How do I do number 12

Answer:

(a) 25%

(b) 50%

(c) 18.75%

Step-by-step explanation:

Since there are two possible outcomes for each coin the number of total possible permutations is given by:

[tex]N=2*2*2*2=16[/tex]

(a) The first and last flips come up heads.

4 possible outcomes meet this requirement:

{HHHH}, {HHTH}, {HTHH},{HTTH}

P=4/16 = 25%

(b) There are at least two consecutive flips that come up heads.

8 possible outcomes meet this requirement:

{HHHH}, {HHHT},{HHTH} {HTHH},{THHH},{TTHH},{HHTT}, {THHT}

P=8/16 = 50%

(c) The first flip comes up tails and there are at least two consecutive flips that come up heads.

3 possible outcomes meet this requirement:

{THHH},{TTHH}, {THHT}

P=3/16 = 18.75%

Answer:

the anwser is c

Step-by-step explanation:

Answer:

The correct answer is A

Step-by-step explanation:

A) CA because corresponding parts of congruent triangles are congruent.

The function S(t) that represents the number of computers sold each month is S(t) = -18t^(5/3) + 2,440.

The company will continue to manufacture this computer for 5.656 months.

We have,

To find the function S(t), which represents the number of computers sold each month, we need to integrate the rate of change function S'(t).

Given:

S'(t) = -30t^(2/3)

Integrating S'(t) with respect to t will give us S(t):

∫S'(t) dt = ∫-30t^(2/3) dt

Using the power rule for integration:

S(t) = -30 * (3/5) * t^(5/3) + C

Simplifying:

S(t) = -18t^(5/3) + C

Now, we need to find the value of C, which represents the constant of integration.

Given that monthly sales at t = 0 (S(0)) are 2,440 computers, we can substitute this value into the equation:

S(0) = -18(0)^(5/3) + C

2,440 = 0 + C

C = 2,440

Substituting the value of C back into the equation, we get:

S(t) = -18t^(5/3) + 2,440

To find how long the company will continue to manufacture this computer, we need to solve for t when S(t) = 1,000:

-18t^(5/3) + 2,440 = 1,000

Simplifying the equation and rearranging:

-18t^(5/3) = -1,440

Dividing by -18:

t^(5/3) = 80

Taking the 3rd root of both sides to isolate t:

t = (80)^(3/5)

t  = 5.656

Therefore,

The function S(t) that represents the number of computers sold each month is S(t) = -18t^(5/3) + 2,440.

The company will continue to manufacture this computer for 5.656 months.

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The function S(t) is obtained via integration of S'(t) with respect to t. We find that S(t) = -45t^(5/3) + 2440. When the computer sales fall to a 1000, it takes approximately 5.71 months, given the decline rate.

To find the function S(t) from its derivative, S'(t), we need to perform an integral of S'(t) which is -30t^(2/3). By performing the integral:

∫ S'(t)dt = ∫ -30t^(2/3) dt = -45t^(5/3) + C

Where C is the constant of integration.

To find C, we know from the question that at t=0, S(t) = 2440. So:
2440 = -45*(0)^(5/3) + C
2440 = C

Now we have the function: S(t) = -45t^(5/3) + 2440

We also know from the question that the company will stop manufacturing the computer when the monthly sales reach 1000 computers. We set our function equal to 1000 to find the time t:
1000 = -45t^(5/3) + 2440

We can solve for t and we get: t = ((2440-1000)/45)^(3/5), which is approximately 5.71 months.

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Answer:

Step-by-step explanation:

a. The pidgeons/objects here are the cards and the pigeonholes/boxes here are the suits. We are trying to draw cards/pidgeon that have the same suits/pidgeonholes. Since there are 4 different suits, we need to draw at least 5 cards to guarantee 2 cards of the same suit.

b. Let there be 3 holes, which represent 3 pairs of number with sum equals to 7. They are:

- {1,6} as 1 + 6 = 7

- {2,5} as 2 + 5 = 7

- {3,4} as 3 + 4 = 7

You can see that if we pick randomly 4 numbers, we will always end up with 2 numbers in the same hole, which means there will be at least a pair and add up to 7.

Answer: The study used a total of N 36 participants.

Step-by-step explanation:

There are 3 treatment conditions were compared in the study and the total number of participants in the study is 39.

Degree of freedom, in mathematics, any of the number of independent quantities necessary to express the values of all the variable properties of a system.

