Solve the problem. The scores on a certain test are normally distributed with a mean score of 60 and a standard deviation of 5. What is the probability that a sample of 90 students will have a mean score of at least 60.527? Write your answer as a decimal rounded to 4 places.

Answer:

x = 7

Step-by-step explanation:

This problem can be solved using angular bisector theorem.

It states that if any angle of triangle is bisected by a line , then that line

divides the opposite side of that angle in same proportion as that of two other sides which contain the angle.

__________________________________

Here one angle is is divided into parts theta

Thus,

using angular bisector theorem

14/21 = 6/3x-12

=> 14(3x-12) = 21*6

=> 3x-12 = 21*6/14 = 9

=> 3x = 12+9 = 21

=> x = 21/3 = 7

Thus, x = 7

Answer: (3y - 10)*2÷4

Step-by-step explanation:

Because 10 less than 3 times a number, y, is done first, it is in parenthesis.  The 3 is there to represent the "three times" and the -10 is there to represent the "ten less".  The *2 is there to represent the "multiplied by two" and the ÷4 is there to represent the "divided by 4"

Hope it helps, and tyvm <3

Answer:

[tex]\displaystyle \frac{2(3y - 10)}{4}[/tex]

Step-by-step explanation:

10 less than 3 times y.

The variable y is multiplied by 3, 10 is subtracted from 3 × y.

The result 3y - 10 is then multiplied by 2.

2(3y - 10) is then divided by 4.

Answer:

Yes based on the numbers .

Step-by-step explanation:

Answer:Yes

Step-by-step explanation:Based on the number given, it shows that there is a hypotenuse (The longest side of a right triangle, in this case being 12), And opposite (Another part of the right triangle, that could be either 9 or 7), and the adjacent (The line next to the opposite, which could be 9 or 7)

━━━━━━━☆☆━━━━━━━

▹ Answer

V ≈ 160.22

▹ Step-by-Step Explanation

V = πr²[tex]\frac{h}{3}[/tex]

V = π3²[tex]\frac{17}{3}[/tex]

V ≈ 160.22

Hope this helps!

- CloutAnswers ❁

Brainliest is greatly appreciated!

━━━━━━━☆☆━━━━━━━

Answer:

Step-by-step explanation:

We would set up the hypothesis test.

For the null hypothesis,

p = 0.32

For the alternative hypothesis,

p ≠ 0.32

This is a two tailed test

Considering the population proportion, probability of success, p = 0.32

q = probability of failure = 1 - p

q = 1 - 0.32 = 0.68

Considering the sample,

Sample proportion, P = x/n

Where

x = number of success = 261

n = number of samples = 750

P = 261/750 = 0.35

We would determine the test statistic which is the z score

z = (P - p)/√pq/n

z = (0.35 - 0.32)/√(0.32 × 0.68)/750 = 1.8

Recall, population proportion, p = 0.32

The difference between sample proportion and population proportion(P - p) is 0.35 - 0.32 = 0.03

Since the curve is symmetrical and it is a two tailed test, the p for the left tail is 0.32 - 0.03 = 0.29

the p for the right tail is 0.32 + 0.03 = 0.35

These proportions are lower and higher than the null proportion. Thus, they are evidence in favour of the alternative hypothesis. We will look at the area in both tails. Since it is showing in one tail only, we would double the area

From the normal distribution table, the area above the z score in the right tail 1 - 0.9641 = 0.0359

We would double this area to include the area in the right tail of z = 0.44 Thus

p = 0.0359 × 2 = 0.07

Since alpha, 0.05 < the p value, 0.07 then we would fail to reject the null hypothesis. Therefore, this is not sufficient evidence to conclude that the actual percentage is different from 32% at the 5% significance level.

Answer:

A velocity of 120km/h is needed to cover the distance in one hour

Step-by-step explanation:

The velocity formula is:

[tex]v = \frac{d}{t}[/tex]

In which v is the velocity, d is the distance and t is the time.

A car travelling from Ibadan to Lagos at 90 km/hr takes 1 hour 20 min.

