Angles opposite to one another at the intersection of two lines are called ____ angles.

[tex]f(t)=(t-5)u_2(t)-(t-2)u_5(t)[/tex]

The Laplace transform is

[tex]F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt=\int_2^\infty(t-5)e^{-st}\,\mathrm dt-\int_5^\infty(t-2)e^{-st}\,\mathrm dt[/tex]

Integrate by parts; in the first integral, take

[tex]u=t-5\implies\mathrm du=\mathrm dt[/tex]

[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]

[tex]\implies\displaystyle\int_2^\infty(t-5)e^{-st}\,\mathrm dt=-\frac{e^{-st}}s(t-5)\bigg|_2^\infty+\frac1s\int_2^\infty e^{-st}\,\mathrm dt[/tex]

[tex]=-\dfrac{3e^{-2s}}s-\dfrac{e^{-st}}{s^2}\bigg|_2^\infty[/tex]

[tex]=-\dfrac{3e^{-2s}}s+\dfrac{e^{-2s}}{s^2}=-\dfrac{(3s-1)e^{-2s}}{s^2}[/tex]

For the second integral, take

[tex]u=t-2\implies\mathrm du=\mathrm dt[/tex]

[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]

[tex]\implies\displaystyle\int_5^\infty(t-2)e^{-st}\,\mathrm dt=-\dfrac{(t-2)e^{-st}}s\bigg|_5^\infty+\frac1s\int_5^\infty e^{-st}\,\mathrm dt[/tex]

[tex]=\dfrac{3e^{-5s}}s-\dfrac{e^{-st}}{s^2}\bigg|_5^\infty[/tex]

[tex]=\dfrac{3e^{-5s}}s+\dfrac{e^{-5s}}{s^2}=\dfrac{(3s+1)e^{-5s}}{s^2}[/tex]

So we have

[tex]F(s)=\dfrac{(3s+1)e^{-5s}-(3s-1)e^{-2s}}{s^2}[/tex]

Answer:

Step-by-step explanation:

Hello!

The objective is to test if the purchase frequencies of new-car buyers follow the distribution of the shares of the U.S. automobile market for 1990.

You have one variable of interest:

X: Brand a new-car buyer prefers, categorized: GM, Japanese, Ford, Chrysler and Other

n= 1000

Observed frequencies

GM 193

Japanese 384

Ford 170

Chrysler 90

Other 163

a) The test to use to analyze if the observed purchase frequencies follow the market distribution you have to conduct a Goodness to Fit Chi-Square test.

The conditions for this test are:

- Independent observations

In this case, we will assume that each buyer surveyed is independent of the others.

- For 3+ categories: each expected frequency (Ei) must be at least 1 and at most 20% of the Ei are allowed to be less than 5.

In our case we have a total of 5 categories, 20% of 5 is 1, only one expected frequency is allowed to have a value less than 5.

I'll check this by calculating all expected frequencies using the formula: Ei= n*Pi (Pi= theoretical proportion that corresponds to the i-category)

E(GM)= n*P(GM)= 1000*0.36= 360

E(Jap)= n*P(Jap)= 1000*0.26= 260

E(Ford)= n*P(Ford)= 1000*0.21= 210

E(Chrys)= n*P(Chrys)= 1000*0.09= 90

E(Other)= n*P(Other)= 1000*0.08= 80

Note: If all calculations are done correctly then ∑Ei=n.

This is a quick way to check if the calculations are done correctly.

As you can see all conditions for the test are met.

b) The hypotheses for this test are:

H₀: P(GM)= 0.36; P(Jap)= 0.26; P(Ford)= 0.21; P(Chrys)= 0.09; P(Other)= 0.08

H₁: At least one of the expected frequencies is different from the observed ones.

α: 0.05

[tex]X^2= sum \frac{(Oi-Ei)^2}{Ei} ~~X^2_{k-1}[/tex]

k= number of categories of the variable.

This test is one-tailed right this mean you'll reject the null hypothesis to high values of X²

[tex]X^2_{k-1;1-\alpha }= X^2_{4;0.95}= 9.488[/tex]

Decision rule using the critical value approach:

If [tex]X^2_{H_0}[/tex] ≥ 9.488, reject the null hypothesis

If [tex]X^2_{H_0}[/tex] < 9.488, don't reject the null hypothesis

[tex]X^2_{H_0}= \frac{(193-360)^2}{360} + \frac{(384-260)^2}{260} + \frac{(170-210)^2}{210} + \frac{(90-90)^2}{90} + \frac{(163-80)^2}{80} = 230.34[/tex]

The value of the statistic under the null hypothesis is greater than the critical value, so the decision is to reject the null hypothesis.

