As theta increases from -theta/2 to 0 radians, the value of cos theta will
1. decrease form 1 to 0
2. decrease from 0 to -1
3. increase from -1 to 0
4. increase from 0 to 1

The equation -9x - 3y^2 = 6 does not represent a function of x because each x-value has two possible y-values.

In order to determine if an equation represents a function of x, we need to check if each value of x produces a unique y-value. This means that for any x-value, there cannot be more than one y-value.

When we rearrange the given equation, we can solve for y in terms of x: y = sqrt((-9x - 6)/3). Since the square root of a positive number always has two possible values (positive and negative), we have two possible y-values for each x-value. Therefore, the given equation does not represent a function of x.

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Answer:

a) [tex]P(X<3)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-5.1}{0.9})=P(z<-2.33)[/tex]

And we can find this probability using the normal standard table and we got:

[tex]P(z<-2.33)=0.010[/tex]

b) [tex]P(X>7)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z>\frac{7-5.1}{0.9})=P(z>2.11)[/tex]

And we can find this probability using the complement rule and the  normal standard table and we got:

[tex]P(z>2.11)=1-P(Z<2.11) = 1-0.983 = 0.017[/tex]

Step-by-step explanation:

Previous concepts

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".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(5.1,0.9)[/tex]  

Where [tex]\mu=5.1[/tex] and [tex]\sigma=0.9[/tex]

Part a

We are interested on this probability

[tex]P(X<3)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X<3)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-5.1}{0.9})=P(z<-2.33)[/tex]

And we can find this probability using the normal standard table and we got:

[tex]P(z<-2.33)=0.010[/tex]

Part b

We are interested on this probability

[tex]P(X>7)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X>7)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z>\frac{7-5.1}{0.9})=P(z>2.11)[/tex]

And we can find this probability using the complement rule and the  normal standard table and we got:

[tex]P(z>2.11)=1-P(Z<2.11) = 1-0.983 = 0.017[/tex]

According to the American Red Cross, 9.2% of all Connecticut residents have Type B blood. If 18 Connecticut residents are taken at random, the standard deviation of the binomial distribution is about 1.214.

The question is asking for the standard deviation of a binomial distribution. To calculate the standard deviation for a binomial distribution, the formula used is √npq, where n is the number of trials, p is the probability of success, and q is the probability of failure (1 - p).

In this case, we have n equals to 18 (number of Connecticut residents randomly sampled), p equals to 0.092 (since 9.2% have Type B blood), and then, by subtraction, q equals to 0.908.

Therefore, the standard deviation would be √18*0.092*0.908, which is about 1.214 when evaluated.

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The standard deviation of the random variable X is approximately 0.067.

The standard deviation of the random variable X can be calculated using the formula:

Standard Deviation (SD) = sqrt(p(1-p)/n)

Given that p is the probability of success (9.2% or 0.092) and n is the sample size (18), we can substitute these values into the formula to find:

SD = sqrt(0.092(1-0.092)/18) = 0.067

Therefore, the standard deviation of the random variable X is approximately 0.067.

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Answer:

13

Step-by-step explanation:

Since Lupe is 3 times as many as Frances

39 ÷ 3= 13

Answer:

13

Step-by-step explanation:

Make an equation. Let x represent the bags that Frances sold. Since Lupe sold 3 times the popcorn Frances did, use multiplication.

[tex]39=3x[/tex]

Solve for x to find the amount Frances sold:

Divide both sides by 3:

[tex]\frac{39}{3}=\frac{3x}{3} \\\\13=x[/tex]

[tex]x=13[/tex]

Frances sold 13 bags.

Answer:

0.15

Step-by-step explanation:

To subtract the matrices [5 0 - 2] and [-1 3 8], subtract each corresponding element, resulting in the matrix [6 -3 -10].

The question asks for the result of the subtraction of two matrices: [5 0 - 2] minus [-1 3 8]. When we subtract matrices, we subtract the corresponding elements from each matrix. Therefore, this subtraction will result in a new matrix where each element is the subtraction of the corresponding elements from the given matrices.

To perform the subtraction, we subtract each element in the second matrix from the corresponding element in the first matrix:

So, the result of the matrix subtraction is [6 -3 -10].

