Answer:
Amount invested in first account = $1,000
Amount invested in second account = $29,000
Step-by-step explanation:
Jina had total amount of $30,000 to invest last year.
Let x be the amount that Jina invested in the first account at interest rate of 9%
Mathematically,
0.09x
she invested the remaining amount in the second account at an interest rate of 5%
Mathematically,
0.05(30,000 - x)
Jina received $1540 in interest.
0.09x + 0.05(30,000 - x) = 1540
0.09x + 1500 - 0.05x = 1540
0.04x = 1540 - 1500
0.04x = 40
x = 40/0.04
x = $1,000
Therefore, Jina invested an amount of $1,000 in the first account at interest rate of 9%
The remaining amount that she invested in the second account is
Amount invested in second account = $30,000 - $1,000
Amount invested in second account = $29,000
Therefore, Jina invested an amount of $29,000 in the second account at interest rate of 5%
Verification:
0.09x + 0.05(30,000 - x) = 1540
0.09(1000) + 0.05(30,000 - 1000) = 1540
90 + 1450 = 1540
1540 = 1540 (satisfied)
a box contains 8 red balls, 5 brown balls, 4 purple balls, and 3 green balls. what is the probability that a purple ball will be selected from the box after a red ball is taken out and not replaced?
Write the probability as a percent. Round to the nearest tenth of a percent as needed.
Inclusive of a 7.6% sales tax, a diamond ring sold for $1721.60. Find the price of the ring before the tax was
added (Round to the nearest cent, if necessary)
Answer:
$159075.84
Step-by-step explanation:
100-7.6=92.4
1721.60*92.4=$159075.84
Tanya has captured many purple-footed bog frogs. She weighs each one and
then counts the number of yellow spots on its back. This trend line is a
fit for these data.
Answer:it’s strong the person under me is slow
Step-by-step explanation: cause I said so
find the amount each contestant would carry if the total amount carried (5 cups) was redistributed equally among all the contestants.
Complete Question
The line plot shows the amount of green ooze (in cup) carried by ten contestants. Find the amount each contestant would carry if the total amount carried (5 cups) was redistributed equally among all the contestants.
Answer:
1/2 Cup
Step-by-step explanation:
This is a Line Plot in which:
1/4 had two Xs3/8 had two Xs1/2 had two Xs5/8 had three Xs7/8 had one XsTotal
[tex]=(\frac{1}{4}X2) +(\frac{3}{8}X2) +(\frac{1}{2}X2) +(\frac{5}{8}X3) +(\frac{7}{8}X1) \\\\=\frac{2}{4} +\frac{6}{8} +1 +\frac{15}{8} +\frac{7}{8}\\\\$=5 Cups[/tex]
If this amount was redistributed equally among the ten contestants:
Each contestant would carry (5/10) cups =1/2 cup.
A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position has the same duties and voting rights. There are 9 faculty members and 10 students eligible to serve on the committee. In how many ways can the committee be formed
Answer:
The committee can be formed in 31,752 ways.
Step-by-step explanation:
Every committee position has the same duties and voting rights. This means that the order in which the members are chosen is not important to solve this question. So the combinations formula is used.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In how many ways can the committee be formed
4 faculty members from a set of 9
5 students from a set of 10.
Then
[tex]T = C_{9,4} \times C_{10,5} = \frac{9!}{4!(9-4)!} \times \frac{10!}{5!(10-5)!} = 31752[/tex]
The committee can be formed in 31,752 ways.
Please, I need help on this problem!
Answer:
$676
Step-by-step explanation:
26 percent of 2600 is 676.
2600 x .26
someone help me please
Answer:
2.5 or 5/2
Step-by-step explanation:
The formula for the area of a triangle is \dfrac{bh}{2}, where b represents the length of the base and h represents the length of the height. With this in mind, you can set up the following equation:
[tex]\dfrac{15}{8}=\dfrac{1.5b}{2}[/tex]
[tex]\dfrac{15}{4}=1.5b[/tex]
b=2.5 or [tex]\frac{5}{2}[/tex]
Hope this helps!
