Answer: 1/6
Step-by-step explanation:
The vertical angle to the top of a flagpole from point A on the ground is observed to be 37°11'. The observer walks 17 m directly away from point A and the flagpole to point B and finds the new angle to be 25°43'. What is the approximate height of the flagpole?
Answer:
22m
Step-by-step explanation:
Let height of flagpole=h
AB==17 m
[tex]\angle CAD=37^{\circ}11'=37+\frac{11}{60}=37.183^{\circ}[/tex](1 degree= 60 minute)
[tex]\angle B=25^{\circ}43'=25+\frac{43}{60}=25.72^{\circ}[/tex]
We have to find the approximate height of the flagpole.
In triangle CDA,
[tex]\frac{CD}{DA}=tan\theta=\frac{Perpendicular\;side}{Base}[/tex]
[tex]\frac{h}{DA}=tan37.183^{\circ}[/tex]
[tex]h=DA(0.759)[/tex]
In triangle CDB,
[tex]tan 25.72^{\circ}=\frac{CD}{DB}[/tex]
[tex]0.482=\frac{h}{DA+17}[/tex]
[tex]0.482DA+8.194=h[/tex]
Substitute the value
[tex]0.482DA+8.194=0.759DA[/tex]
[tex]8.194=0.759DA-0.482DA[/tex]
[tex]8.194=0.277DA[/tex]
[tex]DA=\frac{8.194}{0.277}=29.58[/tex]
Substitute the value
[tex]h=29.58 \times 0.759=22.45 m\approx 22m[/tex]
Hence, the height of the flagpole=22 m
The approximate height of the flagpole is 21.33 meters, calculated using trigonometry and the observed angles and distances.
To find the height of the flagpole, we can use trigonometry, specifically the tangent function, along with the given angles and the distance from the observer's initial position to the flagpole.
Step 1:
Calculate the tangent of the observed angles:
Let (h) be the height of the flagpole.
For the first observation:
[tex]\[ \tan(37^\circ 11') = \frac{h}{d} \][/tex]
For the second observation:
[tex]\[ \tan(25^\circ 43') = \frac{h}{d + 17} \][/tex]
Step 2:
Solve for (h):
First, convert the angles to decimal degrees:
[tex]\[ 37^\circ 11' = 37 + \frac{11}{60} \approx 37.18^\circ \][/tex]
[tex]\[ 25^\circ 43' = 25 + \frac{43}{60} \approx 25.72^\circ \][/tex]
Now, substitute the angles into the tangent equations:
[tex]\[ \tan(37.18^\circ) = \frac{h}{d} \][/tex]
[tex]\[ \tan(25.72^\circ) = \frac{h}{d + 17} \][/tex]
Step 3:
Solve the equations for (h):
[tex]\[ h = d \times \tan(37.18^\circ) \][/tex]
[tex]\[ h = (d + 17) \times \tan(25.72^\circ) \][/tex]
Step 4:
Set the expressions equal to each other:
[tex]\[ d \times \tan(37.18^\circ) = (d + 17) \times \tan(25.72^\circ) \][/tex]
Step 5:
Solve for (d):
[tex]\[ d = \frac{17 \times \tan(25.72^\circ)}{\tan(37.18^\circ) - \tan(25.72^\circ)} \][/tex]
Step 6:
Calculate the value of (d) and then use it to find (h):
[tex]\[ d \approx 27.83 \text{ meters} \][/tex]
[tex]\[ h = 27.83 \times \tan(37.18^\circ) \][/tex]
Step 7:
Calculate (h):
[tex]\[ h \approx 27.83 \times 0.7656 \][/tex]
[tex]\[ h \approx 21.33 \text{ meters} \][/tex]
Therefore, the approximate height of the flagpole is 21.33 meters.
A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner. He made this decision based on a test using a = 0.05. Would he have made the same decision at a = 0.10? How about a = 0.01? Explain.
Answer:
He can make the same decision at a = 0.10, but he may not make the same decision at a = 0.01
Step-by-step explanation:
In hypothesis tests, critical regions are ranges of the distributions where the values represent statistically significant results. Analysts define the size and location of the critical regions by specifying both the significance level (alpha) and whether the test is one-tailed or two-tailed.
As the significance level gets bigger, the range of the critical region increases.