Given the ratio F(2, 36)

F takes in degree of freedom values; degree of freedom of groups ; degree of freedom of error

Hence,

df group or treatment = 2

df error = 36

df treatment = k - 1

k = Number of groups

2 = k - 1

k = 3

Number of treatment condition = 3

df error = N-k

N = total number of observations

36 = N - 3

N = 39

Hence, there are 3 treatment conditions were compared in the study and the total number of participants in the study is 39.

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Answer:

[tex]z=-1.12[/tex]  

[tex]p_v =2*P(z<-1.12)=0.263[/tex]  

So with the p value obtained and using the significance level given [tex]\alpha=0.01[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we don't have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the population proportion of extroverts among college student government leaders is not different from 0.82.  

Step-by-step explanation:

1) Data given and notation

n=74 represent the random sample taken

X=57 represent the number of people found extroverts

[tex]\hat p=\frac{57}{74}=0.770[/tex] estimated proportion of people found  extroverts.

[tex]p_o=0.82[/tex] is the value that we want to test

[tex]\alpha=0.01[/tex] represent the significance level

Confidence=99% or 0.959

z would represent the statistic (variable of interest)

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

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test if population proportion of extroverts among college student government leaders is different (either way) from 82%, the system of hypothesis would be on this case:  

Null hypothesis:[tex]p= 0.82[/tex]  

Alternative hypothesis:[tex]p \neq 0.82[/tex]  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.770 -0.82}{\sqrt{\frac{0.82(1-0.82)}{74}}}=-1.12[/tex]  

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level provided [tex]\alpha=0.01[/tex]. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z<-1.12)=0.263[/tex]  

So with the p value obtained and using the significance level given [tex]\alpha=0.01[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we don't have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the population proportion of extroverts among college student government leaders is not different from 0.82.  

The P-value of 0.0072, we can conclude that there is statistically significant evidence to support the claim that the proportion of wives married less than two years who planned to have children is significantly higher than the proportion of wives married five years.

To determine whether the proportion of wives married less than two years who planned to have children is significantly higher than the proportion of wives married five years, we can conduct a hypothesis test, following these steps:

1. Define the null and alternative hypotheses:

Null hypothesis (H0): The proportion of wives planning to have children is the same in both groups, regardless of the length of marriage.

Alternative hypothesis (H1): The proportion of wives planning to have children is higher in the group married less than two years compared to the group married five years.

2. Calculate the proportions and standard errors:

Proportion of wives planning children (<2 years): 240/300 = 0.8

Proportion of wives planning children (5 years): 288/400 = 0.72

Pooled standard error: sqrt((0.8*(1-0.8))/300 + (0.72*(1-0.72))/400) = 0.0298

3. Calculate the test statistic:

z = (0.8 - 0.72) / 0.0298 = 2.68

4. Find the P-value:

Using a z-score table or statistical software, look up the P-value associated with z = 2.68. This gives us a P-value of approximately 0.0072.

5. Make a decision:

Typically, we use a significance level of alpha (α) = 0.05. Since the P-value (0.0072) is less than alpha, we reject the null hypothesis.

This means there is statistically significant evidence to conclude that the proportion of wives planning to have children is higher in the group married less than two years compared to the group married five years.

Therefore, based on the P-value of 0.0072, we can conclude that there is statistically significant evidence to support the claim that the proportion of wives married less than two years who planned to have children is significantly higher than the proportion of wives married five years.

Answer:

We conclude that the storm is not increasing above the severe rating.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 16.4 feet

Sample mean, [tex]\bar{x}[/tex] = 17.3 feet

Sample size, n = 34

Alpha, α = 0.01

Population standard deviation, σ = 3.5 feet

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 16.4\text{ feet}\\H_A: \mu > 16.4\text{ feet}[/tex]

We use One-tailed(right) z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{17.3 - 16.4}{\frac{3.5}{\sqrt{34}} } = 1.49[/tex]

Now, [tex]z_{critical} \text{ at 0.01 level of significance } = 2.33[/tex]

Since,  

[tex]z_{stat} < z_{critical}[/tex]

We fail to reject the null hypothesis and accept null hypothesis. Thus, the  the storm is not increasing above the severe rating.

The level of significance is 0.01 and the sample test statistic is 1.30.

The level of significance is given as ? = 0.01.

To test if the storm is increasing above the severe rating, we can calculate the z-score.

The formula for the z-score is: z = (x - ?) / (σ / √n), where x is the sample mean, ? is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, x = 17.3 feet, ? = 16.4 feet, σ = 3.5 feet, and n = 34 waves.

Plugging in these values into the formula, we get: z = (17.3 - 16.4) / (3.5 / √34) ≈ 1.30 (rounded to two decimal places).