Write the equation of the circle with a center (5,4) and a radius of 7.
Answer:(x-5)^2+(y-4)^2=49
Step-by-step explanation:
equation of a circle is:
(x-h)^2+(y-k)^2=r^2
(x-5)^2+(y-4)^2=7^2
(x-5)^2+(y-4)^2=7x7
(x-5)^2+(y-4)^2=49
The equation of the circle of center (5, 4) and radius 7, is:
(x - 5)^2 + (y - 4)^2 = 49
How to find the equation of the circle?
The general equation for a circle of radius R centered at the point (a, b) is:
(x - a)^2 + (y - b)^2 = R^2
In this case, the center is (5, 4) and the radius is 7, so we need to replace:
a = 5
b = 4
R = 7
Then the equation is:
(x - 5)^2 + (y - 4)^2 = 7^2 = 49
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a) Spouses are asked about the number of hours of sleep they get each night. Upper A researcher wants to see if husbands get more sleep than wives. Choose the correct answer below. A. This scenario should not be analyzed using paired data because the groups are dependent but do not have a natural pairing. B. This scenario should not be analyzed using paired data because the groups are independent and do not have a natural pairing. C. This scenario should not be analyzed using paired data because the groups have a natural pairing but are independent. D. This scenario should be analyzed using paired data because the groups are dependent and have a natural pairing.
Answer:
D. This scenario should be analysed using paired data because the groups are dependent and have a natural pairing.
Step-by-step explanation:
Paired Data can be defined as data set, when sample units of one data set are related to sample units in other data set.
Research Study about spouses number of hours of sleep each night is paired data. As, each man in the male sample group is related to a corresponding woman (his wife) in the female sample group. So, they have a natural paring & they, their sleep schedules are dependent on each other.
Based on the information given, this scenario should be analyzed using paired data because the groups are dependent and have a natural pairing.
It should be noted that paired data is when sample units of one data set are related to sample units in another data set.Based on the information given, the case should be analyzed using paired data because the groups are dependent and have a natural pairing.In conclusion, the correct option is D.
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2. A random sample of 29 employees of a large company has their systolic blood pressure checked. Summary statistics are provided in the table below. Assume that the systolic blood pressure of all U.S. adults follows a normal distribution with a mean of 122 mm Hg and a standard deviation of 20 mm Hg.
(a) Approximately what percent of all U.S. adults have systolic blood pressure greater than 142 mm Hg?
(b) Describe the distribution of systolic blood pressure for the 29 employees of this company that was sampled.
(c) The company CEO wants to know if the mean systolic blood pressure of employees at her company is higher than the national average. State the hypotheses for testing this concern.
(d) The conditions for the hypothesis test in part (c) were satisfied. The hypothesis test resulted in a t-score of 5.495 and a p-value of 3.591 × 10−6. Interpret the p-value in the context of this hypothesis test. What would this p-value lead you to conclude?
(e) Explain in context what it would mean to make a type II error for the hypothesis test in part (c).
In addition to recording the systolic blood pressure of the 29 employees at the company, their ages were also recorded. A linear regression model was fit to these data. Graphical and numerical summaries of this analysis are given below. Use this information to answer the questions that follow. (Image is attached below)
(f) Interpret the slope of the regression line in this context.
(g) Comment on the strength, direction, and form of the relationship between age and systolic blood pressure.
Answer:
Step-by-step explanation:
Hello!
The variable of interest is
X: Systolic blood pressure of a U.S. adult. (mmHg)
X~N(μ;σ²)
μ= 122 mmHg
σ= 20 mmHg
For all calculations you have to work under the standard normal distribution, because it is tabulated. It is easier and faster to standardize or "translate" all values of X into values of Z and look for the corresponding probabilities in the table than to manually calculate them using the density function of the normal distribution.
The standard normal distribution is derived from the normal distribution. Considering a random variable X with normal distribution, mean μ and variance δ², the variable Z =(X-μ)/δ ~N(0;1) is determined.