Therefore significant results at lower significance levels are still significant at higher significance levels. Thus significant result at a=0.05 is always significant at a=0.10
But significant result at a=0.05 may not be significant at a=0.01 since critical region shrinks, therefore the result may not fall in the critical region at a=0.01
The area under a particular normal curve between 6 and 8 is 0.695. A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie between 6 and 8?
Answer:
69.5%
Step-by-step explanation:
A feature of the normal distribution is that this is completely determined by its mean and standard deviation, therefore, if two normal curves have the same mean and standard deviation we can be sure that they are the same normal curve. Then, the probability of getting a value of the normally distributed variable between 6 and 8 is 0.695. In practice we can say that if we get a large sample of observations of the variable, then, the percentage of all possible observations of the variable that lie between 6 and 8 is 100(0.695)% = 69.5%.
Which system of linear equations is graphed below
Answer:
D
Step-by-step explanation:
Seventy-six percent of products come off the line within product specifications. Your quality control department selects 15 products randomly from the line each hour. Looking at the binomial distribution, if fewer than how many are within specifications would require that the production line be shut down (unusual) and repaired?A. Fewer than 12
B. Fewer than 11
C. Fewer than 10
D. Fewer than 9.
Answer:
A
Step-by-step explanation:
change 76% to decimal and multiply by 15
.76*15 = 11.4
11.4 is fewer than 12.
A2 = [1 2 3; 4 5 6; 7 8 9; 3 2 4; 6 5 4; 9 8 7]
b2 = [1 1 1 1 1 1]′
(a) Find a basis for the row space of A2.
(b) How many solutions does A2 ?
Answer:
Remember, a basis for the row space of a matrix A is the set of rows different of zero of the echelon form of A.
We need to find the echelon form of the matrix augmented matrix of the system A2x=b2
[tex]B=\left[\begin{array}{cccc}1&2&3&1\\4&5&6&1\\7&8&9&1\\3&2&4&1\\6&5&4&1\\9&8&7&1\end{array}\right][/tex]
We apply row operations:
1.
To row 2 we subtract row 1, 4 times.To row 3 we subtract row 1, 7 times.To row 4 we subtract row 1, 3 times.To row 5 we subtract row 1, 6 times.To row 6 we subtract row 1, 9 times.We obtain the matrix
[tex]\left[\begin{array}{cccc}1&2&3&1\\0&-3&-6&-3\\0&-6&-12&-6\\0&-4&-5&-2\\0&-7&-14&-5\\0&-10&-20&-8\end{array}\right][/tex]
2.
We subtract row two twice to row three of the previous matrix.we subtract 4/3 from row two to row 4.we subtract 7/3 from row two to row 5.we subtract 10/3 from row two to row 6.We obtain the matrix
[tex]\left[\begin{array}{cccc}1&2&3&1\\0&-3&-6&-3\\0&0&0&0\\0&0&3&2\\0&0&0&2\\0&0&0&2\end{array}\right][/tex]
3.
we exchange rows three and four of the previous matrix and obtain the echelon form of the augmented matrix.
[tex]\left[\begin{array}{cccc}1&2&3&1\\0&-3&-6&-3\\0&0&3&2\\0&0&0&0\\0&0&0&2\\0&0&0&2\end{array}\right][/tex]
Since the only nonzero rows of the augmented matrix of the coefficient matrix are the first three, then the set
[tex]\{\left[\begin{array}{c}1\\2\\3\end{array}\right],\left[\begin{array}{c}0\\-3\\-6\end{array}\right],\left[\begin{array}{c}0\\0\\3\end{array}\right] \}[/tex]
is a basis for Row (A2)
Now, observe that the last two rows of the echelon form of the augmented matrix have the last coordinate different of zero. Then, the system is inconsistent. This means that the system has no solutions.
Determine whether the lines L1:x=18+6t,y=7+3t,z=13+3t and L2:x=−14+7ty=−12+5tz=−8+6t intersect, are skew, or are parallel. If they intersect, determine the point of intersection; if not leave the remaining answer blanks empty.
Answer:
skew lines
Step-by-step explanation:
we are given 2 lines in parametric form as
L1:x=18+6t,y=7+3t,z=13+3t and L2:x=−14+7ty=−12+5tz=−8+6t[tex]L1:x=18+6t,y=7+3t,z=13+3t \\ L2:x=-14+7t,y=-12+5t,z=-8+6t[/tex]
If the lines intersect then the two points must be equal for one value of t.
Let us try equating x,y and z coordinate.