Answer:

The point estimate for p is 0.5637.

Step-by-step explanation:

Point estimate for the proportion of all judges who are introverts.

We have a big sample.

So the proportion of the sample proportion of introverted judges can be used as the estimation of the population proportion.

In the sample:

502 judges

283 introverts.

So

p = 283/502 = 0.5637

The point estimate for p is 0.5637.

The point estimate for the proportion of judges who are introverts, denoted by p, is found by dividing the number of introverts (283) by the total number of judges in the sample (502), which gives a point estimate of 0.5637 when rounded to four decimal places.

To estimate the proportion of judges who are introverts, we use the given data from the random sample. The point estimate for p, representing the proportion of all judges who are introverts, can be calculated by dividing the number of introverts in the sample by the total number of judges in the sample. The formula for the point estimate p' is:

p' = X / n

where:

Using the given data:

p' = 283 / 502

Calculating this gives us a point estimate for p' of 0.5637 after rounding to four decimal places.

This result represents the estimated proportion of the population of judges who are introverts based on the sample.

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Answer:

a function

Step-by-step explanation:

a function is where u plug something in and get another thing out. the entire function is dependent on another independent variable

Answer: The top of the ladder is moving towards the floor at a rate of 0.75 ft/sec

Step-by-step explanation: Please see the attachments below

Answer:

between 120 and 153 mph in race A between 125 and 145 mph in race B

Step-by-step explanation:

i got it right on my test

Answer:

its

C.between 120 and 153 mph in race A; between 125 and 145 mph in race B

Step-by-step explanation:

Answer:

The student who did the German test scored 2 standard deviations above the mean and the student who did the French test scored 1.6 standard deviations above the mean. Relative to their classmates, the student who did the German test scored better due to the higher z-score.

Step-by-step explanation:

Z-score

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

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

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

German test

Mean was 66 and the standard deviation was 8, scored an 82.

So

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

[tex]Z = \frac{82 - 66}{8}[/tex]

[tex]Z = 2[/tex]

French test:

Mean was 27 and the standard deviation was 5, scored a 35.

So

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

[tex]Z = \frac{35 - 27}{5}[/tex]

[tex]Z = 1.6[/tex]

The student who did the German test scored 2 standard deviations above the mean and the student who did the French test scored 1.6 standard deviations above the mean. Relative to their classmates, the student who did the German test scored better due to the higher z-score.

Final answer:

The German test taker scored 2 standard deviations above the mean, while the French test taker scored 1.6 standard deviations above the mean. Thus, the German test taker scored comparatively higher relative to their peers.

Explanation:

To compare the scores of the two high school students who took equivalent language tests in German and French, we need to calculate the z-scores for each. The z-score is a measure that describes how far an individual test score is from the mean of the respective test, in terms of standard deviations. Let's calculate:

For the German test:

ZGerman = (Score - Mean) / Standard Deviation

ZGerman = (82 - 66) / 8

ZGerman = 16 / 8

ZGerman = 2.0

For the French test:

ZFrench = (Score - Mean) / Standard Deviation

ZFrench = (35 - 27) / 5

ZFrench = 8 / 5

ZFrench = 1.6

Comparing these z-scores, the student who took the German test scored higher relative to their peers (2 standard deviations above the mean) compared to the student who took the French test (1.6 standard deviations above the mean).

Answer:

Timmy is 18 and Tommy is 5.

Step-by-step explanation:

Step-by-step explanation:

So you have to find out tha ages which add up to 23 so if Tommy is 9 years old and when we add 5

So Timmy age = 9 + 5 = 14

Now if we add 14 and 9 the answer is 23

It means Timmy is 14 years old and Tommy is 9 years old

The question deals with the mechanism of reactions between ketones and amines, acid-base reactions, and alkylation reactions. The primary step of a ketone-amine reaction involves the nitrogen attacking the carbonyl carbon to eventually form an imine or enamine. The acid-base reaction between acetic acid and ammonia results in the ammonium and acetate ions.

