answer:

Music Place

Step-by-step explanation:

Find the unit rates by dividing the price by the number of CDs

CD Express:

price/CDs

$60/4 CDS

60/4=15

$15 per CD

Music Place:

price/CDs

$75/6 CDs

75/6=12.5

$12.50 per CD

Music Place is the Better deal because 12.50 is less than 15

Daniel appears to be paying more for auto insurance compared to the average amount his friends pay based on the data from nine friends. However, as the data only represents a sample, and auto insurance rates can vary widely, additional comparison or the advice of an insurance specialist is recommended.

To determine if Daniel is overpaying for his auto insurance, we can compare his insurance cost to the average price paid by his friends. To do this, we need to calculate the mean (average) of his friends' insurance amounts.

Here are the amounts his friends pay: 564, 578, 478, 507, 621, 564, 489, 612, 538.

Adding these together gets a total of 4951. There are nine friends, so we divide 4951 by 9 to get an average cost of 550.

Since Daniel is paying $600, which is more than the average of $550, it seems he's spending more than his friends for auto insurance.

However, we only have the data from a sample of his friends. The insurance amounts can have a wide range, depending on several factors like age, driving records, the type of vehicle insured, and geographic location. Therefore, it's recommended that Daniel compare his rate with more people or consult with an insurance specialist for a more accurate conclusion.

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Answer:

0.8 gallons

Step-by-step explanation:

4 gallons of juice divided into 5 pitchers equally, 4/5=0.8 per pitcher.

Answer:

a) The large sample size 'n' = 1320.59

b) If the standard deviation and the margin of error are both doubled also the sample size is not changed.

Step-by-step explanation:

Explanation:-

a)

Given data the standard deviation of the population

σ = 70.5

Given the margin error = 5 units

We know that the estimate of the population mean is defined by

that is margin error = [tex]\frac{z_{\alpha } S.D }{\sqrt{n} }[/tex]

            [tex]M.E = \frac{z_{\alpha } S.D }{\sqrt{n} }[/tex]

cross multiplication , we get

[tex]M.E (\sqrt{n} ) = z_{\alpha } S.D[/tex]

[tex]\sqrt{n} = \frac{z_{\alpha } S.D }{M.E }[/tex]

[tex]\sqrt{n} = \frac{2.578 X 70.5}{5} }[/tex]

√n = 36.34

squaring on both sides , we get

n = 1320.59

b) The margin error of the mean

   [tex]\sqrt{n} = \frac{z_{\alpha } S.D }{M.E }[/tex]

the standard deviation and the margin of error are both doubled

  √n = zₓ2σ/2M.E

  √n = 36.34

squaring on both sides , we get

     n = 1320.59

If the standard deviation and the margin of error are both doubled also the sample size is not changed.

Answer:

(a)(C)[tex]D^c \cup A[/tex]

(b)8 elements

Step-by-step explanation:

Ina toss of a red and green dice, given the events:

[tex]D^c[/tex]=The numbers do add up to 11.

Therefore, the event: Either the numbers add to 11 or the red die shows a 1 is written as: [tex]D^c \cup A[/tex]

(b)

Sample Space of A={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)}

Sample Space of [tex]D^c[/tex]={(5,6),(6,5)}

[tex]D^c \cup A[/tex]={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(5,6),(6,5)}

[tex]D^c \cup A[/tex] contains 8 elements

Final answer:

The event "Either the numbers add to 11 or the red die shows a 1" is represented by the symbol D' ∪ A. This event contains 7 elements in the context of rolling two six-sided dice.

Explanation:

To express the event "Either the numbers add to 11 or the red die shows a 1" in symbolic form, we consider the symbols for the events defined in the question. Event A denotes the red die shows 1, and D denotes the event the numbers do not add to 11. The complement of D, represented as D', would then denote the event that the numbers do add to 11. The symbol ‘∪’ denotes the union of sets, meaning 'or' in the context of probability. Therefore, the symbolic form for the given event is D' ∪ A.

Regarding how many elements this event contains, we must consider the sample space when rolling two six-sided dice. There are a total of 36 different outcomes (6 possible outcomes for the first die multiplied by 6 outcomes of the second die). Event A (the red die shows a 1) has 6 elements (1 can be paired with any of the 6 outcomes on the green die). Event D' (the numbers add up to 11) can happen in two ways: (5,6) or (6,5), one for each die, making it 2 elements. Therefore, D' ∪ A will consist of all unique elements from both events without double-counting any pair. So, we combine 6 outcomes from A and 2 from D', but we need to ensure to not count the outcome (1,6) twice, hence we have 6 (from A) + 2 (from D') - 1 (overlap of (1,6)) = 7 elements in event D' ∪ A.

Answer:

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

Step-by-step explanation:

From the diagram, our goal is to minimize the length of Rope AC passing through B.

First, we determine the length of the rope AC.