Using a 5% level of significance, there is significant evidence to conclude that the current market greatly differs from the preference distribution of 1990.

The chi-square test can be used to determine if there is a significant difference between the observed frequencies and the expected frequencies based on the shares of the U.S. automobile market in 1990. A calculated chi-square statistic greater than the critical value would indicate a significant difference between the current market shares and those of 1990.

The first step in the process is to calculate the expected counts for each category based on the shares of the U.S. automobile market in 1990. The expected counts for GM, Japanese manufacturers, Ford, Chrysler, and other amounts to 360, 260, 210, 90, and 80, respectively.

Given the observed frequencies, the chi-square statistic X² can be calculated. The formula for the chi-square test is X² = Σ[(O-E)²/E], where O represents the observed frequency and E represents the expected frequency. Using this formula, the chi-square statistic is calculated.

Finally, using a chi-square distribution table with 4 degrees of freedom (5 categories - 1), and an alpha level of 0.05, you can compare the calculated chi-square statistic to the critical value (9.488). If the chi-square statistic is larger than the critical value, you would reject the null hypothesis and conclude that the current market shares do differ from those of 1990. If the chi-square statistic is smaller than the critical value, you would not reject the null hypothesis and conclude that the current market shares do not significantly differ from those of 1990.

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same size...hope this helps :)))))

Answer:

Option D) field notes recorded during participant observation.

Step-by-step explanation:

We are given the following in the question:

Qualitative data and quantitative data:

A) countywide census of speakers of more than one language.

This is a quantitative data as it can be expressed in numbers. The census would give a numerical value of speakers of more than one language.

B) average community income levels, by block.

This is a quantitative data. Average income levels are expressed in numbers.

C) ethnic composition of a community, by percentage.

Again it is quantitative data as expressed in numbers.

D) field notes recorded during participant observation.

Thus is an example of qualitative data as field notes are expressed in name, text.

Thus, the best example of qualitative data is

Option D) field notes recorded during participant observation.

The best example of qualitative data among the choices is field notes recorded during participant observation. These notes are subjective descriptions of experiences, offering insights into the observed party's behaviors and interactions, which are qualitative in nature.

Qualitative data is a type of data that is non-numerical and used to understand concepts, thoughts, or experiences. It goes beyond counts or numbers to capture the nature, quality, or meaning of a thing. Out of the options provided, D) field notes recorded during participant observation are the best example of qualitative data. The field notes are subjective descriptions of observations and experiences, rather than precise measurements or calculations. The other options (A to C) represent quantitative data, which is information about quantities and hard numbers.

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Answer:

The answer is  D.

Step-by-step explanation:

Answer:

Only truck 1 can pass under the bridge.

Step-by-step explanation:

So, first of all, we must do a drawing of what the situation looks like (see attached picture).

Next, we can take the general equation of an ellipse that is centered at the origin, which is the following:

[tex] \frac{x^2}{a^2}+\frac{y^2}{b^2}[/tex]

where:

a= wider side of the ellipse

b= shorter side of the ellipse

in this case:

[tex] a=\frac{38}{2}=19ft[/tex]

and

b=12ft

so we can go ahead and plug this data into the ellipse formula:

[tex] \frac{x^2}{(19)^2}+\frac{y^2}{(12)^2}[/tex]

and we can simplify the equation, so we get:

[tex] \frac{x^2}{361}+\frac{y^2}{144}[/tex]

So, we need to know if either truk will pass under the bridge, so we will match the center of the bridge with the center of each truck and see if the height of the bridge is enough for either to pass.

in order to do this let's solve the equation for y:

[tex] \frac{y^{2}}{144}=1-\frac{x^{2}}{361}[/tex]

[tex] y^{2}=144(1-\frac{x^{2}}{361})[/tex]

we can add everything inside parenthesis so we get:

[tex] y^{2}=144(\frac{361-x^{2}}{361})[/tex]

and take the square root on both sides, so we get:

[tex] y=\sqrt{144(\frac{361-x^{2}}{361})}[/tex]

and we can simplify this so we get:

[tex] y=\frac{12}{19}\sqrt{361-x^{2}}[/tex]

and now we can evaluate this equation for x=4 (half the width of the trucks) so:

[tex] y=\frac{12}{19}\sqrt{361-(8)^{2}}[/tex]

y=11.73ft

this means that for the trucks to pass under the bridge they must have a maximum height of 11.73ft, therefore only truck 1 is able to pass under the bridge since truck 2 is too high.