The sample test statistic is 1.30.

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The null hypothesis is that the population mean is equal to or less than 76, while the alternative hypothesis is that it is greater than 76. The standardized test statistic is 2.089, and the P-value is approximately 0.022. Therefore, we reject the null hypothesis and conclude that there is enough evidence to support the claim.

The null hypothesis, denoted as H0, states that the population mean (mu) is equal to or less than 76. The alternative hypothesis, denoted as H1, states that the population mean (mu) is greater than 76. To test the claim, we can perform a one-sample t-test.

The standardized test statistic, also known as the t-value, can be calculated using the formula t = (x - mu) / (s / sqrt(n)), where x is the sample mean, mu is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Plugging in the values x = 77.5, mu = 76, s = 3.3, and n = 29 into the formula gives us a t-value of 2.089 (rounded to two decimal places).

The P-value of the test statistic can be determined by comparing the t-value to the critical value of the t-distribution with (n - 1) degrees of freedom at the given level of significance (alpha). Since the alternative hypothesis is one-sided (mu > 76), we need to find the right-tail area of the t-distribution. Consulting a t-table or using statistical software, we find that the P-value is approximately 0.022 (rounded to three decimal places).

To make a decision, we compare the P-value to the significance level (alpha). If the P-value is less than alpha, we reject the null hypothesis. In this case, the P-value is 0.022, which is less than alpha = 0.01. Therefore, we reject the null hypothesis and conclude that there is enough evidence to support the claim that the population mean (mu) is greater than 76.

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The null and alternative hypotheses can be set, and the t-value can be calculated using the provided sample statistics. The P-value and the decision to reject or fail to reject the null hypothesis cannot be determined without further calculations.

The null and alternative hypotheses can be defined as follows:

The standardized test statistic (t-value) can be calculated using the formula:

t = (x - mu) / (s / sqrt(n))

Using the given sample statistics:

t = (77.5 - 76) / (3.3 / sqrt(29)) = 1.69 (rounded to two decimal places)

The P-value for the test statistic can be found using a t-distribution table or a statistical software. The P-value represents the probability of obtaining a t-value as extreme as the calculated one, assuming the null hypothesis is true. Since the P-value is not provided in the question, it cannot be determined without further calculations.

To decide whether to reject or fail to reject the null hypothesis, compare the P-value to the significance level (alpha). If the P-value is less than alpha, reject the null hypothesis. If the P-value is greater than or equal to alpha, fail to reject the null hypothesis.

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Answer: a) 2002, b) 840, c) 10010.

Step-by-step explanation:

Since we have given that

Number of students = 14

Number of students senior = 8

Number of students junior = 6

(a) How many ways are there to select a committee of 5 students?

Here, n = 14

r = 5

We will use "Combination" for choosing 5 students from 14 students.

[tex]^{14}C_5=2002[/tex]

(b) How many ways are there to select a committee with 3 seniors and 2 juniors?

Again we will use "combination":

[tex]^8C_3\times ^6C_2\\\\=56\times 15\\\\=840[/tex]

(c) Suppose the committee must have five students (either juniors or seniors) and that one of the five must be selected as chair. How many ways are there to make the selection?

So, number of ways would be

[tex]^{14}C_5\times ^5C_1\\\\=2002\times 5\\\\=10010[/tex]

Hence, a) 2002, b) 840, c) 10010.

There are 2002 ways to select a committee of any 5 students, 840 ways to select a committee specifically with 3 seniors and 2 juniors, and 10010 ways to select a committee and designate one as chair.

The subject of this question is combinatorics, which is part of mathematics. It relates to counting, specifically concerning combinations and permutations. Combinations represent the number of ways a subset of a larger set can be selected, while permutations are the number of ways to arrange a subset.

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Answer:

Confidence Interval: (0.44,1.00)

Step-by-step explanation:

We are given the following data set:

0.60, 0.74, 0.09, 0.89, 1.31, 0.51, 0.94

Formula:

[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]  

where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  

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

[tex]Mean =\displaystyle\frac{:5.08}{7} = 0.725[/tex]

Sum of squares of differences = 0.8809

[tex]S.D = \sqrt{\frac{0.8809}{6}} = 0.383[/tex]

90% Confidence interval:  

[tex]\bar{x} \pm t_{critical}\displaystyle\frac{s}{\sqrt{n}}[/tex]  

Putting the values, we get,  

[tex]t_{critical}\text{ at degree of freedom 6 and}~\alpha_{0.10} = \pm 1.943[/tex]  

[tex]0.725 \pm 1.943(\frac{0.383}{\sqrt{7}} ) =0.725 \pm 0.2812 = (0.44,1.00)[/tex]

No, it does not appear that there is too much mercury in tuna​ sushi.