[tex]18+6t = -14+7t\\t=32\\[/tex]
when we equate y coordinate we get
[tex]7+3t =-12+5t\\2t =19\\t =9.5[/tex]
Since we get two different t we find that these two lines cannot intersect.
Comparing direction ratios we have
I line has direction ratios as (6,3,3) and second line (7,5,6)
These two are not proportional and hence not parallel
So these lines are skew lines
After analyzing the direction vectors of lines L1 and L2, it is clear that they are neither parallel nor do they intersect, which means the lines are skew.
Explanation:To determine whether the lines L1 and L2 intersect, are skew, or are parallel, we need to compare the direction vectors and check if they can be expressed as multiples of each other. If we consider the components of each direction vector given by the 't' terms in the equations of L1 and L2, they are (6, 3, 3) for L1 and (7, 5, 6) for L2 respectively.
To check for parallelism, we find a constant 'k' such that (6, 3, 3) = k*(7, 5, 6). After trying to solve for 'k', we immediately see that no such constant can exist, as 6/7 is not equal to 3/5 or 3/6. Therefore, these lines are not parallel.
To check for intersection, we need to find a common point that satisfies both line equations for some value of the parameter 't'. However, since an attempt to find such values results in inconsistent equations, it suggests that no such point exists and these lines do not intersect.
Given that the lines are neither parallel nor do they intersect, they must be skew lines. Skew lines are lines that do not intersect and are not parallel, lying in different planes.
You have a fair coin. You flip the coin two times. Let T1T1 be the event that the first flip (Flip 1) results in Tails. Let T2T2 be the event that the second flip (Flip 2) results in Tails. Are the events T1T1 and T2T2 independent?
Answer:
Yes, the events T1 and T2 are independent.
Step-by-step explanation:
When flipping a fair coin, for each flip, there is a 50/50 chance that it will result in heads or tails.
For the first flip, P(T1) = 0.5
For the second flip, P(T2) = 0.5 regardless of the outcome of the first flip. Therefore, T1 and T2 are independent events.
*Note that if the question asked for the event of both flips resulting in tails, then the events would be dependent.
A typist makes an average of 5 typo errors per page that she types. If the number of errors per each typed page can be modeled according to the Poisson distribution, what is the probability that she will make more than 3 typo errors in the next page? Express your answer in decimals and round to two significant places after the decimal. For example 0.1678 should be entered as 0.17.
Answer:
Step-by-step explanation:
Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 24 customers per hour or 0.4 customers per minute.
(a) What is the mean or expected number of customers that will arrive in a five-minute period?
(b) Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a five-minute period. If required, round your answers to four decimal places.
(c) Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur? If required, round your answer to four decimal places.
Answer:
2, 0.135, 0.270, 0.270, 0.180, 0.145
Step-by-step explanation:
1. To calculate the mean of people arriving in 5 minute period:
We know the arrival rate of minute is 0.4, so people arriving in 5 minutes will be
0.4 x 5 = 2
2. From part 1 it is known thay mean arrival time=2
For this we will use poisson's probability formula that is
P(X=x) = (2^x) x Exp^(-2/x!)
For X=0
P(X=0) = (2^0) x Exp^(-2/0!) = 0.135
For X = 1
P(X=1) = (2^1) x Exp^(-2/1!) = 0.270
For X = 2
P(X=2) = (2^2) x Exp^(-2/2!) = 0.270
For X = 3
P(X=3) = (2^3) x Exp^(-2/3!) = 0.180
3. For delay expected if more than 3 customer arrive in 5 minutes.
P(X>3) = 1 - P(X=0) - P(X=1) - P(X=2) - P(X=3)
P(X>3) = 1-0.135-0.270-0.270-0.180
P(X>3) = 0.145
Which of the following is the purpose of a confidence interval for the fitted (predicted value) from a
regression model?
A) To estimate the value of a future observation for a given value of x.
B) To estimate the population mean of Y for a given value of x
C) To estimate the population slope of the regression model
D) To estimate the sample mean of Y for a given value of x.
Answer: option A is correct
Step-by-step explanation:
the prediction interval is to cover a “moving target”, the random future value of y,
while the confidence interval is to cover the “fixed target”,
the average (expected) value of y, E(y).