Truck 1 and Truck 2 can both pass under the bridge.

To determine which truck can pass under the bridge, we need to compare the height of the truck to the height of the arch over the center of the roadway. The height of the arch is given as 12 feet. Truck 1 has an overall height of 11 feet, so it can pass under the bridge. Truck 2 has an overall height of 12 feet, which is equal to the height of the arch, so it can also pass under the bridge.

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Answer:

can i getttt a pictture

Step-by-step explanation:

Answer:

  4x^2−12x+9

Step-by-step explanation:

The form of a perfect square trinomial is ...

  (a +b)² = a² +2ab +b²

The first and last terms must be positive and perfect squares. The middle term must be twice the product of their roots (possibly with a minus sign).

  x^2 -10x -25 . . . . -25 is not a positive perfect square

  4x^2 -12x +9 . . . . 12x = 2√(4x^2·9) = 2·6x . . . . perfect square

  9x^2 +12x +16 . . . 12x ≠ 2√(9x^2·16) = 2·12x

  16x^2 +16x +1 . . .  16x ≠ 2√(16x^2·1) = 2·4x

At noon, the Shady Farm Milk Company has 10,000 gallons of milk in the queue to be processed given the demand and the processing rate.

The Shady Farm Milk Company can process 7500 gallons of milk per hour. Given that the company operates from 8 a.m. to 6 p.m., this is a total of 10 hours of operation in a day. Therefore, in 10 hours, the company can process 7500 × 10 = 75,000 gallons of milk.

However, the demand for milk is 100,000 gallons over the course of the day. Therefore, by noon, the company has been operating for 4 hours, meaning they can process 7500 × 4 = 30,000 gallons.

The demand over the same 4 hours period (from 8 a.m. to noon) is calculated by dividing the total demand over the entire course of the day (which is evenly spread) by the number of operating hours. Thus: 100,000 / 10 = 10,000 gallons/hour.

Consequently, the demand from 8 a.m. to noon is: 10,000 × 4 = 40,000 gallons. So, the amount of milk in the queue at noon would be the demand minus what the company has processed at that time.

Hence: 40,000 (demand from 8 a.m. to noon) - 30,000 (processed milk from 8 a.m. to noon) = 10,000 gallons. Therefore, at noon, the company has 10,000 gallons of milk in the queue to be processed.

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At noon, there are 20,000 gallons of milk in the queue to be processed.

To find out how many gallons of milk are in the queue to be processed at noon, we first need to calculate how many gallons of milk have been processed by noon.

The company can process milk at a fixed rate of 7500 gallons per hour. From 8 a.m. to noon, there are 4 hours.

Total gallons processed by noon = Rate of processing Time

[tex]\[ = 7500 \, \text{gallons/hour} \times 4 \, \text{hours} = 30000 \, \text{gallons} \][/tex]

Now, we need to find out how many gallons of milk are still in demand by noon. The total demand over the course of the day is 100,000 gallons, and it is spread out uniformly from 8 a.m. to 6 p.m.

This means that by noon, half of the day has passed.

So, the total demand by noon = Total demand / 2

[tex]\[ = \frac{100000}{2} = 50000 \, \text{gallons} \][/tex]

Now, to find out how many gallons are in the queue to be processed at noon, we subtract the gallons already processed from the total demand:

Gallons in the queue at noon = Total demand by noon - Gallons processed by noon

[tex]\[ = 50000 \, \text{gallons} - 30000 \, \text{gallons} = 20000 \, \text{gallons} \][/tex]

So, at noon, there are 20,000 gallons of milk in the queue to be processed.

The population proportion of people from Hawaii who exercised for at least 30 minutes a day three days a week, according to the 2012 Gallup survey, is 0.622.

The question is seeking to find the population proportion of people from Hawaii who said they exercised for at least 30 minutes a day for at least three days in a week. This is given directly in the data from the survey by Gallup in 2012. The population proportion is a fundamental concept in statistics and refers to the fraction of individuals in a group, or population, possessing a certain characteristic. In this case, the characteristic is exercising for at least 30 minutes a day for three days in a week.

