Using the combination formula, the student can make 165 different possible schedules for next semester if she is planning to take three classes from the eleven available that she has met the prerequisites for.
Explanation:This problem is a combination problem where students need to select 3 classes from 11 available classes to plan her schedule. The combination formula is used when the order of election does not matter. Here, the formula to find a combination is written as C(n, r) = n! / [r!(n - r)!].
In this particular case, the student has 11 classes (n=11) and plans to take 3 classes (r=3). So, the combination becomes C(11,3) = 11! / [3!(11-3)!] = (11*10*9) / (3*2*1) = 165.
This means, the student can make 165 different possible schedules for next semester assuming she is planning to take three classes from the eleven available classes.
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The student could make 165 different schedules.
The formula for combinations is given by:
[tex]\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \][/tex]
where n! (n factorial) is the product of all positive integers up to n, and r is the number of items to choose.
Given that the student has 11 classes left and she plans to take 3 classes next semester, we substitute n = 11 and r = 3 into the combination formula:
[tex]\[ \binom{11}{3} = \frac{11!}{3!(11-3)!} \][/tex]
Now, we simplify the factorial expressions:
[tex]\[ \binom{11}{3} = \frac{11!}{3! \times 8!} \][/tex]
We can cancel out the common terms in the numerator and the denominator:
[tex]\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} \][/tex]
Now, we perform the arithmetic:
[tex]\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \][/tex]
Thus, the student has 165 different possible schedules for next semester if she is to take 3 classes out of the remaining 11.
For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.(a) X = the number of unbroken eggs in a randomly chosen standard egg carton Describe the set of possible values for the variable. (0, 12] 1, 2, 3, ..., 12 0, 1, 2, 3, ..., 12 [0, 12] State whether the variable is discrete. discrete not discrete.
Answer:
X is a discrete random variable.
X can take values from 0 to 12:
[tex]X\in[0,1,2,3,4,5,6,7,8,9,10,11,12][/tex]
Step-by-step explanation:
(a) X = the number of unbroken eggs in a randomly chosen standard egg carton
X is a discrete random variable.
The minimum amount of eggs broken is 0 and the maximum amount of eggs broken is 12 (assuming a dozen egg carton).
Then, X can take values from 0 to 12:
[tex]X\in[0,1,2,3,4,5,6,7,8,9,10,11,12][/tex]
About the probability ditribution nothing can be said, because there is no information about it (it can be a binomial, uniform or non-standard distribution).
The set of possible values for the random variable X, representing the number of unbroken eggs in an egg carton, is {0, 1, 2, 3, ..., 12}. X is a discrete random variable because it takes on distinct, countable values with no gaps in between.
The random variable X represents the number of unbroken eggs in a randomly chosen standard egg carton. To describe the set of possible values for this variable, we can consider the number of eggs in a standard carton, which is typically 12. Therefore, the set of possible values for X would be {0, 1, 2, 3, ..., 12}, as it encompasses all possible outcomes, ranging from having no unbroken eggs (0) to having all 12 unbroken eggs.
Now, let's determine whether the variable X is discrete or not. A discrete random variable is one that can take on a countable number of distinct values with gaps in between.
In this case, X is indeed a discrete random variable because it can only take on integer values from 0 to 12, and there are no intermediate values between these whole numbers. Each value represents a distinct and countable outcome based on the number of unbroken eggs, making it a discrete random variable.
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A physics class has 40students. Of these, 10students are physics majors and 17students are female. Of the physics majors, fourare female. Find the probability that a randomly selected student is female or a physics major.The probability that a randomly selected student is female or a physics major is___. (Round to 3 decimal places)
Answer:
The probability that a randomly selected student is female or a physics major is 0.575.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
Desired outcomes:
Students that are female or physics majors.
17 female
10 physics majors, of which 4 are female.
This means that there are 10 total physics majors and 17-4 = 13 non physics majors female. So
[tex]D = 13 + 10 = 23[/tex]
Total outcomes:
The class has 40 students, so [tex]T = 40[/tex]
Probability
[tex]P = \frac{23}{40} = 0.575[/tex]
The probability that a randomly selected student is female or a physics major is 0.575.
