A group of n people enter an elevator in a building with k floors. Each person independently selects a floor uniformly at random (each with equal probability 1/k). The elevator stops at each floor selected (by at least one person). Let N be the number of stops. (a) [3 marks] Find a simple expression for EN, the average number of stops. Bonus [2 marks] Suppose that n

To find the perimeter of the regular octagon, we use the distance formula to calculate the length of one side given the endpoints and then multiply that length by eight, as there are eight equal sides in a regular octagon.

The question revolves around finding the perimeter of a regular octagon given the coordinates of one of its sides. First, we must determine the length of the side using the distance formula between the two given endpoints (-2,-4) and (4,-6). The distance formula is √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints. After calculating the length of one side, we multiply this by 8 (since an octagon has eight equal sides) to get the perimeter.

The calculation is as follows:

The perimeter of the octagon is 16√10 units.

Answer:

Step-by-step explanation:

What are you looking for as an answer?

We are 90% confident that the true proportion of all Connecticut baseball fans who say their favorite team is the Yankees is between 0.511 and 0.589.

A confidence interval is a range of values that is likely to contain the true value of an unknown parameter, such as a population means or proportion, based on a sample of data from that population.

It is a statistical measure of the degree of uncertainty or precision associated with a statistical estimate.

We have,

To calculate a 90% confidence interval for the proportion of all Connecticut baseball fans who say their favorite team is the Yankees, we can use the formula:

CI = p ± z √((p(1-p))/n)

where:

p is the sample proportion (0.55)

z* is the critical z-value for a 90% confidence level (1.645)

n is the sample size (803)

Plugging in the values,

CI = 0.55 ± 1.645 √((0.55(1-0.55))/803)

CI = 0.55 ± 0.042

Therefore,

The 90% confidence interval for the proportion of all Connecticut baseball fans who say their favorite team is the Yankees is (0.508, 0.592).

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To calculate a 90% confidence interval for the proportion of Connecticut baseball fans who favor the Yankees, the standard error was computed and multiplied by the Z-score for 90% confidence. The confidence interval was found to be between 52.11% and 57.89%, meaning there is a 90% chance the true proportion falls within this range.

The question requires the calculation of a 90% confidence interval for the proportion of all Connecticut baseball fans who favor the New York Yankees. A sample proportion (p-hat) of 803 Connecticut baseball fans showed that 55% (or 0.55) favored the Yankees. To calculate the confidence interval, we use the formula for the confidence interval of a proportion:

CI = p-hat ± Z*sqrt[(p-hat)(1-p-hat)/n], where Z is the Z-score that corresponds to the level of confidence and n is the sample size. For a 90% confidence level, the Z-score is approximately 1.645.

Calculating the standard error (SE):
SE = sqrt[(0.55)(0.45)/803] = sqrt[0.2475/803] = sqrt[0.000308344] = 0.017562

Then, the margin of error (ME):
ME = Z*SE = 1.645 * 0.017562 = 0.028894

The 90% confidence interval (CI) is:
CI = 0.55 ± 0.028894
CI = [0.521106, 0.578894]

Interpreting this confidence interval, we are 90% confident that the true proportion of all Connecticut baseball fans who favor the Yankees falls between 52.11% and 57.89%.

Answer:

The standard deviation of the scores on a recent exam is 6.

The sample size required is 25.

Step-by-step explanation:

Let X = scores of students on a recent exam.

It is provided that the random variable X is normally distributed.

According to the Empirical rule, 99.7% of the normal distribution is contained in the range, μ ± 3σ.

That is, P (μ - 3σ < X < μ + 3σ) = 0.997.

It is provided that the scores on a recent exam were normally distributed with a range from 51 to 87.

This implies that:

P (51 < X < 87) = 0.997

So,

μ - 3σ = 51...(i)

μ + 3σ = 87...(ii)

Subtract (i) and (ii) to compute the value of σ as follows:

  μ -     3σ =    51

(-)μ + (-)3σ = (-)87

______________

-6σ = -36

σ = 6

Thus, the standard deviation of the scores on a recent exam is 6.