From the data given, we know that from a random sample of 100 respondents from Hawaii, 62.2% said yes to the question posed by the survey. In terms of population proportion, this can be expressed as 0.622. This means that if we were to select one person at random from the population of Hawaii, the probability that this person exercises for at least 30 minutes a day, three days a week is estimated to be 0.622.

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The interest for 8 years is $2.10

Step-by-step explanation:

Principal amount (p) = $6.57

Rate of interest (r) = 4%

Time (t) = 8 years

Interest = (p x r x t) /100

= (6.57 x 4 x 8) /100

= 210.24/100

= 2.10

The interest for 8 years is $2.10

Answer:

35 and 15

Step-by-step explanation:

Process of elimination:

Try 105 and 5 - 105 is way too much and 5 is way too little. We know that they are between 5 and 105.

Try 50 and 10.5 - This is closer, but 50 is still too much and 10.5 is too little. We now know that the two numbers are between 10.5 and 50

Try 30 and 17.5 - This is very close, but now 30 is not enough and 17.5 is too much. We now know that the smaller number is between 10.5 and 17.5 as well as the larger number is between 30 and 50.

Try 40 and 13.125 - This is also very close, but 40 is too much and 13.125 is too little. We now know that the lower number is between 13.125 and 17.5 as well as the larger number is between 30 and 40.

Try 35 and 15 - Success: 35 and 5 make the simplified ratio of 7:3! They also are products of 525!

Final answer:

The two positive numbers with a ratio of 7 to 3 and a product of 525 are found to be 35 and 15, by assigning a constant to the numbers, forming and solving a quadratic equation to find the value of the constant, and then using it to find the two numbers.

Explanation:

To find two positive numbers where their ratio is 7 to 3, and their product is 525, we can assign values x and y to the numbers such that x/y = 7/3, and x*y = 525. We can solve these equations simultaneously to find the values of x and y.

Step-by-step solution:

Therefore, the two numbers are 35 and 15.

Answer: [tex]3\frac{1}{2}[/tex]

Multiply

[tex]3.5/1*10/10=35/10[/tex]

Divide each side by 5

[tex]35/10[/tex]  ÷  [tex]5/5=7/2[/tex]

[tex]7/2=3\frac{1}{2}[/tex]

Answer:

[tex] \frac{3.5}{1} \frac{ \times }{ \times } \frac{10}{10} = \frac{35}{10} \\ \frac{35}{10} \frac{ \div }{ \div } \frac{5}{5} \\ = \frac{7}{2} = 3 \frac{1}{2} [/tex]

hope this helps you...

Answer:

Check below

Step-by-step explanation:

That's too bad you haven't attached a rectangle.

Here's an example, with the data you've typed in.

1) When we dilate a rectangle we either grows it or shrink it through a scale factor.

Check the first picture below.

The New Dilated Rectangle A'B'C'D' will follow its coordinates, when the Center of Dilation is at its origin(Middle):

[tex]D_{A'B'C'D'}=\frac{1}{2}(x,y)[/tex]

2) But In this question, B is the center of Dilation. So, Since B is the Dilation Point B=B' .  And More importantly:

[tex]\bar{AB}=\frac{1}{2}\bar{A'B'}\\\bar{CD}=\frac{1}{2}\bar{C'D'}\\\bar{AC}=\frac{1}{2}\bar{A'C'}\\\bar{CD}=\frac{1}{2}\bar{C'D'}\\[/tex]

3) So check the pictures below for a better understanding.

Answer:

Pretty sure its B

Answer:

Option D) 0.014

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 1200

Sample proportion =

[tex]\hat{p} = 0.47[/tex]

We have to make a 95% confidence interval.

Formula for standard error:

[tex]S.E = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

Putting the values, we get:

[tex]S.E = \sqrt{\dfrac{0.47(1-0.47)}{1200}} = 0.014[/tex]

Thus, the correct answer is

Option D) 0.014

Step-by-step explanation:

4x + 7° = 5x - 1°

5x - 4x = 7° + 1°

x = 8°

As OR~=OS

Answer:

The answer to the last one is regular

Step-by-step explanation:

A polygon is a planar figure defined by a finite number of straight-line segments. The polygon that best describes a stop sign is a regular polygon.

A polygon is a planar figure defined by a finite number of straight-line segments that are joined to form a closed polygonal chain in geometry. A polygon can be defined as a bounded planar region, a bounding circuit, or both.