The life in hours of a biomedical device under development in the laboratory is known to be approximately normally distributed. A random sample of 15 devices is selected and found to have an average life of 5323.8 hours and a sample standard deviation of 220.9 hours.
Test the hypothesis that the true mean life of a biomedical device is greater than 5200.
Answer:
We conclude that the true mean life of a biomedical device is greater than 5200 hours.
Step-by-step explanation:
We are given that the life in hours of a biomedical device under development in the laboratory is known to be approximately normally distributed. For this a random sample of 15 devices is selected and found to have an average life of 5323.8 hours and a sample standard deviation of 220.9 hours.
We have to test that the true mean life of a biomedical device is greater than 5200 or not.
Let, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \leq[/tex] 5200 {means that the true mean life of a biomedical device is less than or equal to 5200 hours}
Alternate Hypothesis, [tex]H_1[/tex] : [tex]\mu[/tex] > 5200 {means that the true mean life of a biomedical device is greater than 5200 hours}
The test statistics that will be used here is;
T.S. = [tex]\frac{Xbar-\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]
where, Xbar = sample average life = 5323.8 hours
s = sample standard deviation = 220.9 hours
n = sample devices = 15
So, test statistics = [tex]\frac{5323.8-5200}{\frac{220.9}{\sqrt{15} } }[/tex] ~ [tex]t_1_4[/tex]
= 2.171
Since, we are not given with the significance level, so we assume it to be 5%, now the critical value of t at 14 degree of freedom in t table is given as 1.761. Since our test statistics is more than the critical value of t which means our test statistics will lie in the rejection region. So, we have sufficient evidence to reject our null hypothesis.
Therefore, we conclude that the true mean life of a biomedical device is greater than 5200 hours.
Jim began a 226-mile bicycle trip to build up stamina for a triathlete competition. Unfortunately, his bicycle chain broke, so he finished the trip walking. The whole trip
took 7 hours. If Jim walks at a rate of 4 miles per hour and rides at 40 miles per hour, find the amount of time he spent on the bicycle
Answer: 5.5hours
Step-by-step explanation:
Total distance = 226miles
Total time = 7 hours
Let b represent total time spent while walking.
Distance (walking) = 4b
Distance ( riding) = 40(7-b)
Total distance 226 = 4b + 40(7-b)
226 = 4b + 280 - 40b
226 = 280-36b
b = 54/36
b = 1.5hours
Amount of time spent walking = 1.5hours
Amount of time spent riding = 7-1.5 = 5.5hours
Amount of time spent on bicycle = 5.5hours
12x - 5y = - 20,
12x - 5y = -20y = x + 4
y = x +4
12x-5y=-20
y=x+4
12x-5(x+4)=-20
12x-5x-20=-20
7x=0,
So, we get: x=0 and y=4
Suppose a liquor store sells beer for a net profit of $1 per unit and wine for a net profit of $2 per unit. Let x equal the amount of beer sold and y equal the amount of wine sold. An algebraic formulation of the profit function is:_______
a. max(2x + y)
b. min(2x + y)
c. max(x + 2y)
d. min(x + 2y)
Answer:
The correct optiion is C
Step-by-step explanation:
Beer =$1 and amount sold is x
so $1×x= $x which is the profit on beer
Wine=$2 and amount sold is y
so, $2×y= $2y which is the profit on wine
so an algebraic formulation of the profit function will be,
the sum off both the profit of the beer and wine which is
x+2y= max(x+2y)
An outbreak of the deadly Ebola virus in 2002 and 2003 killed 91 of the 95 gorillas in 7 home ranges in the Congo. To study the spread of the virus, measure distance by the number of home ranges separating a group of gorillas from the first group infected. Here are data on distance and number of days until deaths began in each later group:
Distance x 1 4 6 6 6 7
Days y 2 20 29 38 44 50
Find the correlation r between x and y.
r = ?
Answer:
Correlation between x and y is 0.9508
Step-by-step explanation:
We are given the following in the question:
Distance(x): 1 4 6 6 6 7
Days(y): 2 20 29 38 44 50
We have to find the correlation between x and y.