Any value of any random variable X with normal distribution can be "converted" by subtracting the variable from its mean and dividing it by its standard deviation.
Since the distribution is centered in zero, there are two entries for its table, the left entry shows the cumulated probabilities corresponding to negative values of Z: P(Z≤z)=α and the right entry show the cumulated probabilities corresponding to positive values of Z: P(Z≤z)= 1 - α
a)
You need to calculate the percentage/ proportion of U.S. adults that have a systolic pressure greater than 142 mmHg, symbolically:
P(X>142) = 1 - P(X≤142)
To standardize this value od the variable you have to do the following calculation:
Z =(X-μ)/δ= (142-122)/20= 1
Now you look in the right entry for the cumulative probability until z= 1.00
(Remember, the first column of the table shows you the integer and first digit, the first row of the table shows you the second integer of the z value)
P(Z≤1)= 0.84134
Now you calculate the asked value:
P(X>142) = 1 - P(X≤142)= 1 - P(Z≤1)= 1 - 0.84134= 0.15866
b)
The sample mean is derived from a random variable with a normal distribution it shares that distribution with the exception that its variance is directly affected by the sample size:
X[bar]~N(μ;σ²/n)
μ= 122 mmHg
σ/√n= 20/√29= 3.71 mmHg
c)
The claim is that the mean systolic pressure of the employees is higher than the national average, symbolically μ> 122
The statistical hypotheses are:
H₀: μ≤ 122
H₁: μ> 122
d)
t= 5.495 p-value: 0.000003591
In this example the test statistic depends on the mean and the p-value is 3.591*10⁻⁶. This value indicates that 0.0003591% of the samples with size n=29 taken from a population with mean 122 mmHg, will produce a mean that provides evidence as (or stronger) than the current sample that μ is not at most 122 mmHg.
e)
The type II error is the scenario when you fail to reject the null hypothesis when the hypothesis is false. In this case, it is to fail to reject that the average systolic pressure of the company's employees is at most as the national average.
f)
The variable "X: Age of an employee" was recorded and linear regression of the systolic blood pressure as a function of the employee's ages estimated.
^Y= 97.0771 + 0.9493Xi
0.9493mmHg/years is the modification of the estimated average systolic blood pressure of the company's employees when their age increases one year.
g)
To determine the type of linear regression between the two variables, you have to analyze the slope of the equation:
As you can see in the graphic, the slope of the regression is positive, which means there is a positive regression between the systolic pressure and the age of the employees. I.e. each time the age of the employee increases, his systolic pressure also increases.
To determine the strength of the regression between these two variables you have to analyze the coefficient of determination R²:
The coefficient of determination gives you an idea of how much of the variability of the dependent variable (Y) is due to the explanatory variables under the estimated regression. It takes values between 0 to 1 or 0 to 100% if expressed in percentage. The closer the coefficient is to zero, the weaker the relationship between these two variables.
The closer the coefficient is to 100%, the stronger the relationship between these two variables.
R²= 0.712
71.2% of the variability of the systolic pressure is explained by the age fo the employees under this estimated model ^Y= 97.0771 + 0.9493Xi
The relationship between these variables is strong enough to consider the regression.
I hope this helps!
A six sided number cube has
one number, from 1 through 6,
on each cube. Determine the
probability of the following:
Pleven #)
7.SP.5
Answer:
Probability of even = 0.5
Step-by-step explanation:
Probability of even = total number of even numbers/ total number
Total number of Eve number = [ 2,4,6] = 3
Total numbers = [ 1,2,3,4,5,6] = 6
Probability of even = 3/6
Probability of even = 1/2
Probability of even = 0.5
1. Which of the following is a binomial?
A 5b2 + 3b - 8
B-2x} + 14x=1
C -9a - 15
D-v3 + 4uv+'lly - 1
Answer:
C -9a - 15
Step-by-step explanation:
bionomial means two terms
Shring's dining room table is round
and has a radius of 45 feet What is
the tabletop's area?