The polygon that best describes a stop sign is a regular octagon because a stop sign has 8 sides and the length of each side is the same, therefore, the polygon will be a regular polygon.

Thus, the polygon that best describes a stop sign is a regular polygon.

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Answer:

Period T of the given function f(x) = sin(2x) + cos(4x)

   =  π

Step-by-step explanation:

Given that y(x) is a sum of two trigonometric functions. The period T of sin 2x would be (2π÷2) = π. Period T of cos4x would be (2π÷4) that is π/2

Find the LCM of π and π/2 . That would be  π. Hence the period of the given function would be π

The fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex].

To determine the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \)[/tex], we need to find the smallest positive value of ( T ) such that [tex]f(x + T) = f(x) \) for all \( x \).\\[/tex]
Let's start by analyzing the periods of the individual components of [tex]\( f(x) \).[/tex]
1. Period of [tex]\( \sin(2x) \):[/tex]
The standard period of [tex]\( \sin(x) \) is \( 2\pi \). For \( \sin(2x) \),[/tex] the argument ( 2x ) scales the period. To find the period of [tex]\( \sin(2x) \)[/tex], we set:
[tex]\[ 2x = 2x + 2\pi \]\\[/tex]

Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{2} \][/tex]
Thus, the period of [tex]\( \sin(2x) \)[/tex] is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
2. Period of [tex]\( \cos(4x) \):[/tex]
The standard period of [tex]\( \cos(x) \) is \( 2\pi \). For \( \cos(4x) \)[/tex], the argument ( 4x ) scales the period. To find the period of [tex]\( \cos(4x) \),[/tex] we set:
[tex]\[ 4x = 4x + 2\pi \][/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{4} \][/tex]
Thus, the period of [tex]\( \cos(4x) \)[/tex]is:
[tex]\[ \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Next, we need to find the smallest positive ( T ) such that both [tex]\( \sin(2x) \)[/tex] and [tex]\( \cos(4x) \)[/tex] have the same period ( T ). This means that ( T ) must be a common multiple of the periods of the two components, [tex]\( \pi \)[/tex] and [tex]\( \frac{\pi}{2} \).[/tex]
To find the fundamental period ( T ), we determine the least common multiple (LCM) of [tex]\( \pi \) and \( \frac{\pi}{2} \):[/tex]
- [tex]\( \pi \)[/tex]can be written as [tex]\( \pi \times 1 \).[/tex]
- [tex]\( \frac{\pi}{2} \)[/tex] can be written as [tex]\( \pi \times \frac{1}{2} \).[/tex]
The LCM of [tex]\( 1 \) and \( \frac{1}{2} \) is \( 1 \) since \( 1 \)[/tex] is the smallest number that both [tex]\( 1 \) and \( \frac{1}{2} \)[/tex] can divide without leaving a remainder.
Thus, the LCM of [tex]\( \pi \) and \( \frac{\pi}{2} \) is:[/tex]
[tex]\[ \text{LCM}\left(\pi, \frac{\pi}{2}\right) = \pi \][/tex]
Therefore, the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex]

Answer:

Step-by-step explanation:

A translation 3 units left and 2 units up

A translation 3 units left and 2 units up best describes the graph of  f(x) = log2(x + 3) + 2 as a transformation of the graph of g(x) = log2x

The following steps are shown to describe the graph.

The general equation of f(x) = log2(x-h)+k

The graph of the base of the function shift to the right

The graph of the base function shifts to the left.

The graph of the base function shifts upward.

 The graph of the base function shifts downward

Here h = 3 , k = 2

Hence , a translation 3 units left and 2 units up describes the graph.

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Answer:

6

Step-by-step explanation:

Answer:

2x^2-9x-5

Step-by-step explanation:

Just multiply

Answer:

2x^2-9x-5

Step-by-step explanataion:

multiply

Answer:

g(6)= 6^2= 36

2(36)+3= 72+3= 75

Answer:

The volume is 192.96

Step-by-step explanation:

You multiply length x width x height to find the volume.

Answer:

5.2161,7.5,7.508,7.58

Step-by-step explanation:

Answer:

5.2161; 7.5; 7.508; 7.58

Step-by-step explanation:

-First you will look at the first digit of each number given, obviously 5 is the smallest so it will go first

-Then you will check the following number after the first digit to see which one is smaller of the "7" values. In this case, they all have the number "5" has their second value, so we move on to the third digit.