At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years, and the distribution is approximately normal. If an employee is picked at random, what is the probability that the employee has worked at the store for over 10 years? 99.2% 0.8% 49.2% 1.7%
Answer:
option 0.8%
Step-by-step explanation:
Data provided in the question:
Mean = 5.7 years
Standard deviation, s = 1.8 years
Now,
P(the employee has worked at the store for over 10 years)
= P(X > 10 years)
= [tex]P (Z > \frac{X-Mean}{\sigma})[/tex]
or
= [tex]P (Z > \frac{10-5.7}{1.8})[/tex]
= P (Z > 2.389 )
or
= 0.008447 [from standard z table]
or
= 0.008447 × 100% = 0.84% ≈ 0.8%
Hence,
the correct answer is option 0.8%
To find the probability that an employee has worked at a large department store for over 10 years, we can use the z-score formula and a standard normal distribution table or calculator.
Explanation:To find the probability that an employee has worked at the store for over 10 years, we can use the z-score formula:
z = (x - μ) / σ
where x is the value we are interested in (10 years), μ is the mean (5.7 years), and σ is the standard deviation (1.8 years).
Plugging in the values, z = (10 - 5.7) / 1.8 = 2.39
Using a standard normal distribution table or a calculator, we can find that the probability of a z-score greater than 2.39 is approximately 0.008, or 0.8%.
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A fair coin is tossed three times and the events A, B, and C are defined as follows: A:{ At least one head is observed } B:{ At least two heads are observed } C:{ The number of heads observed is odd } Find the following probabilities by summing the probabilities of the appropriate sample points (note that 0 is an even number):
(a) P(B) =
(b) P(A or B) =
(c) P(A or B or C)
Answer:
(a) 1/2
(b) 1/2
(c) 1/8
Step-by-step explanation:
Since, when a fair coin is tossed three times,
The the total number of possible outcomes
n(S) = 2 × 2 × 2
= 8 { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT },
Here, B : { At least two heads are observed } ,
⇒ B = {HHH, HHT, HTH, THH},
⇒ n(B) = 4,
Since,
[tex]\text{Probability}=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}[/tex]
(a) So, the probability of B,
[tex]P(B) =\frac{n(B)}{n(S)}=\frac{4}{8}=\frac{1}{2}[/tex]
(b) A : { At least one head is observed },
⇒ A = {HHH, HHT, HTH, THH, HTT, THT, TTH},
∵ A ∩ B = {HHH, HHT, HTH, THH},
n(A∩ B) = 4,
[tex]\implies P(A\cap B) = \frac{n(A\cap B)}{n(S)} = \frac{4}{8}=\frac{1}{2}[/tex]
(c) C: { The number of heads observed is odd },
⇒ C = { HHH, HTT, THT, TTH},
∵ A ∩ B ∩ C = {HHH},
⇒ n(A ∩ B ∩ C) = 1,
[tex]\implies P(A\cap B\cap C)=\frac{1}{8}[/tex]
Marie sees a leather jacket kn sale for $73.00. The sign says 20% off. How much was the jacket originally?
Answer:
It was originally $91.25
A toy company is modeling a real home as a doll house. They decide to use a 2/3 inch to 1 foot scale, the traditional scale for doll house built in the 20th century. If the actual house is 30 feet high, how high will the dollhouse be in inches?
Answer:
20 inches
Step-by-step explanation:
Take 2/3 of an inch 30 times.
The height of the model is then 30 times 2/3 inches = 10*2 = 20 inches
The average life a manufacturer's blender is 5 years, with a standard deviation of 1 year. Assuming that the lives of these blenders follow approximately a normal distribution, find the probability that the mean life a random sample of 25 such blenders falls between 4.6 and 5.1 years.
Answer:
The probability that the mean life a random sample of 25 such blenders falls between 4.6 and 5.1 years is 0.6687.
Step-by-step explanation:
We have given :
The average life a manufacturer's blender is 5 years i.e. [tex]\mu=5[/tex]
The standard deviation is [tex]\sigma=1[/tex]
Number of sample n=25.
To find : The probability that the mean life falls between 4.6 and 5.1 years ?
Solution :
Using z-score formula, [tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
The probability that the mean life falls between 4.6 and 5.1 years is given by, [tex]P(4.6<X<5.1)[/tex]
[tex]P(4.6<X<5.1)=P(\frac{4.6-5}{\frac{1}{\sqrt{25}}}<Z<\frac{5.1-5}{\frac{1}{\sqrt{25}}})[/tex]
[tex]P(4.6<X<5.1)=P(\frac{-0.4}{\frac{1}{5}}<Z<\frac{0.1}{\frac{1}{5}})[/tex]
[tex]P(4.6<X<5.1)=P(-2<Z<0.5)[/tex]
[tex]P(4.6<X<5.1)=P(Z<0.5)-P(Z<-2)[/tex]
Using z-table,
[tex]P(4.6<X<5.1)=0.6915-0.0228[/tex]
[tex]P(4.6<X<5.1)=0.6687[/tex]
Therefore, the probability that the mean life a random sample of 25 such blenders falls between 4.6 and 5.1 years is 0.6687.