AC=AB+BC

In the first triangle,

[tex]|AB|^2=16^2+x^2\\|AB|=\sqrt{16^2+x^2}[/tex]

Similarly, in the second triangle,

[tex]|BC|^2=24^2+(30-x)^2\\|BC|=\sqrt{x^2-60x+1476}[/tex]

Length of the Rope, AC

[tex]L=\sqrt{16^2+x^2}+\sqrt{x^2-60x+1476}[/tex]

First, to minimize L,we find its derivative.

[tex]L'=\dfrac{x\sqrt{x^2-60x+1476}+(x-30)\sqrt{16^2+x^2}}{(\sqrt{16^2+x^2})(\sqrt{x^2-60x+1476})}[/tex]

Setting the derivative to zero

[tex]x\sqrt{x^2-60x+1476}+(x-30)\sqrt{16^2+x^2}=0\\-x\sqrt{x^2-60x+1476}=(x-30)\sqrt{16^2+x^2}\\$Square both sides\\x^2(x^2-60x+1476)=(x-30)^2(16^2+x^2)\\x^4-60x^3+1476x^2=x^4-60x^3+1156x^2-15360x+230400\\1476x^2=1156x^2-15360x+230400\\1476x^2-1156x^2+15360x-23040=0\\320x^2+15360x-23040=0\\x=1.46,-49.46[/tex]

The rope should be placed at 1.46 from the 16 ft pole to minimize the length.

The least amount of the rope is the smallest length that can be gotten from the pole

The rope should be placed at 12 feet from the 16 ft pole to use the least amount of rope.

The heights are given as:

[tex]\mathbf{h_1 = 16}[/tex]

[tex]\mathbf{h_2 = 24}[/tex]

The distance is given as:

[tex]\mathbf{d = 30}[/tex]

See attachment for the illustrating diagram

Considering the two right-angled triangles on the diagram, we have the following equations, using Pythagoras theorem

[tex]\mathbf{L_1 = \sqrt{x^2 + 16^2}}[/tex]

[tex]\mathbf{L_2 = \sqrt{(30 - x)^2 + 24^2}}[/tex]

Expand

[tex]\mathbf{L_1 = \sqrt{x^2 + 256}}[/tex]

[tex]\mathbf{L_2 = \sqrt{900 - 60x +x^2 + 576}}[/tex]

[tex]\mathbf{L_2 = \sqrt{1476- 60x +x^2 }}[/tex]

The length (L) of the pole is:

[tex]\mathbf{L = L_1 + L_2}[/tex]

So, we have:

[tex]\mathbf{L = \sqrt{x^2 + 256} + \sqrt{1476 - 60x + x^2}}[/tex]

Differentiate

[tex]\mathbf{L' = \frac{x}{\sqrt{x^2 + 256}} + \frac{x - 30}{\sqrt{1476 - 60x + x^2}}}[/tex]

Set to 0

[tex]\mathbf{\frac{x}{\sqrt{x^2 + 256}} + \frac{x - 30}{\sqrt{1476 - 60x + x^2}} = 0}[/tex]

Take LCM

[tex]\mathbf{\frac{x\sqrt{1476 - 60x + x^2} +(x - 30)\sqrt{x^2 + 256}}{\sqrt{x^2 + 256} \times \sqrt{1476 - 60x + x^2}} = 0}[/tex]

Simplify

[tex]\mathbf{x\sqrt{1476 - 60x + x^2} +(x - 30)\sqrt{x^2 + 256} = 0}[/tex]

Rewrite as:

[tex]\mathbf{x\sqrt{1476 - 60x + x^2} =-(x - 30)\sqrt{x^2 + 256} }[/tex]

Square both sides

[tex]\mathbf{x^2(1476 - 60x + x^2) =(x^2 - 60x + 900)(x^2 + 256) }[/tex]

Expand

[tex]\mathbf{1476x^2 - 60x^3 + x^4 =x^4 - 60x^3 + 900x^2 + 256x^2 - 15360x + 230400}[/tex]

Simplify

[tex]\mathbf{1476x^2 - 60x^3 + x^4 =x^4 - 60x^3 + 1156x^2 - 15360x + 230400}[/tex]

Evaluate like terms

[tex]\mathbf{1476x^2 = 1156x^2 - 15360x + 230400}[/tex]

Rewrite as:

[tex]\mathbf{1476x^2 - 1156x^2 + 15360x - 230400 = 0}[/tex]

[tex]\mathbf{320x^2 + 15360x - 230400 = 0}[/tex]

Divide through by 320

[tex]\mathbf{x^2 + 48x - 720 = 0}[/tex]

Using a calculator, we have:

[tex]\mathbf{x = \{12,-60\}}[/tex]

The value of x cannot be negative.

So, we have:

[tex]\mathbf{x = 12}[/tex]

Hence, the rope should be placed at 12 feet from the 16 ft pole

Read more about minimizing lengths at:

https://brainly.com/question/15174196

Given:

The given figure ABCD is a trapezoid.

The measure of ∠A is (2x + 32).

The measure of ∠B is 112°

The measure of ∠C is y.

The measure of ∠D is 46°

We need to determine the value of x and y.

Value of x:

We know the property that the adjacent angles in a trapezoid are supplementary.