The (1 - α)% confidence interval for population mean is given by:

[tex]CI=\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]

The margin of error of this interval is:

[tex]MOE = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]

Given:

MOE = 2

σ = 6

Confidence level = 90%

Compute the z-score for 90% confidence level as follows:

[tex]z_{\alpha/2}=z_{0.10/2}=z_{0.05}=1.645[/tex]

*Use a z-table.

Compute the sample required as follows:

[tex]MOE = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\\2=1.645\times \frac{6}{\sqrt{n}}\\n=(\frac{1.645\times 6}{2})^{2}\\n=24.354225\\n\approx 25[/tex]

Thus, the sample size required is 25.

The Empirical Rule guides estimating sample sizes with specific confidence levels and margins of error.

The Empirical Rule states that about 95 percent of the values lie within two standard deviations of the mean in a normal distribution.

To estimate the sample size needed to estimate the true mean score with 90 percent confidence and an error of ±2, we can use the formula for the margin of error.

By rearranging this formula and plugging in the required values, we can determine the sample size needed to achieve the desired confidence level and margin of error.

Answer:

N/A

Step-by-step explanation:

what is line l?

Answer: Its C, 9.1

Step-by-step explanation: Did it on usa prept :)

Answer:

Probability that their mean height is less than 68 inches is 0.0764.

Step-by-step explanation:

We are given that in the united states, the height of men are normally distributed with the mean 69 inches and standard deviation 2.8 inches.

Also, 16 men are randomly selected.

Let [tex]\bar X[/tex] = sample mean height

The z-score probability distribution for sample mean is given by;

              Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]  ~ N(0,1)

where, [tex]\mu[/tex] = population mean height = 69 inches

            [tex]\sigma[/tex] = population standard deviation = 2.8 inches

            n = sample of men = 16

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

So, probability that the mean height of 16 randomly selected men is less than 68 inches is given by = P([tex]\bar X[/tex] < 68 inches)

 P([tex]\bar X[/tex] < 68 inches) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{68-69}{\frac{2.8}{\sqrt{16} } }[/tex] ) = P(Z < -1.43) = 1 - P(Z [tex]\leq[/tex] 1.43)

                                                           = 1 - 0.9236 = 0.0764

Now, in the z table the P(Z  x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1.43 in the z table which has an area of 0.92364.

Therefore, probability that their mean height is less than 68 inches is 0.0764.

The total distance covered is 6 m

Step-by-step explanation:

Let the total distance be '2d'

Total time = 1.25hrs

Downhill speed = 6 mphr

Uphill speed = 4 mphr

(d/6) +(d/4) = 1.25

(2d + 3d) /12 = 1.25

5d/12 = 1.25

d = 3m

So, 2d = 6 m

The total distance covered is 6 m

Answer:

Semicircle of radius of 1.6803 meters

Rectangle of dimensions 3.3606m x 1.6803m

Step-by-step explanation:

Let the radius of the semicircle on the top=r  

Let the height of the rectangle =h  

Since the semicircle is on top of the window, the width of the rectangular portion =Diameter of the Semicircle =2r

The Perimeter of the Window

=Length of the three sides on the rectangular portion + circumference of the semicircle

[tex]=h+h+2r+\pi r=2h+2r+\pi r=12[/tex]

The area of the window is what we want to maximize.

Area of the Window=Area of Rectangle+Area of Semicircle

[tex]=2hr+\frac{\pi r^2}{2}[/tex]

We are trying to Maximize A subject to [tex]2h+2r+\pi r=12[/tex]

[tex]2h+2r+\pi r=12\\h=6-r-\frac{\pi r}{2}[/tex]

The first and second derivatives are,

Area, A(r)[tex]=2r(6-r-\frac{\pi r}{2})+\frac{\pi r^2}{2}}=12r-2r^2-\frac{\pi r^2}{2}[/tex]

Taking the first and second derivatives

[tex]A'\left( r \right) = 12 - r\left( {4 + \pi } \right)\\A''\left( r \right) = - 4 - \pi[/tex]

From the two derivatives above, we see that the only critical point  of r

[tex]A'\left( r \right) = 12 - r\left( {4 + \pi } \right)=0[/tex]

[tex]r = \frac{{12}}{{4 + \pi }} = 1.6803[/tex]

Since the second derivative is a negative constant, the maximum area must occur at this point.