The probability that the mean life of a random sample of 25 blenders falls between 4.6 and 5.1 years is approximately 0.3585, or 35.85%.
Explanation:To find the probability that the mean life of a random sample of 25 blenders falls between 4.6 and 5.1 years, we can use the standard normal distribution. First, we need to standardize the values using the z-score formula. The z-score for 4.6 years is (4.6 - 5) / 1 = -0.4, and the z-score for 5.1 years is (5.1 - 5) / 1 = 0.1. We can then look up the corresponding probabilities in the standard normal distribution table or use a calculator to find the area under the curve between these two z-scores. The probability that the mean life falls in this range is approximately 0.3585, or 35.85%.
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Suppose that we are testing a coin to see if it is fair, so our hypotheses are: H0: p = 0.5 vs Ha: p ≠ 0.5. In each of (a) and (b) below, use the "Edit Data" option on StatKey to find the p-value for the sample results and give a conclusion in the test. a. We get 56 heads out of 100 tosses. b. We get 560 heads out of 1000 tosses. c. Compare the sample proportions in parts (a) and (b). Compare the p-values. Why are the p-values so different?
For testing if a coin is fair, a binomial test is used to compare the observed heads to what's expected under a fair coin scenario. Differences in p-values for the same sample proportion across different sample sizes are due to the increased precision of larger samples.
In hypothesis testing, when we want to determine whether a coin (or in this case, a hypothetical lizard) is fair, we employ a binomial test. This test assesses the number of 'successes' (heads in our case), comparing it to what we would expect under the null hypothesis of p=0.5, indicating a fair coin.
For part (a), where we get 56 heads out of 100 tosses, we would calculate the p-value by looking at both the proportion of heads obtained and the standard error for a binomial distribution. Although not calculated here directly, if this p-value is less than the chosen significance level, typically 0.05, we reject the null hypothesis, concluding the coin is not fair.
For part (b), getting 560 heads out of 1000 tosses yields the same sample proportion as part (a). However, the larger sample size increases the power of the test, often resulting in a smaller p-value if the observed proportion is different from the expected 0.5.
Comparing p-values from (a) and (b) reveals that despite having the same sample proportions, the p-values differ due to the sample sizes. Larger sample sizes yield more precise estimates of the population proportion, thus potentially resulting in smaller p-values for the same sample proportion deviation from the null hypothesis.
If Data A has a correlation coefficient of r = - 0.991, and Data B has a correlation coefficient of r = 0.991, which correlation is correct? a. Data A has a stronger linear correlation than Data B. b. Data A and Data B have the same strength in linear correlation. c. Data A has a weaker linear correlation than Data B.
Answer:
Option B) Data A and Data B have the same strength in linear correlation.
Step-by-step explanation:
Correlation:
Correlation is a technique that help us to find or define a relationship between two variables. It is a measure of linear relationship between two quantities. A positive correlation means that an increase in one quantity leads to an increase in another quantity A negative correlation means with increase in one quantity the other quantity decreases. +1 tells about a a perfect positive linear relationship and −1 indicates a perfect negative linear relationship.Data A correlation = -0.991
Data B correlation = 0.991
Data A and data B have same strength of correlation but they are opposite to each other. There absolute value is same but the data A shows negative correlation and data B shows positive correlation. Data A shows indirect or inverse linear relationship and data B shows direct linear relationship. Both data have same magnitude of correlation.
Let A and B be n x n matrices.
The determinant of A is the product of the diagonal entries in A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The statement is true because the determinant of any triangular matrix A is the product of the entries on the main diagonal of A.
B.
The statement is false because the determinant of the 2×2 matrix A = __ is not equal to the product of the entries on the main diagonal of A.
(Type an integer or simplified fraction for each matrixelement.)
C.
The statement is true because the determinant of any square matrix A is the product of the entries on the main diagonal of A.
Final answer:
The statement regarding the determinant of a square matrix A being the product of its diagonal entries is incorrect. The determinant of a 2×2 matrix is calculated as the product of diagonal entries minus the product of off-diagonal entries.