-Since "7.5" does not have a third digit, it will be the smallest of the number "7" values.

-We then have a remaining of two values, 7.508 and 7.58. Still looking at the third numbers of each of the two, we see "0" and "8," and obviously 0 is smaller than 8, so 7.508 is smaller than 7.58

       Equation representing the cups of flour for every teaspoon of yeast will be → c = 2t

          y ∝ x

          y = kx [Here, k = proportionality constant]

It has been given in the question,

"A bread recipe calls for 1 teaspoon of yeast for every 2 cups of flour."

If the number of teaspoons of yeast is represented by 't' and number of cups of flour by 'c',

c = kt

From the given statement,

2 = k × 1

k = 2

    Therefore, equation for the given statement will be → c = 2t

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The correct option is B because the discriminant is [tex]-4[/tex], so the equation has no real solutions.

Given:

The equation is:

[tex]0=x^2-4x+5[/tex]

To find:

The discriminant of the given equation.

Explanation:

In a quadratic equation [tex]ax^2+bx+c=0[/tex], the discriminant is:

[tex]D=b^2-4ac[/tex]

If [tex]D>0[/tex], then the equation has 2 real solutions.

If [tex]D=0[/tex], then the equation has 1 real solution.

If [tex]D<0[/tex], then the equation has no real solutions.

In the given equation, we have [tex]a=1,b=-4,c=5[/tex].

[tex]D=(-4)^2-4(1)(5)[/tex]

[tex]D=16-20[/tex]

[tex]D=-4[/tex]

Since [tex]D<0[/tex], therefore the equation has no real solutions.

Hence, the correct option is B.

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Answer:

Estimate of the standard error of the mean = 0.38 lb

Step-by-step explanation:

We are given the following in the question:

Sample mean, [tex]\bar{x}[/tex] = 10.87 lb

Sample size, n = 131

Standard deviation, σ = 4.31 lb

We have ti find the estimate of the standard error of the mean.

Formula for standard error:

[tex]S.E= \dfrac{\sigma}{\sqrt{n}}[/tex]

Putting values, we get,

[tex]S.E = \dfrac{4.31}{\sqrt{131}} = 0.3766 \approx 0.38[/tex]

0.38 lb is the standard error of the mean.

Answer:

Please see the attached file for the complete answer.

Step-by-step explanation:    

Please use mean = 3.5 instead of 3.1 and varience = 0.015 instead of 0.018 in the solution.

Step-by-step explanation:

Total books sold = 412

So if there are 103 maths books then biology books should be three times = 103 × 3 = 309

So when we add 103 + 309 = 412

It means the number of biology books is 309 and maths book is 103

Answer:

Step-by-step explanation:

We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean

For the null hypothesis,

µ = 7463

For the alternative hypothesis,

µ ≠ 7463

This is a 2 tailed test

Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is

z = (x - µ)/(σ/√n)

Where

x = sample mean

µ = population mean

σ = population standard deviation

n = number of samples

From the information given,

µ = 7463 hours

x = 7163 hours

σ = 1080 hours

n = 81

b) z = (7163 - 7463)/(1080/√81) = - 2.5

Looking at the normal distribution table, the probability corresponding to the z score is 0.02

Recall, population mean is 7463

The difference between sample sample mean and population mean is 7463 - 7163 = 300

Since the curve is symmetrical and it is a two tailed test, the x value for the left tail is 7463 - 300 = 7163

the x value for the right tail is 7463 + 300 = 7763

These means are higher and lower than the null mean. 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. We already got the area below z as 0.02

We would double this area to include the area in the right tail of z = 2.5. Thus

p = 0.02 × 2 = 0.04

It means that in a sample of size 81 light bulbs, we would observe a sample mean of 300 hours or more away from 7463 about 4% of the time by chance alone.

c) Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Margin of error = z × σ/√n

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.95 = 0.05

α/2 = 0.05/2 = 0.025

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.025 = 0.975

The z score corresponding to the area on the z table is 1.96. Thus, z score for 95% confidence level is 1.96

Margin of error = 1.96 × 1080/√81

= 235.2

Confidence interval = 7163 ± 23.2

a) Since alpha, 0.05 > than the p value, 0.04, then we would reject the null hypothesis. Therefore, at a 5% level of significance, there is evidence that the mean life is different from 7463 hours

Comparing the results of a and c, it is true that the population mean life is not 7463 hours.