[tex]h=6-1.6803-\frac{\pi X1.6803}{2}=1.6803[/tex]

So, for the maximum area the semicircle on top must have a radius of 1.6803 meters and the rectangle must have the dimensions 3.3606m x 1.6803m ( Recall, The other dimension of the window = 2r)

The problem is to maximize the area of a window consisting of a rectangle and a semi-circle on top, given a fixed perimeter of framing material, which is a high school level optimization problem in geometry and calculus.

The question addresses the problem of finding the dimensions of a window with the most amount of light passing through, given a fixed amount of framing material. This is a classic problem in mathematics involving optimization under constraints, specifically related to geometry and calculus.

Let the width of the rectangle be x meters, and its height be y meters. Since the top of the window is a semi-circle, its diameter is equal to the width of the rectangle, meaning the radius of the semi-circle is x/2 meters. The perimeter of the entire window consists of the two sides and the bottom of the rectangle, and the circumference of the semi-circle. The total length of framing material is 12 meters, hence:

2y + x + (π(x/2)/2) = 12

Since the area of rectangle A = x*y and the area of the semi-circle is (x/2)²)/2, we want to maximize the total area A = x*y + (x/2)²)/2.

Using calculus, one can differentiate the area with respect to x or y and set the derivative equal to zero to find the maximum value. Assuming the student knows basic differentiation and solving equations, they can arrive at the optimal dimensions to let in the most light.

Answer:

11

Step-by-step explanation:

8x - 3(-z - y)

Substitute

8(1) - 3(-(-2) - 3)

Simplify

8 - 3(-1)

Multiply

8 + 3

Add

11

Answer:

Here; 8x-3(-z-y) =

so your asking 8x-3(-z-y) right?

Let's simplify step-by-step.

8x−3(−z−y)

awnser again xD

=8x+3y+3z

Given:

The system of equations is [tex]y=-\frac{3}{2}x+12[/tex] and [tex]y=5x+28[/tex]

We need to determine the solution to the system of equations.

Solution:

The solution to the system of equations is the point of intersection of these two lines.

Let us solve the system of equations using substitution method.

Thus, we have;

[tex]5x+28=-\frac{3}{2}x+12[/tex]

Simplifying, we get;

[tex]\frac{13}{2}x+28=12[/tex]

       [tex]\frac{13}{2}x=-16[/tex]

       [tex]13x=-32[/tex]

          [tex]x=-2.462[/tex]

Thus, the value of x is -2.462

Substituting x = -2.462 in the equation [tex]y=5x+28[/tex], we get;

[tex]y=5(-2.462)+28[/tex]

[tex]y=-12.31+28[/tex]

[tex]y=15.69[/tex]

Thus, the value of y is 15.69.

Therefore, the solution to the system of the equations is (-2.462, 15.69)

Answer:

The correct option is (c).

Step-by-step explanation:

According to the Central Limit Theorem if we have a population with a known mean and standard deviation and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Then, the mean of the distribution of sample means is given by, the population mean.

And the standard deviation of the distribution of sample means is given by,

[tex]SD_{\bar x}=\frac{SD}{\sqrt{n}}[/tex]

So, the most basic and main objective of the Central limit theorem is to approximate the sampling distribution of a statistic by the Normal distribution even when we do not known the distribution of the population.

Thus, the correct option is (c).

The Central Limit Theorem is crucial in statistics because it provides a powerful tool for dealing with data from various populations by allowing us to rely on the normal distribution for making statistical inferences, even when the underlying population is not normally distributed. This property makes it a cornerstone of statistical theory and practice.

The correct answer is: c) Because for a large sample size n, it says the sampling distribution of the sample mean is approximately normal regardless of the shape of the population.

The Central Limit Theorem (CLT) is a fundamental concept in statistics with widespread applications.

It is important for several reasons:

Approximation of the Sampling Distribution:

The CLT states that for a sufficiently large sample size (n), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population from which the samples are drawn.

This is crucial because it allows statisticians to make inferences about population parameters based on the normal distribution, which simplifies statistical analysis.

Widespread Applicability:

The CLT is not limited to specific populations or data types.

It holds true for a wide range of data distributions, making it a versatile tool in statistical analysis.

Whether the underlying population distribution is normal, uniform, exponential, or any other shape, the CLT assures us that the distribution of sample means will tend to be normal for sufficiently large samples.