Explanation:
The determinant of a square matrix A is not always the product of the diagonal entries in A. For instance, a 2×2 matrix has the determinant given by det(A) = a11a22 - a12a21, which involves the product of the diagonal entries minus the product of the off-diagonal entries. Therefore, the correct choice is:
B. The statement is false because the determinant of the 2×2 matrix A = [[a, b], [c, d]] is not equal to the product of the entries on the main diagonal of A. In fact, the determinant is ad - bc.
Example: For matrix A = [[1,2], [3,4]], the determinant is 1×4 - 2×3, which equals -2, not the product of the diagonal entries (1 and 4), which would be 4.
A random sample of size 100 was taken from a population. A 94% confidence interval to estimate the mean of the population was computed based on the sample data. The confidence interval for the mean is: (107.62, 129.75). What is the z-value that was used in the computation. Round your z-value to 2 decimal places. Select one: A. 1.45 B. 1.55 C. 1.73 D. 1.88 E. 1.96 E-mail fraud (phishing) is becoming an increasing problem for users of the internet. Suppose that 70% of all internet users experience e-mail fraud. If 50 internet users were randomly selected, what is the probability that no more than 25 were victims of e-mail fraud? Answer:
Answer:
Step-by-step explanation:
1) The z value was determined using a normal distribution table. From the normal distribution table, the corresponding z value for a 94% confidence interval is 1.88
The correct option is D
2) if 70% of all internet users experience e-mail fraud. It means that probability of success, p
p = 70/100 = 0.7
q = 1 - p = 1 - 0.7 = 0.3
n = number of selected users = 50
Mean, u = np = 50×0.7 = 35
Standard deviation, u = √npq = √50×0.7×0.3 = 3.24
x = number of internet users
The formula for normal distribution is expressed as
z = (x - u)/s
We want to determine the probability that no more than 25 were victims of e-mail fraud. It is expressed as
P(x lesser than or equal to 25)
The z value will be
z = (25- 35)/3.24 = - 10/3.24 = -3.09
Looking at the normal distribution table, the corresponding z score is 0.001
P(x lesser than or equal to 25) = 0.001
A supervisor records the repair cost for 25 randomly selected dryers. A sample mean of $93.36 and standard deviation of $19.95 are subsequently computed. Determine the 98% confidence interval for the mean repair cost for the dryers. Assume the population is approximately normal.
Answer:
The 98% confidence interval for the mean repair cost for the dryers is (83.4161, 103.3039).
Step-by-step explanation:
We have a small sample size n = 25, [tex]\bar{x} = 93.36[/tex] and s = 19.95. The confidence interval is given by [tex]\bar{x}\pm t_{\alpha/2}(\frac{s}{\sqrt{n}})[/tex] where [tex]t_{\alpha/2}[/tex] is the [tex]\alpha/2[/tex]th quantile of the t distribution with n - 1 = 25 - 1 = 24 degrees of freedom. As we want the 98% confidence interval, we have that [tex]\alpha = 0.02[/tex] and the confidence interval is [tex]93.36\pm t_{0.01}(\frac{19.95}{\sqrt{25}})[/tex] where [tex]t_{0.01}[/tex] is the 1st quantile of the t distribution with 24 df, i.e., [tex]t_{0.01} = -2.4922[/tex]. Then, we have [tex]93.36\pm (-2.4922)(\frac{19.95}{\sqrt{25}})[/tex] and the 98% confidence interval is given by (83.4161, 103.3039).
Answer:
Step-by-step explanation:
Our aim is to determine a 98% confidence interval for the mean repair cost for the dryers
Number of samples. n = 25
Mean, u = $93.36
Standard deviation, s = $19.95
For a confidence level of 98%, the corresponding z value is 2.33. This is determined from the normal distribution table.
We will apply the formula,
Confidence interval
= mean ± z × standard deviation/√n
It becomes
93.36 ± 2.33 × 19.95/√25
= 93.36 ± 2.33 × 3.99
= 93.36 ± 9.2967
The lower boundary of the confidence interval is 93.36 - 9.2967 =84.0633
The upper boundary of the confidence interval is 93.36 + 9.2967 = 102.6567
Therefore, with 98% confidence interval, the mean repair costs for the dryers is between $84.0633 and $102.6567
Which of the following functions computes a value such that 2.5% of the area under the standard normal distribution lies in the upper tail defined by this value? ○ 0 ° ○ a.-NORMSINV(0.975) b.-NORM·S.INV(0.05) C.-NORM.S-INV(0.95) d.-NORMSINV(0.025)
Answer:
a) NORM.S.INV(0.975)
Step-by-step explanation:
1) Some definitions
The standard normal distribution is a particular case of the normal distribution. The parameters for this distribution are: the mean is zero and the standard deviation of one. The random variable for this distribution is called Z score or Z value.