This means that [tex]v = 90, t = 1 + \frac{20}{60} = 1.3333[/tex]

We use this to find d.

[tex]v = \frac{d}{t}[/tex]

[tex]90 = \frac{d}{1.3333}[/tex]

[tex]d = 90*1.3333[/tex]

[tex]d = 120[/tex]

The distance is 120 km.

How fast must one travel to cover the distance in one hour?

Velocity for a distance of 120 km(d = 120) in 1 hour(t = 1). So

[tex]v = \frac{d}{t}[/tex]

[tex]v = \frac{120}{1}[/tex]

[tex]v = 120[/tex]

A velocity of 120km/h is needed to cover the distance in one hour

Answer:

4

Step-by-step explanation:

1, 3, 4, 6, 2, 4, 5, 6, 2, 3, 1, 4, 0, 4, 4, 4, 8, 9, 7, 4

Arrange the numbers from smallest to largest

0,1, 1,2,2, 3,3, 4, 4,4,4,4,4 , 4, 5, 6, 6,   7, 8, 9,

There are 20 numbers

The middle number is between 10 and 11

0,1, 1,2,2, 3,3, 4, 4,4   ,4,4,4 , 4, 5, 6, 6,   7, 8, 9,

The median is 4

Solution,

Arranging the data in ascending order:

0,1,1,2,2,3,3,4,4,4,4,4,4,4,5,6,6,7,8,9

N(total number of items)= 20

Now,

Median:

[tex] (\frac{n + 1}{2)} ) ^{th \: item} \\ = (\frac{20 + 1}{2} ) ^{th \: item} \\ = \frac{21}{2} \\ = 10.5 \: th \: \: item[/tex]

Again,

Median:

[tex] \frac{10 \: th \: item + 11 \: th \: item}{2} \\ = \frac{4 + 4}{2} \\ = \frac{8}{2} \\ = 4[/tex]

Answer:

  $904,510.28

Step-by-step explanation:

If we assume the withdrawals are at the beginning of the month, we can use the annuity-due formula.

  P = A(1 +r/n)(1 -(1 +r/n)^(-nt))/(r/n)

where r is the APR, n is the number of times interest is compounded per year (12), A is the amount withdrawn, and t is the number of years.

Filling in your values, we have ...

  P = $4000(1 +.034/12)(1 -(1 +.034/12)^(-12·30))/(.034/12)

  P = $904,510.28

You need to have $904,510.28 in your account when you begin withdrawals.

Answer:

You need to have $904,510.28 in your account when you begin

Answer:

1. y² - 3x - 18

2. 4x² - 33x + 35

3. 12x² - 11x + 2

Step-by-step explanation:

All we do with these questions are expanding the factored binomials. Use FOIL:

1. y² + 3y - 6y - 18

y² - 3y - 18

2. 4x² - 28x - 5x + 35

4x² - 33x + 35

3. 12x² - 3x - 8x + 2

12x² - 11x + 2

Answer:

1) (y-6) (y+3)

=> [tex]y^2+3y-6y-18[/tex]

=> [tex]y^2-3y-18[/tex]

2) (4x-5) (x-7)

=> [tex]4x^2-28x-5x+35[/tex]

=> [tex]4x^2-33x+35[/tex]

3) (3x - 2) ( 4x - 1)

=> [tex]12x^2-3x-8x+3[/tex]

=> [tex]12x^2-11x+3[/tex]

Answer:

the 3rd option is the answer

Step-by-step explanation:

I hope the attached file is self-explanatory

Answer:

B

Step-by-step explanation:

The answer is B because in order for the square root of a number to be equal to another number, the answer squared should be the number under the square root.

B. [tex](-4)^2\neq -16[/tex].

Hope this helps.

Answer:

[tex]\huge\boxed{x=\sqrt{66}}[/tex]

Step-by-step explanation:

ΔADC and ΔABD are similar (AAA)

Therefore the cooresponging sides are in proportion:

[tex]\dfrac{AD}{AC}=\dfrac{AB}{AD}[/tex]

Substitute

[tex]AD=x;\ AC=6+5=11;\ AB=6[/tex]

[tex]\dfrac{x}{11}=\dfrac{6}{x}[/tex]          cross multiply

[tex](x)(x)=(11)(6)\\\\x^2=66\to x=\sqrt{66}[/tex]

Answer:

The integers are 7 and 14.