[tex]\sum y = 183\\\sum x=30\\\bar{x} = \displaystyle\frac{\sum x}{n} = \frac{30}{6} = 5\\\\\bar{y} = \displaystyle\frac{\sum y}{n} = \frac{183}{6} = 30.5\\\\(x-\bar{x}) = -4,-1,+1,+1,+1,+2\\(y-\bar{y}) = -28.5,-10.5,-1.5,7.5,13.5,19.5\\\sum (x-\bar{x}) (y-\bar{y}) = 183\\\sum(x-\bar{x})^2 = 24\\\sum(y-\bar{y})^2= 1543.5\\[/tex]
Formula:
[tex]r = \dfrac{\sum(x-\bar{x})(y - \bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}}[/tex]
Putting values, we get,
[tex]r = \dfrac{183}{\sqrt{24\times 1543.5}} = 0.9508[/tex]
Thus, correlation between x and y is 0.9508
To find the correlation between the distance and the number of days until deaths began, we can use a statistical measure called the correlation coefficient (r). The formula for calculating r is given, and we can use the provided data to calculate the correlation coefficient, which is approximately -0.994, indicating a strong negative correlation.
Explanation:To find the correlation between the distance and the number of days until deaths began, we can use a statistical measure called the correlation coefficient (r). The formula for calculating r is:
r = (n(Σxy) - (Σx)(Σy)) / √((n(Σx^2) - (Σx)^2)(n(Σy^2) - (Σy)^2))
Using the provided data, we can calculate the correlation coefficient:
Calculate the sum of x, y, xy, x^2, and y^2.Substitute the values into the formula and simplify.Calculate the square root to find the correlation coefficient (r).After performing the calculations, we find that the correlation coefficient (r) is approximately -0.994, indicating a strong negative correlation between distance and the number of days until deaths began.
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There was a special on sweatshirts is sweatshirt was on sale for $9.69 if a customer bought three striped shirt at the regular price of $12.95 a fourth sweatshirt was free which is the better buy for sweatshirts at 9.69 each or three sweatshirts at $12.95 in a fourth one free?
Answer: it is cheaper to buy for sweatshirts at $9.69 each.
Step-by-step explanation:
The regular price for one sweatshirt is $12.95. if a customer bought three shirts at the regular price of $12.95, a fourth sweatshirt was free
It means that the cost of buying 4 shirts is
12.95 × 3 = $38.85
Due to a special, the price of one sweatshirt was $9.69. It means that the cost of buying 4 shirts at this price is
9.69 × 4 = $38.76
Therefore, it is cheaper to buy for sweatshirts at $9.69 each than to buy at $12.95 each and get a free shirt
Answer: it is cheaper to buy for sweatshirts at $9.69 each.
If np is greater than or equal to 15 and n(1-p) is greater than or equal to 15, what is the approximate shape of the sampling distribution of the sample proportion?
A. p-hat
B. true proportion p
C. x-bar
D. sqrt(p*(1-p)/n)
E. Normal
F. sigma/sqrt(n)
G. Binomial
H. Bimodal
Answer:
We need to check the conditions in order to use the normal approximation.
[tex]np \geq 15[/tex]
[tex]n(1-p) \geq 15[/tex]
If we see that we satisfy the conditions and then we can apply the approximation.
If we appply the approximation the new mean and standard deviation are:
[tex]E(X)=np[/tex]
[tex]\sigma=\sqrt{np(1-p)}[/tex]
[tex] X \sim N (\mu = np, \sigma=\sqrt{np(1-p)}) [/tex]
So then the correct answer for this case would be:
E. Normal
Step-by-step explanation:
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Let X the random variable of interest, on this case we assume that:
[tex]X \sim Binom(n, p)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
We need to check the conditions in order to use the normal approximation.
[tex]np \geq 15[/tex]
[tex]n(1-p) \geq 15[/tex]
If we see that we satisfy the conditions and then we can apply the approximation.
If we appply the approximation the new mean and standard deviation are:
[tex]E(X)=np[/tex]
[tex]\sigma=\sqrt{np(1-p)}[/tex]
[tex] X \sim N (\mu = np, \sigma=\sqrt{np(1-p)}) [/tex]
So then the correct answer for this case would be:
E. Normal
The approximate shape of the sampling distribution of the sample will be normal.
Option E is correct.