NORM.S.INV Excel function "is used to find out or to calculate the inverse normal cumulative distribution for a given probability value"
The function returns the inverse of the standard normal cumulative distribution(a z value). Since uses the normal standard distribution by default the mean is zero and the standard deviation is one.
2) Solution for the problem
Based on this definition and analyzing the question :"Which of the following functions computes a value such that 2.5% of the area under the standard normal distribution lies in the upper tail defined by this value?".
We are looking for a Z value that accumulates 0.975 or 0.975% of the area on the left and by properties since the total area below the curve of any probability distribution is 1, then the area to the right of this value would be 0.025 or 2.5%.
So for this case the correct function to use is: NORM.S.INV(0.975)
And the result after use this function is 1.96. And we can check the answer if we look the picture attached.
Bob and Sally mortgage payment increases to $1632.00. Their roof starts to leak and they need to replace it. They budget 500$ per month (replace the old student loan charge with roof expense).
Answer:
...
Step-by-step explanation:
A truck driver operates a delivery service in a southern city. His start-up costs amounted to $2500. He estimates that it costs him (in terms of gasoline, wear and tear on his truck, etc.)$3.00 per delivery. He charges $5.50 per delivery. Let x represent the number of deliveries he makes. (a) Express the cost C as a function of x. (b) Express the revenue R as a function of x. (c) Determine analytically the value of x for which revenue equals cost. (d) Graph y1equalsC(x) and y2equalsR(x) on the same xy-axes and interpret the graphs.
Answer:
(a)C(x) = 2500 + 3x
(b)R(x) = 5.5x
(c)x = 1000
Step-by-step explanation:
(a)His cost function as a function of x
C(x) = 2500 + 3x
(b)His revenue R function as a function of x
R(x) = 5.5x
(c)When revenue R equals to Cost C
C(x) = R(x)
2500 + 3x = 5.5x
2.5x = 2500
x = 1000
(d) Refer to attachment. We can see that the 2 lines intercepts at x = 1000 and y = 2500. That's the point of turning to profit.
Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for metal sheets of a particular type, its mean value and standard deviation are 70 GPa and 2.2 GPa, respectively. Suppose the distribution is normal. (Round your answers to four decimal places.) (a) Calculate P(69 ≤ X ≤ 71) when n = 16.
The P(69 ≤ X ≤ 71) when n = 16 is approximately 0.9312.
Here's the calculation for P(69 ≤ X ≤ 71) when n = 16:
1. Find the standard error (SE):
SE = σ / √n = 2.2 GPa / √16 = 0.55 GPa
2. Standardize the values:
Z1 = (69 - 70) / 0.55 = -1.82
Z2 = (71 - 70) / 0.55 = 1.82
3. Use a standard normal table or calculator to find the probabilities:
P(Z ≤ 1.82) = 0.9656
P(Z ≤ -1.82) = 0.0344
4. Calculate the probability of the interval:
P(69 ≤ X ≤ 71) = P(-1.82 ≤ Z ≤ 1.82) = P(Z ≤ 1.82) - P(Z ≤ -1.82)
= 0.9656 - 0.0344 = 0.9312
Therefore, P(69 ≤ X ≤ 71) when n = 16 is approximately 0.9312.
The cycle time for trucks hauling concrete to a highway construction site is uniformly distributed over the interval 50 to 70 minutes. What is the conditional probability that the cycle time is no more than 65 minutes if it is known that the cycle time exceeds 55 minutes?
Answer:
2/3=0.6667
Step-by-step explanation:
Let X be the cycle time for trucks hauling concrete to a highway construction site
Given that X is U(50,70)
Hence pdf of X is
[tex]f(x) = \frac{1}{20} ,50<x<70[/tex]
Let A be the event that cycle time is no more than 65 minutes and
B the event cycle time exceeds 55 minutes
Required probability
= the conditional probability that the cycle time is no more than 65 minutes if it is known that the cycle time exceeds 55 minutes
= P(A/B)
=[tex]\frac{P(A\bigcapB)}{P(B)} \\=\frac{P(55<x<65)}{P(X>55)} \\=\frac{65-55}{70-55} \\=\frac{2}{3}[/tex]
A family raised $1000 for their initial investment. If they invest the money in an account that earns 5% interest compounded annually, what will be the value of their investment at the end of 15 years?