Step-by-step explanation:

y = 2x

1/y + 1/x = 3/14

1/(2x) + 1/x 3/14

1/(2x) + 2/(2x) = 3/14

3/(2x) = 3/14

1/2x = 1/14

2x = 14

x = 7

y = 2x = 2(7) = 14

Answer: The integers are 7 and 14.

The required two integers are 7 and 14

This is a question on word problems leading to the simultaneous equation:

Let the two unknown integers be x and y. If a positive integer is twice another, then x = 2y .......... 1

Also, if the sum of the reciprocals of the two positive integers is 3/14, then:

[tex]\frac{1}{x}+ \frac{1}{y} =\frac{3}{14}[/tex] ..........2

Substitute equation 1 into 2

[tex]\frac{1}{2y} +\frac{1}{y} =\frac{3}{14} \\[/tex]

Find the LCM of 2y and y

[tex]\frac{1+2}{2y} =\frac{3}{14} \\\frac{3}{2y} =\frac{3}{14} \\\\cross \ multiply\\2y \times 3=3 \times 14\\6y=42\\y=\frac{42}{6}\\y=7[/tex]

Substitute y = 7 into equation 1:

Recall that x = 2y

[tex]x = 2(7)\\x = 14[/tex]

Hence the required two integers are 7 and 14.

Learn more here: https://brainly.com/question/17671977

Answer:

2/7

Step-by-step explanation:

The numbers greater than 7 or less than 3 are 2 and 8.

2 numbers out of 7.

P(greater than 7 or less than 3) = 2/7

Answer:

2/7

Step-by-step explanation:

There are a total of 7 sample spaces also known as 2,3,4,5,6,7,8. Now we have to find a number greater than 7 and less than 3. 2 is less than 3, and 8 is greater than 7, so two numbers are selected. This would become 2/7 because out of all of the 7 outcomes, only two are selected.

Answer:

option D 9x³

Step-by-step explanation:

the monomial 9x³ comes from (3x)³, which gives, 3×3×3×x×x×x= 9x³

9 is 3 times 3 and x³ is 3 times x. So here, 9x³ is a perfect cube

Answer:

(f o g) = x, then, g(x) is the inverse of f(x).

Step-by-step explanation:

You have the following functions:

[tex]f(x)=-\frac{2}{x}-1\\\\g(x)=-\frac{2}{x+1}[/tex]

In order to know if f and g are inverse functions you calculate (f o g) and (g o f):

[tex]f\ o\ g=f(g(x))=-\frac{2}{-\frac{2}{x+1}}-1=x+1-1=x[/tex]

[tex]g\ o\ f=g(f(x))=-\frac{2}{-\frac{2}{x}+1}=-\frac{2}{\frac{-2+x}{x}}=\frac{2x}{2-x}[/tex]

(f o g) = x, then, g(x) is the inverse of f(x).

Answer:

Height at t = 1 sec is 1144 ft

Step-by-step explanation:

Given:

Initial height of object = 1160 feet

Height of object after t seconds is given by the polynomial:

[tex]- 16t ^2+ 1160[/tex]

Let [tex]h(t)=- 16t ^2+ 1160[/tex]

Let us analyze the given equation once.

[tex]t^2[/tex] will always be positive.

and coefficient of [tex]t^2[/tex] is [tex]-16[/tex] i.e. negative value.

It means something is subtracted from 1160 ft (i.e. the initial height).

So, height will keep on decreasing with increasing value of t.

Also, given that the object is dropped from the top of a tower.

To find:

Height of object at t = 1 sec.

OR

[tex]h (1)[/tex] = ?

Solution:

Let us put t = 1 in the given equation: [tex]h(t)=- 16t ^2+ 1160[/tex]

[tex]h(1)=- 16\times 1 ^2+ 1160\\\Rightarrow h(1) = -16 + 1160\\\Rightarrow h(1) = 1144\ ft[/tex]

So, height of object at t = 1 sec is 1144 ft.