Sampling distribution:It is a statistic that determines the probability of an event based on data from a small group within a large population.
Given that, [tex]np[/tex] is greater than or equal to 15 and [tex]n(1-p)[/tex] is greater than or equal to 15.
[tex]np\geq 15[/tex] and [tex]n(1-p)\geq 15[/tex]
So that the new mean and standard deviation will be,
[tex]mean=\mu=np\\\\Deviation=\sigma=\sqrt{np(1-p)}[/tex]
Thus, the approximate shape of the sampling distribution of the sample will be normal.
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The sum of a number and 47 is prime. Which could be the sum
Answer:
53
Step-by-step explanation:
47+6=53
Answer:
53
Step-by-step explanation:
Consider the following events for a driver selected at random from a general population.
A = driver is under 25 years old (1)
B = driver has recieved a speeding ticket (2)
Translate each of the following phrases into symbols.
(a) The probability the driver is under 25 years old and has recieved a speeding ticket.
(b) The probability a driver who is under 25 years old has recieved a speeding ticket.
(c) The probability a driver who has recieved a speeding ticket is 25 years or older.
(d) The probability the driver is under 25 years old or has recieved a speeding ticket.
(e) The probability the driver is under 25 years old or has not recieved a speeding ticket.
Answer:
a. P(AnB)
b. P(B|A)
c. [tex]P(A^I|B)[/tex]
d. P(A or B)
e. [tex]P(B^I or A)[/tex]
Step-by-step explanation:
Since A= driver is under 25 years old (1)
B = driver has received a speeding ticket (2)
a.The probability the driver is under 25 years old and has recieved a speeding ticket.
this simple means the intersection of both set, which can be written as
P(AnB)
b. The probability a driver who is under 25 years old has received a speeding ticket.
This is a conditional probability, probability that B will occur given that A as occur.
P(B|A)
c. the probability a driver who has received a speeding ticket is 25 years or older.
[tex]P(A^I|B)[/tex]
d. The probability the driver is under 25 years old or has received a speeding ticket.
P(A or B)
e. The probability the driver is under 25 years old or has not received a speeding ticket.
[tex]P(B^I or A)[/tex]
A five-card poker hand is dealt at random from a standard 52-card deck. Note the total number of possible hands is C(52,5)=2,598,960. Find the probabilities of the following scenarios: (a) What is the probability that the hand contains exactly one ace?
Answer:
(a) Probability = 0.29947
Step-by-step explanation:
The probability of the hand containing exactly one ace would be:
Number of ways this can happen = 4C1 * 48C4 (using combinations)
Number of ways this can happen = 4 * 194580
Number of ways this can happen = 778,320
Total number possible hands = 2,598,960 (as stated in question)
Total probability of exactly one ace = Number of ways to get an ace / total number of ways
Total probability = 778320 / 2598960 = 0.29947
Thus, the probability of the hand containing exactly one ace will be 0.2994
Another way to solve this:
Probability of one ace and 5 other cards = [tex]\frac{4}{52}*\frac{48}{51}*\frac{47}{50}*\frac{46}{49}*\frac{45}{48}[/tex] = 0.059894
Number of ways to arrange 1 ace and 4 other cards = 5
Total probability = 0.0598 * 5 = 0.29947
The probability of getting exactly one ace in a five-card poker hand is 0.2556 (rounded to four decimal places).
Explanation:To find the probability of getting exactly one ace in a five-card poker hand, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes. In a standard 52-card deck, there are 4 aces, and we need to choose 1 ace out of the 4. The remaining 4 cards in the hand can be chosen from the remaining 48 non-ace cards in the deck. Hence, the number of favorable outcomes is C(4,1) * C(48,4). The probability can be calculated as:
P(exactly one ace) = (C(4,1) * C(48,4)) / C(52,5)
Substituting the values and evaluating the expression, we get:
P(exactly one ace) = (4 * 171,230) / 2,598,960 = 0.2556 (rounded to four decimal places)
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Can someone help me with this question?
Step-by-step explanation:
Hope it helps you in your learning process.