Answer:
it is 6,358.2. hope this helps
Compute: 3:40 - 1:55
Answer:
0 days 3 hours 40 minutes 0 seconds
- 0 days 1 hours 55 minutes 0 seconds
= 0 days 1 hours 45 minutes 0 seconds
= 0.0729167 days
= 1.75 hours
= 105 minutes
= 6,300 seconds
Step-by-step explanation:
Final answer:
To calculate the difference between given times, convert them into minutes and subtract. The result will be the time difference in minutes.
Explanation:
Compute: 3:40 - 1:55
To compute the difference between 3:40 and 1:55, you need to subtract the two times. First, convert both times into minutes: 3:40 = 3 x 60 + 40 = 180 + 40 = 220 minutes, and 1:55 = 1 x 60 + 55 = 60 + 55 = 115 minutes. Then, subtract 115 from 220 to get the result.
The difference between 3:40 and 1:55 is 105 minutes.
A police officer is concerned about speeds on a certain section of Interstate 95. The data accompanying this exercise show the speeds of 40 cars on a Saturday afternoon. Click here for the Excel Data File Click here for the CSV Data File a. The speed limit on this portion of Interstate 95 is 64 mph. Specify the competing hypotheses in order to determine if the average speed is greater than the speed limit.
Answer:
Jo : μ ≤ 64.
Jo : μ > 64.
Step-by-step explanation:
From the question, the Important information to solve this question is that; ''the speed limit on this portion of Interstate 95 is 64 mph''.
The question wants us to find the fact that the average speed is greater or more than the speed limit.
Therefore, In order to solve this question, the formula to use is the ''null and alternative" hypothesis. It states that;
Jo : μ ≤ 64.
Jo : μ > 64.
The above expression is the competing hypotheses in order to determine if the average speed is greater than the speed limit.
kevin started his new job with a salaryof $26,550. every year, he recieves a 3.1% increase in his salary. write and use a continuous exponential growth to find his salary after 20 years?
its $48892.00
Answer:
His salary after 20 years is $48892
Step-by-step explanation:
Kevin started his new job with a salary of $26,550.
We are also given that every year, he receives a 3.1% increase in his salary
So, rate of increase = 3.1%
Formula : [tex]y(t)=y_0(1+r)^t[/tex]
Where y(t)= Amount after t years
[tex]y_0[/tex] = initial amount = 26550
r = rate of increase in decimals = [tex]\frac{3.1}{100}[/tex]
t = time =20
we are supposed to find his salary after 20 years
Substitute the values in the formula :
Salary after 20 years = [tex]26500(1+\frac{3.1}{100})^{20}=48892.00[/tex]
Hence His salary after 20 years is $48892
HELP ME PLEASE, it’s confusing
The number at the end of the equation shifts the graph up or down.
Changing the 1/2 to a -2 would shift the graph down which would change the y-intercept.
The answer is D.
- + 106 = 315 cubic inches
2. Find the volume of the blue rectangular prism first.
v=-
cubic inches
V=
cubic inches
Total volume =
X
x
9 in.
X
x
Sin
Answer:585 cubic inch
Step-by-step explanation:
volume=15 x 10 x 3 + 9 x 3 x 5
Volume=450 + 135
Volume=585
Use the complement to find the probability. Enter your answer in simplified fraction form.
A spinner has 3 equal sections that are white, green, and blue. What is the probability of not landing on blue?
The probability of not landing on blue is
What percentage of the Mountain View polled supported Olunloyo?
Answer: The answer is C (about 47%) because the total amount of Mountain View residents is 0.38, 0.18 people chose Olunloyo, which is about 47 percent of residents.
To find out what percentage of the Mountain View polled supported Olunloyo, divide the number of Olunloyo supporters by the total number of poll participants, and multiply by 100. This reflects the views of the polled sample, not the entire population.
Explanation:To determine what percentage of the Mountain View polled supported Olunloyo, you would first need to know the total number of people who participated in the poll from Mountain View. Then, find out the number of these participants who indicated their support for Olunloyo. Divide the number of supporters by the total number of participants, and multiply the result by 100. This will give you the percentage of Mountain View residents in the poll who supported Olunloyo. Remember that this only applies to the sample of the population that participated in the poll and may not reflect the entire population of Mountain View.