The student's question is about organic reaction mechanisms, specifically the reactions between ketones and amines, and other related reactions such as acid-base reactions between acetic acid and ammonia. In the case of a ketone reacting with a primary or secondary amine, the first step typically involves nucleophilic attack by the nitrogen of the amine on the electrophilic carbonyl carbon of the ketone. This produces a tetrahedral intermediate which, after losing a water molecule, forms an imine or an enamine depending on whether a primary or secondary amine was used respectively.

For the acid-base reaction between acetic acid (CH3CO2H) and ammonia (NH3), the mechanism begins with the lone pair on the nitrogen of ammonia attacking the hydrogen of acetic acid. This is shown with a curved arrow from the nitrogen to the hydrogen. This results in the formation of the ammonium ion (NH4+) and the acetate ion (CH3CO2-).

The alkylation reaction at the alpha-carbon of a ketone or aldehyde and the concept of kinetic versus thermodynamic control are also touched upon. In an alkylation reaction, the first step usually involves the generation of an enolate ion from the ketone or aldehyde, followed by its nucleophilic attack on an alkyl halide.

Answer:x=-15

Step-by-step explanation:You want to isolate x so u would subtract 7 from one side and subtract 7 from the other side

X+7=-8

X=-15

Answer:

B. Information from the sample is typical of information in the population.

Step-by-step explanation:

In Statistical Research, a Bias refers to the tendency of a sample statistic to systematically overestimate or underestimate a population parameter.

(A)When portions of the population are excluded or underrepresented from the sample, it is a Sampling Bias.

(C)When those responding to a survey or poll differ systematically from the nonrespondents, it is referred to as Non-Response Bias.

(D)When certain groups in the population are systematically underrepresented in the sample, it is referred to as Selection Bias.

Therefore option B is not an example of a Bias.

Final answer:

The correct answer is "Information from the sample overemphasizes a particular stratum of the population."

Explanation:

Bias in research refers to systematic errors or deviations from the true population value. It can occur at various stages, including during sampling, data collection, analysis, and interpretation. The options provided describe different forms of bias commonly encountered in research:

1. Exclusion or underrepresentation of portions of the population from the sample is known as sampling bias. This can lead to results that do not accurately reflect the characteristics of the entire population.

2. Overemphasizing a particular stratum of the population is an example of selection bias. This occurs when certain groups within the population are disproportionately represented in the sample, leading to skewed or inaccurate results.

3. Differences between respondents and nonrespondents in a survey or poll can lead to nonresponse bias. This occurs when those who choose to respond to a survey differ systematically from those who do not respond, leading to results that may not be generalizable to the entire population.

4. Systematic underrepresentation of certain groups in the sample can also lead to bias. This is similar to sampling bias but specifically refers to the consistent exclusion or underrepresentation of certain demographic groups.

It is important to recognize and address bias in research to ensure the validity and reliability of the findings. Researchers can use various techniques such as random sampling, stratified sampling, and sensitivity analysis to mitigate bias and improve the quality of their research.

Answer:

a) Transition matrix:

[tex]\left[\begin{array}{ccc}0.90&0.10\\0.25&0.75\end{array}\right][/tex]

b) The long-term fraction of downtime of the computer is 0.286 or 28.6%.

c) The probability of being down 10 hours from now is independent of the inital state and is equal to 0.286.

Step-by-step explanation:

We have two states for the computer: Up and Down.

The rows will represent the actual state and the column the next state, and the numbers within the matrix will be the probabilities of transition from the state of the row to the state of the column.

- If the computer is Up, there is a probability of 0.9 of being Up in the next hour. Then, there is a probability of 0.1 of being Down in the next hour.

- If the computer is Down, there is a probability of 0.75 of being Down in the next hour. Then, there is a probability of 0.25 of being Up in the next hour.

a) The transition matrix becomes:

[tex]\left[\begin{array}{ccc}&U&D\\U&0.90&0.10\\D&0.25&0.75\end{array}\right][/tex]

b) We can consider that we have a long-term state (stable) [πU, πD] when the fraction of each state does not change. This can be expressed as:

[tex]\left[\begin{array}{ccc}\pi_U&\pi_D\end{array}\right] *\left[\begin{array}{ccc}0.90&0.10\\0.25&0.75\end{array}\right]=\left[\begin{array}{ccc}\pi_U&\pi_D\end{array}\right][/tex]

If we develop this multiplication of matrix we get:

[tex]0.90\pi_U+0.25\pi_D=\pi_U\\\\0.10\pi_U+0.75\pi_D=\pi_D[/tex]

As this equations are linear combinations of each other, we need another equation to solve this.