Suppose X is a continuous variable with the following probability density: f(x)={C(10−x)2,0, if 0
Answer:
[tex]C = 1/18[/tex]
Step-by-step explanation:
Remember that for a probability density function
[tex]\int_{-\infty}^{\infty } f(x) dx = 1[/tex]
Since [tex]f(x) = 0[/tex] outside [tex][0,2][/tex] we would have that
[tex]\int_{0}^{2} C(10-x) dx = 1[/tex]
Therefore
[tex]18C = 1 \\C = 1/18[/tex]
A sample of 140 Vopstra customers have had their annual phone charge recorded for the previous calendar year. The data were used to calculate a 92% confidence interval for the mean annual phone charge of all Vopstra customers. The confidence interval was calculated as $470 + $65. According to this confidence interval, it is most reasonable to conclude that:a.you are 92% confident the interval between $405 and $535 contains the mean phone charge of all Vopstra customers b.you are 92% confident the mean phone charge of all Vopstra customers is approximately $470 c.you are 92% confident the mean phone charge of all mobile phone customers is approximately $470 d.you are 92% confident the interval between $405 and $535 contains the mean phone charge of all mobile phone customers
Answer:
Correct option: (a)
Step-by-step explanation:
A confidence interval is an interval estimate of the parameter value.
A (1 - α)% confidence interval implies that the confidence interval has a (1 - α)% probability of consisting the true parameter value.
OR
If 100 such confidence intervals are made then (1 - α) of these intervals would consist the true parameter value.
The 92% confidence interval for the mean annual phone charge of all Vopstra customers is:
[tex]\$470\pm \$65=(\$405, \$535)[/tex]
This confidence interval implies that true mean annual phone charge of all Vopstra customers is contained in the interval ($405, $535) with 0.92 probability.
Thus, the correct option is (a).
Write the Leibniz notation for the derivative of the given function and include units. The cost, , of a steak, in dollars, is a function of the weight, , of the steak, in pounds.
Answer:
note:
please find the attachment
J. P. Morgan Asset Management publishes information about financial investments. Over the past years, the expected return for the S&P was with a standard deviation of and the expected return over that same period for a Core Bonds fund was with a standard deviation of (J. P. Morgan Asset Management, Guide to the Markets, 1st Quarter, ). The publication also reported that the correlation between the S&P and Core Bonds is . You are considering portfolio investments that are composed of an S&P index fund and a Core Bonds fund. a. Using the information provided, determine the covariance between the S&P and Core Bonds. Round your answer to two decimal places. If required enter negative values as negative numbers.
The covariance between two investments such as the S&P index fund and Core Bonds fund is calculated using the standard deviations of each investment and their correlation coefficient. This measure indicates if their returns move together. However, key data is not provided in the question.
Explanation:In finance, the covariance between two investments, such as the S&P index fund and a Core Bonds fund, is used as a measure of how their returns move together. The formula for covariance is standard deviation of instrument one multiplied by standard deviation of instrument two, multiplied by their correlation coefficient. However, there is missing data in your question: the standard deviations and correlation coefficient were not provided. Once you have those, use the formula: Covariance = (Standard Deviation of S&P) * (Standard Deviation of Core Bonds) * (Correlation coefficient of the S&P and Core Bonds). Now, if you find that the outcome is a large number, it means that the returns move a lot in unison, for both good and bad outcomes. A low positive or negative number means that any movements are not strongly linked.
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When testing for current in a cable with eight color-coded wires, the author used a meter to test five wires at a time. How many different tests are required for every possible pairing of five wires?
Answer:
56 different tests
Step-by-step explanation:
Given:
Number of wires available (n) = 8
Number of wires taken at a time for testing (r) = 5
In order to find the number of different tests required for every possible pairing of five wires, we need to find the combination rather than their permutation as order of wires doesn't disturb the testing.
So, finding the combination of 5 pairs of wires from a total of 8 wires is given as:
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Plug in the given values and solve. This gives,
[tex]^8C_5=\frac{8!}{5!(8-5)!}\\\\^8C_5=\frac{8\times 7\times 6\times 5!}{5!\times 3\times2\times1}\\\\^8C_5=56[/tex]
Therefore, 56 different tests are required for every possible pairing of five wires.
A total of 56 different tests are needed to check every possible combination of five wires out of eight in the cable.