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A manufacturer of t-shirts marks a shirt as "irregular" when it has defects such as crooked seams, stains, rips, or holes. A small number of irregular t-shirts are expected as part of the manufacturing process, but if more than 8% of the t-shirts manufactured at a plant are classified as irregular, the manager has to do an investigation to try to find the source of the increased mistakes in the manufacturing process. In order to test whether his plant is making a higher than expected number of irregular t-shirts, the manager of a plant randomly selects 100 t-shirts and finds that 12 are irregular. He plans to test the hypothesis: H0, P = 0.08, versus Ha, p > 0.08 (where p is the true proportion of irregular t-shirts). What is the test statistic
Answer:
The test statistic value is 1.474.
Step-by-step explanation:
In this case we need to determine whether the plant is making a higher than expected number of irregular t-shirts.
If more than 8% of the t-shirts manufactured at a plant are classified as irregular, the manager has to do an investigation to try to find the source of the increased mistakes in the manufacturing process..
The hypothesis for this test can be defined as follows:
H₀: The proportion of irregular t-shirts is 8%, i.e. p = 0.08.
Hₐ: The proportion of irregular t-shirts is more than 8%, i.e. p > 0.08.
The information provided is:
n = 100
X = number of irregular t-shirts = 12
Compute the sample proportion as follows:
[tex]\hat p=\frac{X}{n}=\frac{12}{100}=0.12[/tex]
Compute the test statistic as follows:
[tex]t=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
[tex]=\frac{0.12-0.08}{\sqrt{\frac{0.08(1-0.08)}{100}}}\\\\=1.47441\\\\\approx 1.474[/tex]
Thus, the test statistic value is 1.474.
help, Which equation should be used to find the volume of the figure?
Answer:
V = (10*6*12)/3
V = 720/3
V = 240 cm^3
Step-by-step explanation:
V = LWH/3
L = base length
W = base width
H = pyramid height
Kyle and Julie are playing a game where they flip a fair coin four times and try to predict the outcomes. ... Using the sample space of possible outcomes listed below, answer each of the following questions. What is P(A)P(A)P, left parenthesis, A, right parenthesis, the probability that the first flip is heads?
Answer: 1/2
Step-by-step explanation:
Students in a statistics class are conducting a survey to estimate the mean number of hours each week that students at their college study. The students collect a random sample of 49 students. The mean of the sample is 12.2 hours. The standard deviation is 1.6 hours. Use the T-distribution Inverse Calculator applet to answer the following question. What is the 95% confidence interval for the number of hours students in their college study?
The 95% confidence interval for the number of hours students at the college study weekly, given a sample size of 49, mean of 12.2 hours, and standard deviation of 1.6, is from 11.62 hours to 12.78 hours.
Explanation:In statistics, the 95% confidence interval for the mean number of hours students study in a week can be calculated using the collected data and the t-distribution. Given that the sample size is 49 (n=49), the sample mean(x) is 12.2 hours, and the standard deviation(s) is 1.6 hours, we need to apply the formula for the confidence interval which is x ± t∗(s/√n). The t-value for 95% confidence with 48 degrees of freedom (49-1) can be found using the t-distribution Inverse Calculator applet, which is about 2.01 (it varies slightly depending on the calculator used).
Now, plug in the given values into the formula: 12.2 ± 2.01 * (1.6 / √49), it simplifies to 12.2 ± 0.58. So, the 95% confidence interval for the number of hours students in the college study each week is from 11.62 hours to 12.78 hours.
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Your small farm encompasses 120 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $3,000 for each acre of tomatoes, $1,400 for each acre of lettuce and $400 for each acre of carrots. You would like to spend at least $480 on fertilizer and your farm laborers can supply up to 600 hours per week. How many acres of each crop should you plant to maximize total profits?
tomatoes acre(s) =
lettuce acre(s) =
carrots acre(s)=
profit $ =
In this event, will you be using all 120 acres of your farm?