Answer: he drove 156 miles

Step-by-step explanation:

Aldo rented a truck for one day. There was a base fee of $20.99. This means that the fee of $20.99 is constant.

There was an additional charge of 86 cents for each mile driven.

Converting 86 cents to dollar, it becomes 86/100 = $0.86

Aldo had to pay $155.15

when he returned the truck. This means that the sum of the base fee and the additional fee is $155.15. The additional fee paid would be the total amount paid minus the base fee. It becomes

155.15 - 20.99 = 134.16

Since there is an additional charge of 0.86 for each mile driven, number of miles for which he was charged $134.16 will be total additional fee divided by cost per mile. It becomes

134.16/0.86 = 156 miles

Answer:

The proof is wrong. The part that says 'take an element b ∈ A such that (a, b) ∈ R' is assuming that such an element b exists, and that may not be the case.  We can give the 'theorem' validation by adding the hypothesis ' each element of A is related to any other element of A', otherwise the 'theorem' is false. An example for that is the following relation on the natural numbers:

'a R b if both numbers are even'

This relation if crearly symmetric and transitive, but it is not reflexive, an odd number does not relate with itself. In fact, odd numbers dont relate with any number.

Final answer:

The error in the given "proof" lies in the assumption that an element b ∈ A exists such that (a, b) ∈ R, which is not necessarily true for all a ∈ A. Symmetry and transitivity of a relation R do not inherently guarantee its reflexivity.

Explanation:

The error in the "proof" that a symmetric and transitive relation R on a set A is also reflexive lies in the initial assumption. The proof assumes the existence of an element b ∈ A such that (a, b) ∈ R for an arbitrary a ∈ A, but this is not guaranteed. Reflexivity requires that (a, a) ∈ R for all a ∈ A, independently of the existence of any such b. The proof fails because it assumes a specific relationship exists (between a and b) to demonstrate reflexivity, which is a general property that must hold without needing to reference any other elements.

The properties of reflexivity, symmetry, and transitivity define an equivalence relation. Symmetry and transitivity alone do not imply reflexivity. Therefore, the statement that a symmetric and transitive relation is reflexive is incorrect without additional assumptions about R.

Answer:

More than 50% would germinate

Step-by-step explanation:

Given that in the production of a plant, a treatment is being evaluated to germinate seed. From a total of 60 seed it was observed that 37 of them germinated

Let us check whether more than 50% will germinate using hypothesis test

[tex]H_0: p = 0.50\\H_a: p>0.50\\[/tex]

(right tailed test)

Sample proportion p =[tex]\frac{37}{60} =0.617\\q = 0.383\\Std error = \sqrt{\frac{pq}{n} } =0.0628[/tex]

p difference = 0.117

Test statistic Z = p difference/std error = 1.864

p value =0.0312

Since p value <0.05 our significance level of 5% we reject null hypothesis

It is  possible to claim that most of the seed will germinate (i.e. more than 50%)

Answer:

Step-by-step explanation:

Answer:

P(Type ll error) = 0.2327

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 15 minutes

Sample size, n = 10

Alpha, α = 0.05

Population standard deviation, σ = 4 minutes

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 18\\H_A: \mu > 18[/tex]

We use One-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

[tex]z_{critical} \text{ at 0.05 level of significance } = 1.645[/tex]

Putting the values, we get,

[tex]z_{stat} = \displaystyle\frac{\bar{x}- 15}{\frac{4}{\sqrt{10}} } > 1.645\\\\\bar{x} -1 5 > 1.645\times \frac{4}{\sqrt{10}}\\\\\bar{x} -15 > 2.08\\\bar{x} = 17.08[/tex]

Type ll error is the error of accepting the null hypothesis when it is not true.

P(Type ll error)

[tex]P(\bar{x}<17.07 \text{ when mean is 18})\\\\= P(z < \frac{\bar{x}-18}{\frac{4}{\sqrt{10}}})\\\\= P(z < \frac{17.08-18}{\frac{4}{\sqrt{10}}})\\\\= P(z<-0.7273)[/tex]

Calculating value from the z-table we have,

[tex]P(z<-0.7273) = 0.2327[/tex]

Thus,

P(Type ll error) = 0.2327

Answer:

(a) [tex](3, -\frac{4\pi}{3})[/tex]

(b) [tex](-3, \frac{5\pi}{3})[/tex]

(c)[tex](3, \frac{8\pi}{3})[/tex]

Step-by-step explanation:

All polar coordinates of point (r, θ ) are

[tex](r,\theta+2n\pi)[/tex] and [tex](-r,\theta+(2n+1)\pi)[/tex]

where, θ is in radian and n is an integer.