Thus, we have;

[tex]\angle A+\angle B=180[/tex]

Substituting the values, we get;

[tex]2x+32+112=180[/tex]

       [tex]2x+144=180[/tex]

                 [tex]2x=36[/tex]

                   [tex]x=18[/tex]

Thus, the value of x is 18.

Value of y:

The value of y can be determined using the property that the adjacent angles of a trapezoid are supplementary.

Thus, we have;

[tex]\angle C+\angle D=180[/tex]

    [tex]y+46=180[/tex]

            [tex]y=134[/tex]

Thus, the value of y is 134.

Hence, Option c is the correct answer.

Answer:

See explanation

Step-by-step explanation:

Solution:-

The random variable, Y be the temperature of chemical reaction in degree fahrenheit be a linear expression of a random variable X : The  temperature in at which a certain chemical reaction takes place.

                             Y = 1.8*X + 32

- The median of the random variate "X" is given to be equal to "η". We can mathematically express it as:

                             P ( X ≤ η ) = 0.5

- Then the median of "Y" distribution can be expressed with the help of the relation given:

                             P ( Y ≤ 1.8*η + 32 )

- The left hand side of the inequality can be replaced by the linear relation:

                             P ( 1.8*X + 32 ≤ 1.8*η + 32 )

                             P ( 1.8*X ≤ 1.8*η )   ..... Cancel "1.8" on both sides.

                            P ( X ≤ η ) = 0.5 ...... Proven

Hence,

- Through conjecture we proved that: (1.8*η + 32) has to be the median of distribution "Y".

b)

- Recall that the definition of proportion (p) of distribution that lie within the 90th percentile. It can be mathematically expressed as the probability of random variate "X" at 90th percentile :

                             P ( X ≤ p_.9 ) = 0.9 ..... 90th percentile

- Now use the conjecture given as a linear expression random variate "Y",

          P ( Y ≤ 1.8*p_0.9 + 32 ) = P ( 1.8*X + 32 ≤ 1.8*p_0.9 + 32 )

                                                 = P ( 1.8*X ≤ 1.8*p_0.9 )

                                                 = P ( X  ≤ p_0.9 )

                                                 = 0.9

- So from conjecture we saw that the 90th percentile of "X" distribution is also the 90th percentile of "Y" distribution.

c)

- The more general relation between two random variate "Y" and "X" is given:

                            Y = aX + b

Where, a : is either a positive or negative constant.

- Denote, (np) as the 100th percentile of the X distribution, so the corresponding 100th percentile of the Y distribution would be : (a*np + b).

- When a is positive,

                   P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )

                                                 = P ( a*X ≤ a*p_% )

                                                 = P ( X  ≤ p_% )

                                                 = np_%        

- When a is negative,

                   P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )

                                                 = P ( a*X ≤ a*p_% )

                                                 = P ( X  ≥ p_% )

                                                 = 1 - np_%        

In a temperature conversion equation Y = 1.8X + 32, medians and percentiles of Y are related to the corresponding values of X through the equation itself. For an arbitrary linear transformation Y = aX + b, percentiles of Y and X are related as aX + b, with ordering depending on the sign of a.

Lets start by talking about the relationship between X and Y. In the context of the temperature conversion between Celsius (X) and Fahrenheit (Y), Y equals to 1.8 times X plus 32.

1. If the median of X is M, substituting M into the equation Y = 1.8X + 32 gives the median of Y as 1.8M + 32, since the transformation is linear.

2. The 90th percentile of the Y distribution will relate to the 90th percentile of X distribution in a similar fashion. If we denote the 90th percentile of X as P, then the 90th percentile of Y will be 1.8P + 32.

3. For a general linear transformation Y = aX + b, where a is non-zero, any percentile of Y is related to the corresponding percentile of X as aX + b. If a is positive, the transformation will preserve the ordering of percentiles (e.g., higher values of X correspond to higher values of Y). If a is negative, it will reverse the ordering of percentiles (e.g., higher values of X will correspond to lower values of Y).

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Answer:

160 plus 35 = 185

Step-by-step explanation:

8×20

hopefully this helps you

Answer:

  classification is based on the measure of the largest angle

Step-by-step explanation:

Every triangle will have angle measures that add up to 180°. The classification as to acute, right, or obtuse is based on the largest angle.

__

If the largest angle is less than 90°, the triangle is acute.

If the largest angle is equal to 90°, the triangle is right.

If the largest angle is greater than 90°, the triangle is obtuse.

__

Because you know the angle sum is always 180°, you can generally figure out what kind of triangle it is from the sum of two of the angles. If both are less than 90° and their sum is more than 90°, then the triangle will be acute, for example.

Answer:

6/5

Step-by-step explanation:

[tex]\frac{3}{5} * 2 = \frac{6}{5}[/tex]

please give me brainliest

Answer:

1.2 [Decimal Form]

[tex]\frac{6}{5}[/tex] [Exact Form]

[tex]1\frac{1}{5}[/tex] [Mixed Number Form]