Foundation for Hypothesis Testing and Confidence Intervals:

Many statistical methods, including hypothesis testing and the construction of confidence intervals, rely on the assumption of a normal distribution.

The CLT's ability to transform non-normally distributed data into a normal sampling distribution is essential for these statistical techniques.

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Answer:

19 + x ≥ 38

Step-by-step explanation:

Margo must sell at least 19 more tubs of cookie dough to meet her goal for the student council fundraiser, represented by the inequality x ≥ 19

The question asks to find an inequality that represents the number of tubs of cookie dough Margo still needs to sell.

Margo needs to sell at least 38 tubs of cookie dough and has already sold 19, so we subtract the tubs sold from the total needed:

38 - 19 = 19

Now, let x represent the number of tubs Margo still needs to sell.

The inequality that best represents this situation is:

x ≥ 19

This inequality shows that Margo needs to sell at least 19 more tubs to meet her minimum goal for the fundraiser.

Answer:

1/x^2

Step-by-step explanation:

When you have a negative sign in the exponent, the integer will become fraction.

Answer:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

Step-by-step explanation:

To find the roots you use:

[tex]z^{\frac{1}{n}}=r^{\frac{1}{n}}[cos(\frac{\theta+2\pi k}{n})+isin(\frac{\theta+2\pi k}{n})][/tex]     ( 1 )

n: the order of the roots

k: 0,1,2,...,n-1

First, you write z in polar notation:

[tex]z=re^{i\theta}\\\\r=\sqrt{(473)^2+(4)^2}=473.01\\\\\theta=tan^{-1}(\frac{4}{473})=0.48\°[/tex]

Thus, by using these values for the angle and r in the expression (1), you obtain:

[tex]k=0\\\\z_1=(473.01)^{1/3}[cos(\frac{0.48+2\pi(0)}{3})+isin(\frac{0.48+2\pi(0)}{3})]\\\\z_1=7.79(0.99+i2.79*10^{-3})=7.71+i0.02\\\\z_2=7.79[cos(\frac{0.48+2\pi(1)}{3})+isin(\frac{0.48+2\pi(1)}{3})]\\\\z_2=7.73+i0.306\\\\z_3=7.79[cos(\frac{0.48+2\pi(2)}{3})+isin(\frac{0.48+2\pi(2)}{3})]\\\\z_3=7.78+i0.59[/tex]

hence, from the previous results you obtain:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

I attached and image of the plot

The diameter of the hemisphere is 20.4 centimeters.

Here's the breakdown of the calculation:

Formula for hemisphere volume: The volume of a hemisphere is equal to two-thirds the volume of a full sphere with the same radius (r).

V_hemisphere = (2/3) * (4/3) * π * r^3

Substituting known values: We know the volume of the hemisphere (V_hemisphere) is 2233 cm^3. Plugging this into the equation:

2233 cm^3 = (2/3) * (4/3) * π * r^3

Solve for radius (r): Isolate r:

r^3 = (2233 cm^3 * 3 / 8 * π) ≈ 1181.36 cm^3

r ≈ 10.3 cm

Calculate diameter: Diameter is twice the radius:

diameter = 2 * radius ≈ 20.6 cm

Therefore, the diameter of the hemisphere is approximately 20.4 centimeters to the nearest tenth of a centimeter.

Option B, pooled samples, is the correct answer, as it relies on the homogeneity of variance assumption for a t-test comparing two means.

Regarding inferences about the difference between two population means, when both populations are assumed to have equal standard deviations (or variances), the sampling design that uses a pooled sample variance is based on independent samples. This method pools the sample variances into a single, blended variance estimate when conducting a t-test for comparing two means.

The correct answer to the original question is B.​pooled samples. This approach uses the assumption that the population variances are equal, an assumption known as homogeneity of variance. The pooled variance is then used to calculate the test statistic which is crucial for the t-test to compare the means from the two populations accurately.

If the assumption of equal population variances is true, using pooled variance gives a more precise estimate of the population variance, which can lead to a more powerful statistical test. If the populations do not have equal variances, another test designed for unequal variances, like Welch's t-test, should be used instead.

the sampling design that uses a pooled sample variance in cases of equal population standard deviations is based on:pooled samples

The correct option is (B).