The given point is [tex](3, \frac{2\pi}{3})[/tex]. So, all polar coordinates of point are

[tex](3, \frac{2\pi}{3}+2n\pi)[/tex] and [tex](-3, \frac{2\pi}{3}+(2n+1)\pi)[/tex]

(a) [tex]r>0,-2\pi\leq \theta <0[/tex]

Substitute n=-1 in [tex](3, \frac{2\pi}{3}+2n\pi)[/tex], to find the point for which [tex]r>0,-2\pi\leq \theta <0[/tex].

[tex](3, \frac{2\pi}{3}+2(-1)\pi)[/tex]

[tex](3, -\frac{4\pi}{3})[/tex]

Therefore, the required point is [tex](3, -\frac{4\pi}{3})[/tex].

(b) [tex]r<0,0\leq \theta <2\pi[/tex]

Substitute n=0 in [tex](-3, \frac{2\pi}{3}+(2n+1)\pi)[/tex], to find the point for which [tex]r>0,-2\pi\leq \theta <0[/tex].

[tex](-3, \frac{2\pi}{3}+(2(0)+1)\pi)[/tex]

[tex](-3, \frac{2\pi}{3}+\pi)[/tex]

[tex](-3, \frac{5\pi}{3})[/tex]

Therefore, the required point is [tex](-3, \frac{5\pi}{3})[/tex].

(c) [tex]r>0,2\pi \leq \theta <4\pi[/tex]

Substitute n=1 in [tex](3, \frac{2\pi}{3}+2n\pi)[/tex], to find the point for which [tex]r>0,2\pi \leq \theta <4\pi[/tex].

[tex](3, \frac{2\pi}{3}+2(1)\pi)[/tex]

[tex](3, \frac{2\pi}{3}+2\pi)[/tex]

[tex](3, \frac{8\pi}{3})[/tex]

Therefore, the required point is [tex](3, \frac{8\pi}{3})[/tex].

Answer:

$ 77.88

Step-by-step explanation:

1 Gallon = 4 Quarts

3 Gallons = 3*4=12 Quarts

thus (19.47/3)*12 =$ 77.88 (we know 3 quarts cost $19.47, so divide by 3 and multiply by 12 gives the result.

Answer:

Hence Proved △ SPT ≅ △ UTQ

Step-by-step explanation:

Given: S, T, and U are the midpoints of Segment RP , segment PQ , and segment QR respectively of Δ PQR.

To prove: △ SPT ≅ △ UTQ

Proof:

∵ T is is the midpoint of PQ.

Hence PT = PQ    ⇒equation 1

Now,Midpoint theorem is given below;

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

By, Midpoint theorem;

TS║QR

Also, [tex]TS = \frac{1}{2} QR[/tex]

Hence, TS = QU (U is the midpoint QR) ⇒ equation 2

Also by Midpoint theorem;

TU║PR

Also, [tex]TU = \frac{1}{2} PR[/tex]

Hence, TU = PS (S is the midpoint QR) ⇒ equation 3

Now in △SPT and △UTQ.

PT = PQ (from equation 1)

TS = QU (from equation 2)

PS = TU (from equation 3)

By S.S.S Congruence Property,

△ SPT ≅ △ UTQ ...... Hence Proved

To prove the congruency of the triangles, we use the Midpoint Theorem and the Side-Side-Side (SSS) criterion. According to the Midpoint Theorem, the line segments are parallel and half the length of the third side of the triangle. Since the triangles share a side and the other sides are proportional, the SSS criterion allows us to conclude that the triangles are congruent.

To prove that the triangles △SPT and △UTQ are congruent, we need to first understand that S, T, and U are midpoints. According to the Midpoint Theorem, a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So, consider the segments ST and TU, which are connecting the midpoints of the sides of △PQR.

By the Midpoint Theorem, we know ST is parallel to UQ and also ST = 0.5*UQ. Similarly, TU is parallel to SP and TU = 0.5*SP.

Moreover, since T is the midpoint of PQ, we know PT = QT. Therefore, △SPT and △UTQ share  a side (PT), and their other corresponding sides are proportional due to the Midpoint Theorem. Consequently, by Side-Side-Side (SSS) criterion, we can conclude that △SPT is congruent to △UTQ.