Answer:
tomatoes acre(s) = 120lettuce acre(s) = 0carrots acre(s)= 0profit $ = $360,000all 120 acres are usedStep-by-step explanation:
You can write the linear system model as follows. Let t, l, c represent acres of tomatoes, lettuce, and carrots, respectively. The we have ...
t + l + c ≤ 120 . . . . . constraint on available land
5t +4l +2c ≥ 480 . . requirement for spending on fertilizer
4t +2l +2c ≤ 600 . . constraint on available labor
t ≥ 0; l ≥ 0; c ≥ 0 . . requirement for non-negative acres
Then the objective function (profit) is ...
p = 3000t +1400l +400c
The linear programming problem is to maximize p subject to the above constraints.
__
Any of a variety of solvers can find the solution to this problem. That solution is ...
(t, l, c) = (120, 0, 0) and p = 360,000
In summary ...
tomatoes acre(s) = 120 (all available acres are used)
lettuce acre(s) = 0
carrots acre(s) = 0
profit $ = $360,000
_____
Additional comments
This solution suggests that it can be found simply by examining the profit associated with each unit of resource.
profit per acre is maximized for tomatoes, at $3000 per acre
profit per fertilizer dollar is maximized for tomatoes, at $600 per dollar
profit per labor hour is maximized for tomatoes, at $750 per hour
That is, the profit per acre is maximized for tomatoes, regardless of the resource being considered. Thus it make sense to put all of the acreage in tomatoes. At $5 per acre for fertilizer, we use $600 worth of fertilizer. At 4 hours per week per acre, we use 480 hours of labor, so not all available labor is used.
To maximize profits, use linear programming to find the number of acres for each crop. The total farm acres used depends on the constraints.
Explanation:To maximize total profits, the number of acres of each crop should be determined. Let's assign the variables t, l, and c to represent the number of acres of tomatoes, lettuce, and carrots, respectively. We have the following constraints:
The total cost of fertilizer must be at least $480: 5t + 4l + 2c ≥ 480The total hours of labor must not exceed 600: 4t + 2l + 2c ≤ 600We want to maximize total profits: Profit = 3000t + 1400l + 400cThe objective is to find the values of t, l, and c that satisfy the constraints and maximize the profit. This is a linear programming problem that can be solved using methods such as the simplex method or graphical method.
To determine if all 120 acres will be used, we need to check if t + l + c = 120 under the constraints.
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The net below represent a container. What solid figure does it show?
How many vertices does the container have?
The solid figure that is shown by the net would be a Rectangular prism.
The number of vertices would be 8 vertices.
How to describe a rectangular prism?A rectangular prism is a 3D shape with six faces, eight vertices, and twelve edges. It is a type of polyhedron, which is a solid figure with flat faces and straight edges.
The net given is a rectangular prism because the dashed lines show that the sides would fold up along with the triangles to form a rectangular prism.
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What is the mean of 3, 2, 26, 9
Answer:
10
Step-by-step explanation:
3+2=5+26=31+9=40/4=10
Step-by-step explanation:
When finding mean, you have to add up all of the number and divide that answer by the number of numbers used.
3 + 2 + 26 + 9 = 40
Now, we will divide 40 by 4 because there are 4 numbers.
40 / 4 = 10
So, the mean of the set is 10.
Find the value of g(7) for the function below. g(x)= 7/8x -1/2
g(x) = ⅞x - ½
g(7) = ⅞(7) - ½
= 49/8 - ½
= 49/8 - 4/8
= 45/8
= 5 ⅝
So, the value of g(7) for that function is 5 ⅝
Hope it helpful and useful :)
The value of the given function for g(7) is 45/8.
The given function is [tex]g(x)=\frac{7}{8}x-\frac{1}{2}[/tex].
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
We need to find the value of g(7)
Substitute x=7 in the given function and simplify, we get
[tex]g(7)=\frac{7}{8}(7)-\frac{1}{2}[/tex]
= 49/8 - 1/2
= 49/8 - 4/8
= 45/8
Hence, the value of the given function for g(7) is 45/8.