We also know that the sum of the fractions of uptime and downtime is equal to one.

Solving these equation, we can calculate the long-term downtime fraction:

[tex]0.90\pi_U+0.25\pi_D=\pi_U\\\\0.25\pi_D=(1-0.90)\pi_U=0.10\pi_U\\\\\pi_U=(0.25/0.10)\pi_D=2.5\pi_D\\\\\\\pi_D+\pi_U=1\\\\\pi_D+2.5\pi_D=1\\\\3.5\pi_D=1\\\\\pi_D=1/3.5=0.286[/tex]

The long-term fraction of downtime of the computer is 0.286 or 28.6%.

c) To know what is the probability that it will be down 10 hours from now if the computer is now on, we have to compute the transition matrix for 10 hours. This is:

[tex]T^{10}=\left( \left[\begin{array}{ccc}0.90&0.10\\0.25&0.75\end{array}\right]\right)^{10}= \left[\begin{array}{ccc}0.714&0.286\\0.714&0.286\end{array}\right][/tex]

This is considered a steady state already.

If the computer is up, the actual state is [1, 0].

If we multiply this by the transition matrix, we get:

[tex]\left[\begin{array}{ccc}1&0\end{array}\right] *\left[\begin{array}{ccc}0.714&0.286\\0.714&0.286\end{array}\right]=\left[\begin{array}{ccc}0.714&0.286\end{array}\right][/tex]

The probability of being down 10 hours from now is independent of the inital state and is equal to 0.286.

Tonisha must make $114 in daily sales to cover her $34 expenses and meet her $80 profit goal. This is a basic mathematics problem related to costs, revenues, and profits within a business context.

The subject of Tonisha’s lemonade stand involves determining the required earnings to achieve a certain profit goal, which falls under the subject of Mathematics. Specifically, it involves basic arithmetic and understanding of costs and revenues within a business context. To calculate how much Tonisha needs to make in sales to achieve her goal of at least $80 per day, we must add the expenses of $34 to the desired profit of $80. Therefore, Tonisha needs to make $114 per day in sales to cover expenses and meet her profit goal.

The standard matrix of the transformation T, resulting from a composition of the reflection through the vertical x2-axis and the reflection through the line x2 = x1, is: [ 0 -1 ] [ 1 0 ].

Find the standard matrix of the transformation T.

To do this, we can break the transformation T down into two simpler transformations:

Reflection through the vertical x2-axis: This transformation negates the x1 coordinate of each point while leaving the x2 coordinate unchanged. The standard matrix for this transformation is:

[ -1   0 ]

[  0   1 ]

Reflection through the line x2 = x1: This transformation swaps the x1 and x2 coordinates of each point. The standard matrix for this transformation is:

[ 0   1 ]

[ 1   0 ]

Since we are performing these transformations one after the other, we need to multiply the matrices together. The order of multiplication matters here, because matrix multiplication is not commutative. So, the standard matrix for the combined transformation T is:

[ 0   1 ]   [ -1   0 ]   =   [ 0 -1 ]

[ 1   0 ]   [  0   1 ]     [ 1  0 ]

Therefore, the standard matrix of the transformation T is:

[ 0 -1 ]

[ 1  0 ]

Complete question:

Assume that T is a linear transformation. Find the standard matrix of T.

​T:

set of real numbers R squaredℝ2right arrow→set of real numbers R squaredℝ2

first reflects points through the

vertical x 2 dash axisvertical x2-axis

and then reflects points through the

line x 2 equals x 1line x2=x1

The standard matrix of the linear transformation reflects points through the vertical axis and the line[tex]\(x_2 = x_1\) is \([-1\ 1\ 0\ 0]\).[/tex]

To find the standard matrix of the linear transformation  T , we need to determine the images of the standard basis vectors [tex]\( \mathbf{e}_1 \) and \( \mathbf{e}_2 \)[/tex] under  T .