Explanation:The question is asking for the number of different tests required to test every possible pairing of five wires in an eight-wire cable. This is a combination problem, where we're looking for how many different ways we can combine five items from a group of eight. The formula for combinations is C(n, k) = n! / [k!(n - k)!], where n is the total number of items, k is the number of items to choose, and ! represents factorial. Plugging into this formula, we get C(8, 5) = 8! / [5!(8 - 5)!] = 56. Therefore, a total of 56 different tests are needed to check every possible combination of five wires out of eight.
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A. Create a set of 5 points that are very close together and record the standard deviation. Next, add a sixth point that is far away from the original 5 and record the new standard deviation. What is the impact of the new point on the standard deviation?
Answer: The addition of the new point alters the previous standard deviation greatly
Step-by-step explanation:
Let the initial five points be : 2 3 4 5 and 6. In order to calculate the standard deviation for this data, we will need to calculate the mean first.
Mean = summation of scores/number of scores.
The mean is therefore: (2+3+4+5+6)/5 = 20/5 = 4.
We'll also need the sum of the squares of the deviations of the mean from all the scores.
Since mean = 4, deviation of the mean from the score "2" = score(2) - mean (4)
For score 3, it is -1
For 4, it's 0
For 5 it's 1
For 6 it's 2.
The squares for -2, -1, 0, 1, and 2 respectively will be 4, 1 , 0, 1, 4. Summing them up we have 10 i.e (4+1+0+1+4=10).
Calculating the standard deviation, we apply the formula:
√(summation of (x - deviation of mean)^2)/N
Where N means the number of scores.
The standard deviation = √(10/5) = 1.4142
If we add another score or point that is far away from the original points, say 40, what happens to the standard deviation. Let's calculate to find out.
i.e we now have scores: 2, 3, 4, 5, 6 and 40
We calculate by undergoing same steps.
Firstly mean. The new mean = (2+3+4+5+6+40)/6 = 60/6 = 10.
The mean deviations for the scores : 2, 3, 4, 5, 6 and 40 are -8, -7, -6, -5, -4 and 30 respectively. The squares of these deviations are also 64, 49, 36, 25, 16 and 900 respectively as well. Their sum will then be 1090. i.e. (64+49+36+25+16+900 = 1090).
The new standard deviation is then=
√(1090/6)
= √181.67
= 13.478.
It's clear that the addition of a point that's far away from the original points greatly alters the size of the standard deviation as seen /witnessed in this particular instance where the standard deviation rises from 1.412 to 13.478
Ted is making trail mix for a party. He mixes 1 1/2 cups of nuts, 1/4 cup of raisins, and 1/4 cup of pretzels. How many cups of pretzels does Ted need to make 15 cups of trail mix?
Answer:
Ted will need [tex]3\frac{3}{4}[/tex] cups of pretzels to make 15 cups of trail mix.
Step-by-step explanation:
Ted is making trail mix for a party. He mixes 1 1/2 cups of nuts, 1/4 cup of raisins, and 1/4 cup of pretzels.
So, 1/4 cup of pretzels to make 1 trail mix
x cups of pretzels to make 15 trail mix
Using the ratio and proportional
∴ x = (1/4) * 15 = 3.75 = [tex]3\frac{3}{4}[/tex] cups.
Ted will need [tex]3\frac{3}{4}[/tex] cups of pretzels to make 15 cups of trail mix.
Please help? (03.04) What are the coordinates of the vertex for f(x) = x^2 + 4x + 10?
Answer:
(-2, 6)
Step-by-step explanation:
f(x) = x² + 4x + 10
f(x) = x² + 4x + 4 + 6
f(x) = (x + 2)² + 6
The vertex is at (-2, 6).
A lab is testing the amount of a certain active chemical compound in a particular drug that has been recently developed. The manufacturer claims that the average amount of the chemical is 90 mg. It is known that the standard deviation in the amount of the chemical is 6 mg. A random sample of 31 batches of the new drug is tested and found to have a sample mean concentration of 92.2 mg of the active chemical.
Calculate the 95% confidence interval for the mean amount of the active chemical in the drug.