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Answer:

13,8%

Step-by-step explanation:

There are six employees and six cheks, so there are 36 (6x6) possible combinations so if we need to measure the probability that five of them receive the exact check  is only one for each one of them over the 36 possibilities, so 1/36 for one plus 1/36 the second and so on.  5/36 = 13,8%.

Answer:

0

Step-by-step explanation:

If 5 of them are correct, then the 6th one will also be correct. Thus, it is impossible to have exactly 5 correct, therefore 0

Answer:

Step-by-step explanation:

Hello!

You have the difference between two sample means. Remember the sample means are variables with certain distribution, let's say, in this case, both sample means have a normal distribution. X[bar] = sample mean

X₁[bar]~N(μ₁;σ₁²/n₁)

X₂[bar]~N(μ₂;σ₂²/n₂)

Then the difference between these two variables results in a third variable that will also have a normal distribution:

X₁[bar]-X₂[bar]~N(μ₁-μ₂;σ₁²/n₁+σ₂²/n₂)

These are some of the properties of the normal distribution

Centered in μ

Symmetrical

Bell-shaped

[μ - σ; μ + σ]= 68% of the distribution

[μ - 2σ; μ+ 2σ]= 95% of the distribution

[μ - 3σ; μ+ 3σ]= 99.7% of the distribution

Check attachment.

Taking these properties into account, if you where to draw the results of the 100 trials, where μ₁-μ₂=0 would be its center and the standard deviation of the difference is 1.78.

68% of the information will be between (μ₁-μ₂) ± [(σ₁/√n₁)+(σ₂/√n₂)], this is 0 ± 1.78

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Answer:

There is a 26.39% probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 17.

Step-by-step explanation:

We have to find the probability of the three separate events, and then multiply them.

coin comes up heads

There is a [tex]\frac{1}{2}[/tex] probability that the coin comes up heads.

the die comes up less than 4

The are 15 sides on the die, from 0 to 14.

The values that satisfy us are 0,1,2,3.

So the probability is [tex]\frac{4}{15}[/tex].

the roulette wheel comes up with a number greater than 17

There roulette wheel can come from 1 to 36. There are 19 values greater than 17. So this probability is [tex]\frac{19}{36}[/tex]

What is the probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 17 ?

[tex]P = \frac{1}{2}*\frac{4}{15}*\frac{19}{36} = 0.2639[/tex]

There is a 26.39% probability that the coin comes up heads and the die comes up less than 4 and the roulette wheel comes up with a number greater than 17.

The combined probability of the coin landing on heads, the die rolling less than 4, and the roulette wheel landing on a number greater than 17 is found by multiplying the individual probabilities of each event.

The question involves calculating the combined probability of independent events in a probability experiment: flipping a coin, rolling a 15-sided die, and spinning a roulette wheel. Each event's probability is calculated separately and then multiplied together to find the probability of all events occurring simultaneously.

The probability the coin comes up heads is 0.5, because a fair coin has two sides and the event has a theoretical probability of 1 in 2. When rolling a 15-sided die, the probability it comes up less than 4 (that is, obtaining a 1, 2, or 3) is 3 out of 15, as there are 3 favorable outcomes out of 15 possible outcomes. And for the roulette wheel, assuming it has numbers from 0 to 36, the probability of spinning a number greater than 17 is 19 out of 38, since there are 19 numbers from 18 to 36 (excluding 0 and 00 which might also be present).

To find the combined probability, you would multiply these probabilities together:

Combined Probability = Probability(Heads) * Probability(Die < 4) * Probability(Roulette > 17)
= 0.5 * (3/15) * (19/38)

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Answer:

D. Because we would be interested in any difference between running on hard and soft surfaces, we should use a two-sided hypothesis test

Step-by-step explanation:

Hello!

When planning what kind of hypothesis to use, you have to take into account any other studies that were made about that topic so that you can decide the orientation you will give them.

Normally, when there is no other information available to give an orientation to your experiment, the first step to take is to make a two-tailed test, for example, μ₁=μ₂ vs. μ₁≠μ₂, this way you can test whether there is any difference between the two stands. Only after having experimental evidence that there is any difference between the treatments is there any sense into testing which one is better than the other.

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Answer:

a) 8.5

b) between 7.7 and 9.3.

c)Both using a sample of size 400 (instead of 81) and using a 90% level of confidence (instead of 95%) are correct.