Answer:

x=11

Step-by-step explanation:

Since the lines in the middle are parallel, we know that both sides are proportional to each other.

6:48 can be simplified to 1:8

Since we know the left side ratio is 1:8, we need to match the right side with the same ratio

We can multiply the ratio by 5 to match 5:3x+7

5:40

5:3x+7

Now we can set up the equation: 40=3x+7

Subtract 7 from both sides

3x=33

x=11

Answer:

(a) X ~ N([tex]\mu=63, \sigma^{2} = 13^{2}[/tex]).

    [tex]\bar X[/tex] ~ N([tex]\mu=63,s^{2} = (\frac{13}{\sqrt{43} } )^{2}[/tex]).

(b) If a single randomly selected individual is observed, the probability that this person consumes is between 61.4 mL and 62.8 mL is 0.0398.

(c) For the group of 43 pancake eaters, the probability that the average amount of syrup is between 61.4 mL and 62.8 mL is 0.2512.

(d) Yes, for part (d), the assumption that the distribution is normally distributed necessary.

Step-by-step explanation:

We are given that the amount of syrup that people put on their pancakes is normally distributed with mean 63 mL and a standard deviation of 13 mL.

Suppose that 43 randomly selected people are observed pouring syrup on their pancakes.

(a) Let X = amount of syrup that people put on their pancakes

The z-score probability distribution for the normal distribution is given by;

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

where, [tex]\mu[/tex] = mean amount of syrup = 63 mL

            [tex]\sigma[/tex] = standard deviation = 13 mL

So, the distribution of X ~ N([tex]\mu=63, \sigma^{2} = 13^{2}[/tex]).

Let [tex]\bar X[/tex] = sample mean amount of syrup that people put on their pancakes

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

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

where, [tex]\mu[/tex] = mean amount of syrup = 63 mL

            [tex]\sigma[/tex] = standard deviation = 13 mL

            n = sample of people = 43

So, the distribution of [tex]\bar X[/tex] ~ N([tex]\mu=63,s^{2} = (\frac{13}{\sqrt{43} } )^{2}[/tex]).

(b) If a single randomly selected individual is observed, the probability that this person consumes is between 61.4 mL and 62.8 mL is given by = P(61.4 mL < X < 62.8 mL)

   P(61.4 mL < X < 62.8 mL) = P(X < 62.8 mL) - P(X [tex]\leq[/tex] 61.4 mL)

  P(X < 62.8 mL) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{62.8-63}{13}[/tex] ) = P(Z < -0.02) = 1 - P(Z [tex]\leq[/tex] 0.02)

                                                           = 1 - 0.50798 = 0.49202

  P(X [tex]\leq[/tex] 61.4 mL) = P( [tex]\frac{X-\mu}{\sigma}[/tex] [tex]\leq[/tex] [tex]\frac{61.4-63}{13}[/tex] ) = P(Z [tex]\leq[/tex] -0.12) = 1 - P(Z < 0.12)

                                                           = 1 - 0.54776 = 0.45224

Therefore, P(61.4 mL < X < 62.8 mL) = 0.49202 - 0.45224 = 0.0398.

(c) For the group of 43 pancake eaters, the probability that the average amount of syrup is between 61.4 mL and 62.8 mL is given by = P(61.4 mL < [tex]\bar X[/tex] < 62.8 mL)

   P(61.4 mL < [tex]\bar X[/tex] < 62.8 mL) = P([tex]\bar X[/tex] < 62.8 mL) - P([tex]\bar X[/tex] [tex]\leq[/tex] 61.4 mL)

  P([tex]\bar X[/tex] < 62.8 mL) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{62.8-63}{\frac{13}{\sqrt{43} } }[/tex] ) = P(Z < -0.10) = 1 - P(Z [tex]\leq[/tex] 0.10)

                                                           = 1 - 0.53983 = 0.46017

  P([tex]\bar X[/tex] [tex]\leq[/tex] 61.4 mL) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] [tex]\leq[/tex] [tex]\frac{61.4-63}{\frac{13}{\sqrt{43} } }[/tex] ) = P(Z [tex]\leq[/tex] -0.81) = 1 - P(Z < 0.81)

                                                           = 1 - 0.79103 = 0.20897

Therefore, P(61.4 mL < X < 62.8 mL) = 0.46017 - 0.20897 = 0.2512.