When making inferences about the difference between two population means, one common scenario is when the standard deviations of the two populations are equal. In such cases, the pooled sample variance approach is used.

Pooled sample variance involves combining the variances of the two samples into a single pooled estimate. This is done because when the standard deviations are equal, it's reasonable to assume that the underlying population variances are also equal.

The pooled sample variance approach is typically used in independent samples designs, where the samples are taken from two separate populations or groups. These samples are independent because the individuals in one sample are not related to the individuals in the other sample.

Therefore, the correct answer is B. pooled samples, as it accurately describes the sampling design used in cases of equal population standard deviations when making inferences about the difference between two population means.

The 12 inch cake needs 3 cups of flour.

In order to solve a problem for two different values of a quantity, its unit value is first derived. This method is known as unitary method.

Given that,

The amount of cup of flour for 8 inch cake is 2 cups.

The  given problem can be solved using unitary method as follows,

The 8 inch cake needs 2 cups.

Then, 1 inch cake needs 2/8 = 1/4 cups.

Thus, 12 inch cup requires 1/4 × 12 = 3 cups.

Hence, the number of cups needed for 12 inch cake is 3.

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Answer:

2.7

Step-by-step explanation:

This can be modeled using exponential growth/decay.

A = P (1 + r)ⁿ

where A is the final amount,

P is the initial amount,

r is the rate of growth/decay,

and n is the number of cycles.

For half life problems, r = -½, and n = t / T, where t is time and T is the half life.

A = P (1 − ½)^(t/T)

A = P (½)^(t/T)

Given that P = 9, t = 10000, and T = 5730:

A = 9 (½)^(10000/5730)

A ≈ 2.7

There are approximately 2.7 mg of ¹⁴C left.

Answer:

1/60 probabiliity

Step-by-step explanation:

You have two independent events that you want to put together.

Let Pr. mean "probability"

Pr(5 from 10 cards and 2 on a dice ) = Pr(5 from 10 cards) * Pr( 2 on a dice)

Pr(5 from 10 cards and 2 on a dice ) =(1/10) * (1/6)

= 1/60

Pr(5 from 10 cards and 2 on a dice ) =0.0167

or 1.67% probability

Answer:

Its b (2,9)

Step-by-step explanation:

I got it on edge

The points which line on the graph as per the given function will be (2, 9). Hence, option B is correct.

In math, graph science is the theory of geometric structures called graphs that are used to represent pairwise different objects. Vertices—also known as nodes or points—that are joined by edges make form a network in this sense.

Undirected graphs, where edges connect two vertices equally, and focused therapy, where edges connect two vertices unevenly, are distinguished.

As per the given information in the question,

The given function is,

y = 3ˣ

In this issue, we have to form points (x₀, y₀). All the function's points need to be replaced by y = 3ˣ.

Replace x₀ with x and y₀ with y.

Only when the right and left sides of the equality are equal does the point belong to the function.

Now, let's check the options one by one.

(a) (1, 0)

y = 3ˣ

0 = 3¹

0 = 3, it is incorrect.

(b) (2, 9)

y = 3ˣ

9 = 3²

9 = 9, it is correct.

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Answer:

Country ​A = 84

Country​ B = 28

Country C = 18

Step-by-step explanation:

Country ​A, Country​ B, and Country C won a total of 130 medals;

A + B + C = 130 ......1

Country B won 10 more medals than Country C;

B = C + 10 .......2

Country A won 38 more medals than the total amount won by the other two;

A = B + C + 38 ........3

Substituting equation 3 to 1;

(B+C+38) + B+C = 130

2B + 2C + 38 = 130 .......4

Substituting equation 2 into 4;

2(C+10) + 2C + 38 = 130

4C + 58 = 130

4C = 130-58 = 72

C = 72/4 = 18

B = C + 10 = 18 + 10 = 28

A = B + C + 38 = 18 + 28 + 38 = 84

Country ​A = 84

Country​ B = 28

Country C = 18

Answer

equilateral

Step-by-step explanation:

rate and choose brainliest

Step-by-step explanation:

A circular swimming pool has a tacius of 28 ft. There is a path all the way around the pool that is 4 ft wide. A fence is going to be built around the outside edge of the pool path