(e) 144

Step-by-step explanation:

Explanation for a)

The point estimate for the population mean μ is the sample mean, x ¯ . In this case, to estimate the mean number of weekly hours of home-computer use among the population of U.S. adults, we used the sample mean obtained from the sample, therefore x ¯ = 8.5.

Explanation for b)

The 95% confidence interval for the mean, μ, is x ¯ ± 2 ⋅ σ n = 8.5 ± 2 ⋅ 3.6 81 = 8.5 ± . 8 = ( 7.7 , 9.3 ) .

Explanation for c)

In general, a more concise (narrower) confidence interval can be achieved in one of two ways: sacrificing on the level of confidence (i.e. selecting a lower level of confidence) or increasing the sample size

Explanation for d)

We would like our confidence interval to be a 95% confidence interval (implying that z* = 2) and the confidence interval length should be 1.2, therefore the margin of error (m) = 1.2 / 2 = .6. The sample size we need in order to obtain this is: 144.

Answer:

Part (A): The correct option is true.

Part (B): The null and alternative hypothesis should be:

[tex]H_o: \mu =0 ; H_a:\mu \neq0[/tex]

Step-by-step explanation:

Consider the provided information.

Part (A)

A random sample of 100 students from a large university.

Increasing the sample size decreases the confidence intervals, as it increases the standard error.

If the researcher increase the sample size to 150 which is greater than 100 that will decrease the confidence intervals or the researcher could produce a narrower confidence interval.

Hence, the correct option is true.

Part (B)

The researcher wants to identify that whether there is any significant difference between the measurement of the blood pressure.

Therefore, the null and alternative hypothesis should be:

[tex]H_o: \mu =0 ; H_a:\mu \neq0[/tex]

The statement is true. The researcher could produce a narrower confidence interval by increasing the sample size. The correct null hypothesis for the medical researcher's study is that there is no significant difference between the blood pressure measurements with and without coffee.

The statement is true. A wider confidence interval means that there is more uncertainty in the estimate. To make the confidence interval narrower, the sample size needs to be increased. Increasing the sample size to 150 would likely result in a narrower confidence interval.

The correct null hypothesis for the medical researcher's study would be: There is no significant difference between the blood pressure measurements when patients drink coffee and when they don't. The alternative hypothesis would be: There is a significant difference between the blood pressure measurements when patients drink coffee and when they don't.

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Answer:

Step-by-step explanation:

8) looking at the figure at number 8,

It is a quadrilateral. In a quadrilateral, the opposite angles are supplementary. This means that the sum of all the angles is 360 degrees. We have assigned y degrees to the remaining unknown angle.

y = 180 - 71 = 109 degrees( this is because the sum of angles on a straight line is 180 degrees.

Therefore

10x + 6 + 8x - 1 + 13x - 2 + 109 = 360

31x + 112 = 360

31x = 360 - 112 = 248

x = 248/31 = 8

9) looking at the figure, we have assigned alphabets a, b and c to represent the inner angles that form the triangle.

a + 9x + 1 = 180

a = 180 - 1 -9x = 179 - 9x(this is because the sum of the angles on a straight line is 180 degrees)

b + 5x + 12 = 180

b = 180 - 12 - 5x

b = 168 - 5x

c + 10x -37 = 180

c = 180 - 10x + 37

c = 217 - 10x

a + b + c = 180( sum of angles in a triangle is 180 degrees)

179 - 9x + 168 - 5x + 217 - 10x = 180

-9x - 5x - 10x = 180 - 179 - 168 - 217

-24x = -384

x = -384/-24

x = 16

Answer and explanation:

To find : List the sample space and tell whether you think the events are equally likely ?

Solution :

a) Toss 2 coins; record the order of heads and tails.

Let H is getting head and t is getting tail.

When two coins are tossed the sample space is {HH,HT,TH,TT}.

Total number of outcome = 4

As the outcome HT is different from TH. Each outcome is unique.

Events are equally likely since their probabilities [tex]\frac{1}{4}[/tex] are same.

b) A family has 3 children; record the number of boys.

Let B denote boy and G denote girl.

If there are 3 children then the sample space is

{GGG,GGB,GBG,BGG,BBG,GBB,BGB,BBB}

The possible number of boys are 0,1,2 and 3.

Number of boys      Favorable outcome    Probability

           0                      GGG                        [tex]\frac{1}{8}[/tex]

           1                    GGB,GBG,BGG          [tex]\frac{3}{8}[/tex]

           2                   GBB,BGB,BBG           [tex]\frac{3}{8}[/tex]

           3                       BBB                         [tex]\frac{1}{8}[/tex]

Since the probabilities are not equal the events are not equally likely.

c)  Flip a coin until you get a head or 3 consecutive tails; record each flip.