(d) Yes, for part (d), the assumption that the distribution is normally distributed necessary.

Given:

An equilateral triangle JKL inscribed in circle M.

Solution:

To draw an equilateral triangle inscribed in circle follow the steps:

1: Draw a circle with any radius.

2. Take any point A, anywhere on the circumference of the circle.

3.  Place the compass on point A, and swing a small arc crossing the circumference of the circle.

Remember the span of the compass should be the same as the radius of the circle.

4. Place the compass at the intersection of the previous arc and the circumference and draw another arc but don't change the span of the compass.

5. Repeat this process until you return to point A.

6. Join the intersecting points on the circle to form the equilateral triangle.

So the correct option is A. The width must be equal to the radius of circle M.

Answer:

x = 15

y = 90

Step-by-step explanation:

Step 1: Find x

We use Definition of Supplementary Angles

9x + 3x = 180

12x = 180

x = 15

Step 2: Find y

All angles in a triangle add up to 180°

3(15) + 3(15) + y = 180

45 + 45 + y = 180

90 + y = 180

y = 90°

Answer:

[a] y=x^2+3,  vertex, V(0,3)

[b] y=2x^2, vertex, V(0,0)

[c] y=-x^2 +  4, vertex, V(0,4)

[d] y= (1/2)x^2 - 5, vertex, V(0,-5)

Step-by-step explanation:

The vertex, V, of a quadratic can be found as follows:

1. find the x-coordinate, x0,  by completing the square

2. find the y-coordinate, y0, by substituting the x-value of the vertex.

[a] y=x^2+3,  vertex, V(0,3)

y=(x-0)^2 + 3

x0=0, y0=0^2+3=3

vertex, V(0,3)

[b] y=2x^2, vertex, V(0,0)

y=2(x-0)^2+0

x0 = 0, y0=0^2 + 0 = 0

vertex, V(0,0)

[c] y=-x^2 +  4, vertex, V(0,4)

y=-(x^2-0)^2 + 4

x0 = 0, y0 = 0^2 + 4 = 4

vertex, V(0,4)

y = (1/2)(x-0)^2 -5

x0 = 0, y0=(1/2)0^2 -5 = -5

vertex, V(0,-5)

Conclusion:

When the linear term (term in x) is absent, the vertex is at (0,k)

where k is the constant term.

Answer:

The answer is explained below

Step-by-step explanation:

Given that, the equation H*x = c is inconsistent for some c in R^n, we can say that the equation A*x = b has at least one solution for each b in R^n of IMT (Inverse Matrix Theorem) is not fulfilled.

Thanks to this we can say that by equivalence of theorem statement, the equation H*x = 0 will not have only the trivial solution. It will have non-trivial solutions too.

Answer:

The 95% confidence interval for the population mean rating is (5.73, 6.95).

Step-by-step explanation:

We start by calculating the mean and standard deviation of the sample:

[tex]M=\dfrac{1}{n}\sum_{i=1}^n\,x_i\\\\\\M=\dfrac{1}{50}(6+4+6+. . .+6)\\\\\\M=\dfrac{317}{50}\\\\\\M=6.34\\\\\\s=\sqrt{\dfrac{1}{n-1}\sum_{i=1}^n\,(x_i-M)^2}\\\\\\s=\sqrt{\dfrac{1}{49}((6-6.34)^2+(4-6.34)^2+(6-6.34)^2+. . . +(6-6.34)^2)}\\\\\\s=\sqrt{\dfrac{229.22}{49}}\\\\\\s=\sqrt{4.68}=2.16\\\\\\[/tex]

We have to calculate a 95% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=6.34.