The exponential function that expresses the amount of the investment as a function of time when compounded annually is: A(t)=P(1+r)^t
1. Evaluate function: A(15) =
2. Amount of interest earned in dollars:
3. Use the trace function to determine the value of the investment after 18 years.
4. Use the table function to determine how many years it will take for the investment to double.
5. How much more interest would be earned after 15 years if the interest rate were 5.5%?
If the investment is compounded more often than annually, the exponential function becomes ( ) (1 )nt r At P n = + , where n is the number of times the interest is compounded annually. Notice that if n = 1, the formula simplifies to the one used above.
6. If the family’s initial investment is the same with a 5% interest rate, but now it is compounded monthly, what is the value after 15 years of the investment?
7. How many years does it take the investment to double?
8. How much interest would be earned if the initial investment was $1,500 compounded monthly at the same 5% interest rate? How does this compare to the interest earned with the original $1,000 investment?
Answer:
8 or 3
Step-by-step explanation:
In a poll of 1000 adults in July 2010, 540 of those polled said that schools should ban sugary snacks and soft drinks. Complete parts a and b below. a. Do a majority of adults (more than 50%) support a ban on sugary snacks and soft drinks? Perform a hypothesis test using a significance level of 0.05.State the null and alternative hypotheses. Note that p is defined as the population proportion of people who believe that schools should ban sugary foods.
Answer:
[tex]z=2.53[/tex]
[tex]p_v =P(z>2.53)=0.0057[/tex]
The p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so then we have enough evidence to reject the null hypothesis, and we can say that at 5% of significance, the proportion of adults who support a ban on sugary snacks and soft drinks is more than 0.5 or 50%.
Step-by-step explanation:
1) Data given and notation
n=1000 represent the random sample taken
X=540 represent the adults that said that schools should ban sugary snacks and soft drinks
[tex]\hat p=\frac{540}{1000}=0.54[/tex] estimated proportion of adults that said that schools should ban sugary snacks and soft drinks
[tex]p_o=0.5[/tex] is the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level
Confidence=95% or 0.95
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that majority of adults (more than 50%) support a ban on sugary snacks and soft drinks, the system of hypothesis are:
Null hypothesis:[tex]p\leq 0.5[/tex]
Alternative hypothesis:[tex]p > 0.5[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.54 -0.5}{\sqrt{\frac{0.5(1-0.5)}{1000}}}=2.53[/tex]
4) Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a one side right tailed test the p value would be:
[tex]p_v =P(z>2.53)=0.0057[/tex]
So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so then we have enough evidence to reject the null hypothesis, and we can say that at 5% of significance, the proportion of adults who support a ban on sugary snacks and soft drinks is more than 0.5 or 50%.
what are the next two terms in the sequence -16, 4, -1, 1/4
Answer: fifth term = -1/16
Sixth term = 1/256
Step-by-step explanation:
The given sequence is a geometric progression. This is because the ratio of two consecutive terms is constant.
We will apply the formula for determining the nth term of a geometric progression series.
Tn = ar^n-1
Where
Tn = value of the nth term of the geometric series
a = The first term of the series.
r = common ratio(ratio of a term to a consecutive previous term)
n = number of terms in the series
From the in information given,
a = -16
r = 4/-16 = -1/4
The next 2 terms are the 5th and 6th terms.
T5 = -16 × -1/4^(5-1)
T5 = -16 × (-1/4)^4
T5 = -16 × 1/256 = -1/16
The 6th term would be the 5th term × the common ratio. It becomes
T6 = -1/16 ×-1/4 = 1/256
The product of an initial investment, I, and the quantity of the sum of 1 and the annual interest rate, r, raised to the power of n, the number of years of the investment, is equal to M, the current amount of money in an investment account. If an initial investment of $1,423.00 is made to an account with an annual interest rate of 2%, what will be the value of M after 3 years? Round to the nearest cent. A. $1,510.10 B. $1,339.32 C. $1,480.49 D. $1,508.38
Answer:
A. $1,510.10
Step-by-step explanation:
Fill in the given numbers and do the arithmetic.
M = I·(1+r)^n
M = $1423.00·(1 +.02)^3 = $1510.099
M ≈ $1510.10