First, let's understand the transformation described:

1. Reflection through the vertical [tex]\( x_2 \)[/tex]axis:

  This transformation replaces each point [tex]\( (x_1, x_2) \) with \( (-x_1, x_2) \).[/tex]

2. Reflection through the line [tex]\( x_2 = x_1 \):[/tex]

  This transformation replaces each point [tex]\( (x_1, x_2) \) with \( (x_2, x_1) \).[/tex]

To find the images of the standard basis vectors under  T :

- [tex]\( T(\mathbf{e}_1) \) is obtained by reflecting \( \mathbf{e}_1 = (1, 0) \) through the \( x_2 \) axis, resulting in \( (-1, 0) \).[/tex]

- [tex]\( T(\mathbf{e}_2) \) is obtained by reflecting \( \mathbf{e}_2 = (0, 1) \) through the line \( x_2 = x_1 \), resulting in \( (1, 0) \).[/tex]

Now, we can construct the standard matrix of  T  using the images of the standard basis vectors:

[tex]\[ [T] = \begin{bmatrix} -1 & 1 \\ 0 & 0 \end{bmatrix} \][/tex]

This matrix represents the transformation  T  as described.

The Complete Question:

Assume that T is a linear transformation. Find the standard matrix of T.

​T: set of real numbers R squaredℝ2right arrow→set of real numbers R squaredℝ2

first reflects points through the

vertical x 2 dash axisvertical x2-axis

and then reflects points through the

line x 2 equals x 1line x2=x1

To increase the number of customers you talk to daily by 20%, you will need to calculate 20% of the current number of customers you talk to and add it to the original number.

To increase the number of customers you talk to daily by 20%, you will need to calculate 20% of the current number of customers you talk to daily and add it to the original number. In this case, you talk to an average of 8 customers per hour during an 8-hour shift, which means you talk to 8 x 8 = 64 customers per day. To increase this number by 20%, you need to calculate 20% of 64, which is 0.20 x 64 = 12.8. Round this number to the nearest whole number to get an increase of 13 customers. Finally, add this increase to the original number of customers to find the new number of customers you need to talk to per day: 64 + 13 = 77 customers per day.

Step-by-step explanation:

The formula is subtracting by 20 you can count it so the next - 70

Answer:

10-20(n-1)

Step-by-step explanation:

Answer:

[tex]t=\frac{1.17-1.04}{\sqrt{\frac{0.11^2}{8}+\frac{0.09^2}{8}}}}=2.587[/tex]  

[tex]df=n_{1}+n_{2}-2=8+8-2=14[/tex]

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

[tex]p_v =P(t_{(14)}>2.587)=0.0108[/tex]

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, we have enough evidence to reject the null hypothesis on this case and the 25 mil film have a mean greater than the 20 mil film so then the claim is not appropiate

Step-by-step explanation:

Data given and notation

[tex]\bar X_{1}=1.17[/tex] represent the mean for the sample 1 (25 mil film)

[tex]\bar X_{2}=1.04[/tex] represent the mean for the sample 2 (20 mil film)

[tex]s_{1}=0.11[/tex] represent the sample standard deviation for the sample 1

[tex]s_{2}=0.09[/tex] represent the sample standard deviation for the sample 2

[tex]n_{1}=8[/tex] sample size selected for 1

[tex]n_{2}=8[/tex] sample size selected for 2

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

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

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if reducing the film thickness increases the mean speed of the film, the system of hypothesis would be:

Null hypothesis:[tex]\mu_{1} \leq \mu_{2}[/tex]

Alternative hypothesis:[tex]\mu_{1} > \mu_{2}[/tex]

If we analyze the size for the samples both are less than 30 so for this case is better apply a t test to compare means, and the statistic is given by:

[tex]t=\frac{\bar X_{1}-\bar X_{2}}{\sqrt{\frac{s^2_{1}}{n_{1}}+\frac{s^2_{2}}{n_{2}}}}[/tex] (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other".