The sample size is N=50.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

[tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{2.16}{\sqrt{50}}=\dfrac{2.16}{7.071}=0.305[/tex]

The degrees of freedom for this sample size are:

[tex]df=n-1=50-1=49[/tex]

The t-value for a 95% confidence interval and 49 degrees of freedom is t=2.01.

The margin of error (MOE) can be calculated as:

[tex]MOE=t\cdot s_M=2.01 \cdot 0.305=0.61[/tex]

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

[tex]LL=M-t \cdot s_M = 6.34-0.61=5.73\\\\UL=M+t \cdot s_M = 6.34+0.61=6.95[/tex]

The 95% confidence interval for the mean is (5.73, 6.95).

Answer:

TRUE

Step-by-step explanation:

The circumcenter of a triangle is the center of the only circle that can be circumscribed about it

Answer:

False

Step-by-step explanation:

Answer: 151

Step-by-step explanation:

if prior population proportion is unknown , then the formula is used to find the sample size :

[tex]n=0.25(\frac{z_{\alpha/2}}{E})^2[/tex]

, where [tex]z_{\alpha/2}[/tex] = Two tailed critical value for significance level of [tex]\alpha.[/tex]

E = Margin of error.

Given : margin of error = 8%= .08

For 95% confidence level , two tailed critical value = 1.96

Now, the required sample size :

[tex]n=0.25(\frac{1.96}{0.08})^2\\\\=0.25(24.5)^2\\\\=150.0625\approx151[/tex]

Hence, the size of the sample needed = 151.

Answer:

[tex]2169.67-2.624\frac{48.72}{\sqrt{15}}=2136.66[/tex]    

[tex]2169.67+2.624\frac{48.72}{\sqrt{15}}=2202.68[/tex]    

And the confidence interval would be given by (2137, 2203)

Step-by-step explanation:

2051 ,2061 ,2162 ,2167 , 2169 ,2171 , 2180 , 2183 , 2186 , 2195 , 2196 , 2198 , 2205 , 2210  ,2211

We can calculate the mean and deviation with these formulas:

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)  

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)  

And we got:

[tex]\bar X=2169.67[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean

s=48.72 represent the sample standard deviation

n=15 represent the sample size  

Confidence interval

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)

The degrees of freedom are given by:

[tex]df=n-1=15-1=14[/tex]

Since the Confidence is 0.98 or 98%, the significance is [tex]\alpha=0.02[/tex] and [tex]\alpha/2 =0.01[/tex], and using excel we calculate the critical value [tex]t_{\alpha/2}=2.624[/tex]

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

[tex]2169.67-2.624\frac{48.72}{\sqrt{15}}=2136.66[/tex]    

[tex]2169.67+2.624\frac{48.72}{\sqrt{15}}=2202.68[/tex]    

And the confidence interval would be given by (2137, 2203)

Answer:

2699

Step-by-step explanation:

you do all the multiplication first

500×6= 3000

5 ×50 = 250

so it becomes

3000-51-250 = 2699

Answer:

2699

Step-by-step explanation:

Answer:

3/10

Step-by-step explanation:

We have that the Armans last winter had 5/6 of a row of stacked logs and at the end of the winter he had 8/15 of the same row left, therefore:

Ambitious

First we have to do is that the denominator is the same.

in the case of 5/6 it would be 25/30

and for 8/15 it would be 16/30

Now if we can do the subtraction and it would be:

25/30 - 16/30 = 9/30 or what equals 3/10

3/10 was the amount of wood he burned in the winter

Answer:

D) 3/10 row

Step-by-step explanation:

Answer:

The 95% confidence interval for the proportion of college students who get 8 or more hours of sleep per night is (0.133, 0.161).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 2305, \pi = \frac{339}{2305} = 0.147[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.147 - 1.96\sqrt{\frac{0.147*0.853}{2305}} = 0.133[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.147 + 1.96\sqrt{\frac{0.147*0.853}{2305}} = 0.161[/tex]

The 95% confidence interval for the proportion of college students who get 8 or more hours of sleep per night is (0.133, 0.161).