Getting a head in a trial is dependent on the previous toss.

Similarly getting 3 consecutive tails also dependent on previous toss.

Hence, the probabilities cannot be equal and events cannot be equally likely.

d) Roll two dice; record the larger number

The sample space of rolling two dice is

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Now we form a table that the number of time each number occurs as maximum number then we find probability,

Highest number        Number of times         Probability

           1                                   1                     [tex]\frac{1}{36}[/tex]

           2                                  3                    [tex]\frac{3}{36}[/tex]

           3                                  5                    [tex]\frac{5}{36}[/tex]

           4                                  7                    [tex]\frac{7}{36}[/tex]

           5                                  9                    [tex]\frac{9}{36}[/tex]

           6                                  11                    [tex]\frac{11}{36}[/tex]

Since the probabilities are not the same the events are not equally likely.

The sample spaces for different scenarios include (a) {HH, HT, TH, TT}, (b){0, 1, 2, 3}, (c){H, TH, TTH, TTTH}, and (d){1, 2, 3, 4, 5, 6}.

Let's explore the sample spaces and determine if the events are equally likely for each scenario:

(a) Toss 2 Coins; Record the Order of Heads and Tails

The sample space is: {HH, HT, TH, TT}. Each of these outcomes has an equal probability of occurring, since each coin flip is independent and there are two coins, making each combination equally likely.

(b) A Family Has 3 Children; Record the Number of Boys

The sample space can be represented by the number of boys: {0, 1, 2, 3}. Assuming that each child is equally likely to be a boy or a girl, the probabilities of each outcome differ. For example, having 1 boy (and thus 2 girls) has more combinations than having 3 boys or no boys at all.

(c) Flip a Coin Until You Get a Head or 3 Consecutive Tails; Record Each Flip

The sample space includes sequences that stop as soon as we get a head or reach 3 tails: {H, TH, TTH, TTTH}. These outcomes are not equally likely because the sequence lengths vary, affecting their probabilities.

(d) Roll Two Dice; Record the Larger Number

The sample space is: {1, 2, 3, 4, 5, 6}. However, the events are not equally likely. For example, getting a 6 as the larger number is more probable since it can occur in more pairs (e.g., (6,5), (6,4), etc.) compared to lower numbers.

Answer:

Step-by-step explanation:

Given that Advandia is a drug used to treat diabetes, but it may cause an increase in heart attacks among a population already susceptible.

Sample size n =1`456

Sample proportion of persons having heart attack p1= 27/1456 = 0.0186

II sample

Size = 2895

p2 = sample proportion = 41/2895 = 0.0142

a) Not matched pair data but independent proportions

b)

Step 1:  [tex]H_0: p_1=p_2\\H_a: p_1 > p_2[/tex]

(right tailed test at 5% significance level)

Step 2:  Test statistic z = 1.0995

Step 3:  p value = 0.13567

Step 4:  Since p >0.05 accept null hypothesis.

There is no evidence to suggest that there is a significant increase in the proportion of people who suffer heart attacks when using Avandia compared to other treatments.

Step 5:

Answer:

The 95% confidence interval would be given by (29.780;32.020)  

Step-by-step explanation:

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".

[tex]\bar X=30.9[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

s=7.5 represent the sample standard deviation

n=174 represent the sample size  

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=174-1=173[/tex]

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,173)".And we see that [tex]t_{\alpha/2}=1.97[/tex], this value is similar to the obtained with the normal standard distribution since the sample size is large to approximate the t distribution with the normal distribution.  

Now we have everything in order to replace into formula (1):

[tex]30.9-1.97\frac{7.5}{\sqrt{174}}=29.780[/tex]    

[tex]30.9+1.97\frac{7.5}{\sqrt{174}}=32.020[/tex]

So on this case the 95% confidence interval would be given by (29.780;32.020)    

The value 29.6 is not contained on the interval calculated.

Answer: 1.4375

Step-by-step explanation:

Pela has 5 3/4 feet of string to make necklaces. We would convert 5 3/4 feet to improper fraction. It becomes 23/4 feet. So Pela has 23/4 feet of string to make necklaces.

She wants to make 4 necklaces that are the same length. To determine how long should each necklace be,

We would divide the total length of the string by the number of necklaces to be made. It becomes

23/4 ÷ 4 = 23/4 × 1/4 = 23/16

The length of each string is 23/16 or 1.4375 feet