WE need to find the circumference of the circle

radius of circular swimming pool is 28 feet

path is 4 ft wide

so we add 4 with radius

radius of the pool with path is 28+4= 32

[tex]circumference =2\pi (r)[/tex]

where r is the radius

r=32

[tex]circumference =2\pi (r)\\circumference =2\pi (32)=200.96=201[/tex]

Answer:

201 feet

The arc length formed by the 2 radio is 15.24 yards

Step-by-step explanation:

Angle between the 2 radio = 45°

Radius = 64 yards

Arc length = (45/360) (2π) (64)

= (1/8) (2π) (64

= 16π

= 16(3.14)

=15.24 yards

The arc length formed by the 2 radio is 15.24 yards

Stephen's heart beats 68 times in one minute

Step-by-step explanation:

60 sec (one min) divided by 15 sec ( the unit of time we use to track the 17 heartbeats) = 4

4 times 17 =68

Final answer:

The difference in surface areas of the two boxes is 3 square feet.

Explanation:

The difference in surface areas between two boxes is computed by first finding the surface area of each box and then subtracting the smaller surface area from the larger one.

Rectangular Prism:

For the rectangular prism with dimensions 3 ft by 4.5 ft by 2 ft, the surface area (SA) is calculated using the formula SA = 2lw + 2lh + 2wh.

The surface areas of the rectangular prism are:    

Total surface area of the rectangular prism = 27 ft² + 12 ft² + 18 ft² = 57 ft²

Cube:

For the cube with a side of 3 ft, the surface area is found using the formula SA = 6s².

Thus:

Total surface area of the cube = 6(3 ft × 3 ft) = 54 ft²

Difference in Surface Area:

Difference in surface areas = Surface area of the rectangular prism - Surface area of the cube = 57 ft² - 54 ft² = 3 ft².

Answer: Stefan didn’t use only rigid transformations, so the figures are not congruent

Step-by-step explanation:

Stefan didn’t use only rigid transformations, so the figures are not congruent.

Two triangles are said to be congruent if their sides are equal in length, the angles are of equal measure, and they can be superimposed on each other.

In the given, Δ ABC and Δ ABD are not congruent triangles. if they are congruent then This means that the corresponding angles and corresponding sides in both the triangles are equal.

The following are the congruence theorems or the triangle congruence criteria that help to prove the congruence of triangles;

SSS (Side, Side, Side)

SAS (side, angle, side)

ASA (angle, side, angle)

AAS (angle, angle, side)

RHS (Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem)

As, from the given cases the prediction of congruency of two triangles is incorrect. There is no error he made.

Hence, Stefan didn’t use only rigid transformations, so the figures are not congruent

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Answer:

The value of t test statistics is -2.

Step-by-step explanation:

We are given that a study was conducted to determine whether UH students sleep fewer than 8 hours.

The study was based on a sample of 100 students. The sample mean number of hours of sleep was 7 hours and the sample standard deviation was 5 hours.

Let [tex]\mu[/tex] = mean number of hours UH students sleep.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 8 hours     {means that UH students sleep more than or equal to 8 hours}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 8 hours     {means that UH students sleep fewer than 8 hours}

The test statistics that would be used here One-sample t test statistics as we don't know about the population standard deviation;

                         T.S. =  [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean number of hours of sleep = 7 hours

             s = sample standard deviation = 5 hours

            n = sample of students = 100

So, test statistics  =  [tex]\frac{7-8}{\frac{5}{\sqrt{100} } }[/tex]  ~ [tex]t_9_9[/tex]

                              =  -2

Hence, the value of t test statistics is -2.

Final answer:

The approximate probability of winning 4 games in a row when each game has a 1/3 chance of winning is 1.23%.

(Option B)

Explanation:

To find the approximate probability of winning 4 games in a row, we need to multiply the probabilities of winning each game. Since the probability of winning each game is 1/3, we can calculate the overall probability as [tex](1/3)^4[/tex]. Using a calculator, this comes out to be approximately 0.0123 or 1.23%.

Multiplying the individual probabilities of winning each game, given as 1/3, results in the overall probability of winning 4 games in a row, expressed as [tex](1/3)^4[/tex]. Using a calculator, this evaluates to approximately 0.0123, or 1.23%, highlighting the cumulative nature of independent events.