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Solve the logarthmic equation. 2 log3(x + 4) - log39 = 2
Answer:
5
Step-by-step explanation:
2 log₃(x + 4) − log₃9 = 2
2 log₃(x + 4) − log₃3² = 2
2 log₃(x + 4) − 2 = 2
2 log₃(x + 4) = 4
log₃(x + 4) = 2
3^(log₃(x + 4)) = 3^2
x + 4 = 9
x = 5
The solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.
To solve the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2, we can use logarithmic properties to simplify the equation and then solve for x.
First, let's apply the properties of logarithms:
2 log₃(x + 4) - log₃(9) = 2
Using the power rule of logarithms, we can rewrite 2 log₃(x + 4) as log₃((x + 4)²):
log₃((x + 4)²) - log₃(9) = 2
Next, we can combine the logarithms using the quotient rule:
log₃((x + 4)² / 9) = 2
Now, we can eliminate the logarithm by rewriting the equation in exponential form:
3² = (x + 4)² / 9
9 = (x + 4)² / 9
Multiplying both sides of the equation by 9:
81 = (x + 4)²
Taking the square root of both sides:
±√81 = ±√((x + 4)²)
±9 = x + 4
Solving for x, we have two possible solutions:
x + 4 = 9
x = 9 - 4
x = 5
or,
x + 4 = -9
x = -9 - 4
x = -13
Therefore, the solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.
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A math teacher claims that she has developed a review course that increases the scores of students on the math portion of a college entrance exam. Based on data from the administrator of the exam, scores are normally distributed with mu equalsμ=522522. The teacher obtains a random sample of 20002000 students, puts them through the review class, and finds that the mean math score of the 20002000 students is 527527 with a standard deviation of 110110. Complete parts (a) through (d) below. (a) State the null and alternative hypotheses. Let muμ be the mean score. Choose the correct answer below. A. Upper H 0 : mu equals 522H0: μ=522, Upper H 1 : mu not equals 522H1: μ≠522 B. Upper H 0 : mu less than 522H0: μ<522, Upper H 1 : mu greater than 522H1: μ>522 C. Upper H 0 : mu greater than 522H0: μ>522, Upper H 1 : mu not equals 522H1: μ≠522 D. Upper H 0 : mu equals 522H0: μ=522, Upper H 1 : mu greater than 522
Answer: D
H0: μ=522
H1: μ>522
Step-by-step explanation:
The null hypothesis (H0) tries to show that no significant variation exists between variables or that a single variable is no different than its mean. While an alternative Hypothesis (Ha) attempt to prove that a new theory is true rather than the old one. That a variable is significantly different from the mean.
So, for this case;
The null hypothesis is that the mean score equals to 522
H0: μ=522
The alternative hypothesis is that the mean score is greater than 522.
H1: μ>522
The correct answer is D. [tex]Upper H_0: = 522, Upper H_1 > 522[/tex].
In this scenario, the math teacher is claiming that her review course increases the scores of students on the math portion of a college entrance exam. Therefore, the null hypothesis [tex](H_0)[/tex] should represent the claim that there is no increase in the mean score, which would be that the mean score is equal to the population mean of 522. The alternative hypothesis [tex](H_1)[/tex] should represent the teacher's claim that the mean score has increased, which would be that the mean score is greater than 522.The null hypothesis is a statement of no effect or no difference and serves as the baseline for testing. The alternative hypothesis is what we aim to support; it is a statement that there is an effect or a difference. In this case, we are testing whether the mean score of students after taking the review course is greater than the population mean.Therefore, the correct hypotheses are:- Null Hypothesis [tex](H_0): = 522[/tex] , indicating no increase in the mean score.- Alternative Hypothesis [tex](H_1): > 522[/tex] , indicating an increase in the mean score.This is a one-tailed test because we are only interested in whether the mean score has increased, not whether it has changed at all (which would require a two-tailed test).