Calculate the statistic

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

[tex]t=\frac{1.17-1.04}{\sqrt{\frac{0.11^2}{8}+\frac{0.09^2}{8}}}}=2.587[/tex]  

P-value

The first step is calculate the degrees of freedom, on this case:

[tex]df=n_{1}+n_{2}-2=8+8-2=14[/tex]

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

[tex]p_v =P(t_{(14)}>2.587)=0.0108[/tex]

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, we have enough evidence to reject the null hypothesis on this case and the 25 mil film have a mean greater than the 20 mil film so then the claim is not appropiate

Answer:

Both three and six are divisible by three. Therefore, this fraction could be simplified to one-half.

Step-by-step explanation:

Three divided by three is equal to one.

Answer: odd number:1/2

number less than six:5/6

Step-by-step explanation:

Answer:

B is better option

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

Answer:

x = theta = 0°

Step-by-step explanation:

Given the trigonometry function

Tan²x/2-2 cos x = 1

Tan²x-4cosx = 2 ... 1

From trigonometry identity

Sec²x = tan²x+1

tan²x = sec²x-1 ... 2

Substituting 2 into 1, we have:

sec²x-1 -4cosx = 2

Note that secx = 1/cosx

1/cos²x - 1 - 4cosx = 2

Let cosx. = P

1/P² - 1 - 4P = 2

1-P²-4P³ = 2P²

4P³+2P²+P²-1 = 0

4P³+3P² = 1

P²(4P+3) = 1

P² = 1 and 4P+3 = 1

P = ±1 and P = -3/4

Since cosx = P

If P = 1

Cosx = 1

x = arccos1

x = 0°

If x = -1

cosx = -1

x = arccos(-1)

x = 180°

Since our angle must be between 0 and 1 therefore x = 0°

Answer:

31.2

Step-by-step explanation:

Because you only round up if the number next to the 1st decimal place is greater than 5, you just drop the 4 and keep 31.2.

you just drop the 4 and keep 31.2.

Answer:

i am positive he can

Step-by-step explanation:

sorry if this is wrong...also can i pls have brainliest

Answer:

Step-by-step explanation:

Given Parameters

Mean, [tex]x[/tex] = 180

total samples, n = 20

Standard dev, [tex]\sigma[/tex] = 30

[tex]\alpha[/tex] = 1 - 0.95 = 0.05 at 95% confidence level

Df = n - 1 = 20 - 1 = 19

Critical Value, [tex]t_\alpha[/tex], is given by

[tex]t_{c}=t_{\alpha, df} = t_{0.05,19} = 1.729[/tex]

a).

Confidence Interval, [tex]\mu[/tex], is given by the formula

[tex]\mu = x +/- t_c \times \frac{s}{\sqrt{n} }[/tex]

[tex]\mu = 180 +/- 1.729 \times \frac{30}{\sqrt{20} }[/tex]

[tex]\mu = 180 +/-11.5985[/tex]

[tex]191.5985 > \mu > 168.4015[/tex]

b).

Critical Value, [tex]t_{\alpha/2}[/tex], is given by

[tex]t_{c}=t_{\alpha/2, df} = t_{0.05/2,19} = 2.093[/tex]

Confidence Interval, [tex]\mu[/tex], is given by

[tex]\mu = x +/- t_c \times \frac{s}{\sqrt{n} }[/tex]

[tex]\mu = 180 +/- 2.093 \times \frac{30}{\sqrt{20} }[/tex]

    = 180 +/- 14.0403

    = 165.9597 < [tex]\mu[/tex] < 194.0403

Answer:

For this case we have the following info related to the time to prepare a return

[tex] \mu =90 , \sigma =14[/tex]

And we select a sample size =49>30 and we are interested in determine the standard deviation for the sample mean. From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard deviation would be:

[tex]\sigma_{\bar X} =\frac{14}{\sqrt{49}}= 2[/tex]

And the best answer would be

b. 2 minutes

Step-by-step explanation:

Previous concepts

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".  

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Solution to the problem

For this case we have the following info related to the time to prepare a return

[tex] \mu =90 , \sigma =14[/tex]

And we select a sample size =49>30 and we are interested in determine the standard deviation for the sample mean. From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And the standard deviation would be:

[tex]\sigma_{\bar X} =\frac{14}{\sqrt{49}}= 2[/tex]

And the best answer would be

b. 2 minutes