Answer:
95% confidence interval for the mean amount of the active chemical in the drug = [ 90.088 , 94.312 ]
Step-by-step explanation:
We are given that a lab is testing the amount of a certain active chemical compound in a particular drug that has been recently developed. It is known that the standard deviation in the amount of the chemical is 6 mg.
A random sample of 31 batches of the new drug is tested and found to have a sample mean concentration of 92.2 mg of the active chemical i.e.;
Population standard deviation, [tex]\sigma[/tex] = 6 mg
Sample mean, [tex]Xbar[/tex] = 92.2 mg
Sample size, n = 31
Now, the pivotal quantity for 95% confidence interval is given by;
[tex]\frac{Xbar -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
So, 95% confidence interval for the mean amount of the active chemical in the drug is given by;
P(-1.96 < N(0,1) < 1.96) = 0.95
P(-1.96 < [tex]\frac{Xbar -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.96) = 0.95
P(-1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] < [tex]Xbar - \mu[/tex] < 1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.95
P(Xbar - 1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] < [tex]\mu[/tex] < Xbar + 1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.95
95% confidence interval for [tex]\mu[/tex] = [Xbar - 1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] , Xbar + 1.96 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ]
= [ 92.2 - 1.96 * [tex]\frac{6}{\sqrt{31} }[/tex] , 92.2 + 1.96 * [tex]\frac{6}{\sqrt{31} }[/tex] ]
= [ 90.088 , 94.312 ]
To calculate the 95% confidence interval for the mean amount of the active chemical in the drug, substitute the sample mean concentration, standard deviation, and sample size into the formula: 95% Confidence Interval = Sample Mean ± (Z * (Standard Deviation / √Sample Size)).
Explanation:To calculate the 95% confidence interval for the mean amount of the active chemical in the drug, we can use the formula:
95% Confidence Interval = Sample Mean ± (Z * (Standard Deviation / √Sample Size))
Given that the sample mean concentration is 92.2 mg, the standard deviation is 6 mg, and the sample size is 31 batches, we can substitute these values into the formula:
95% Confidence Interval = 92.2 mg ± (1.96 * (6 mg / √31))
Calculating this expression gives us a 95% confidence interval of approximately 90.855 mg to 93.545 mg.
Consider two securities, A and B. Security A and B have a correlation coefficient of 0.65. Security A has standard deviation of 12, and security B has standard deviation of 25. Calculate the covariance between these two securities
Answer:
195
Step-by-step explanation:
The relationship between the covariance (cov_AB), and the correlation coefficient (ρ_AB = 0.65), and the standard deviations (σ_A = 12 and σ_B = 25) for the securities A and B is :
[tex]cov_{A,B} = \rho_{A,B}*\sigma_A*\sigma_B[/tex]
Applying the given data:
[tex]cov_{A,B} = 0.65*12*25\\cov_{A,B} = 195[/tex]
The covariance between these two securities is 195.
One of Shakespeare's sonnets has a verb in 11 of its 18 lines, an adjective in 13 lines, and both in 8 lines. How many lines have a verb but no adjective?
Answer:
Step-by-step explanation:
The total number of lines, n(U) = 18
Let the number of lins with verb be n(V) = 11
Let the number of lines with adjectives be n(A) = 13
n(V n A) = 8
Find the number of lines that have a verb but no adjective, that is, n(V n A')
Mathematically, according to sets theory,
n(V) = n(V n A) + n(V n A')
So,
n(V n A') = n(V) - n(V n A) = 11 - 8 = 3.
Hence, only 3 lines have a verb but no adjectives.
6 Consider a situation in which a random sample of 1000 adults is surveyed and the proportion that primarily buys organic vegetables is found. If a new random sample of 1000 adults is taken from the same population, explain whether each of the following would change: a. The population proportion, p. b. The sample proportion, p^. c. The standard deviation of p^. d. The standard error of p^. e. The sampling distribution of p^, including its shape, mean, and standard deviation.
Answer:
In the Step-by-step explanation.
Step-by-step explanation:
If a new sample is taken out of the same population,
a. The population proportion, p will not change. It is an unknown value that is estimated with the statistics of the samples.
b. The sample proportion, p^ is expected to change, because it is a new sample that has its own statistic value that may or may not be equal to the first sample.
c. The standard deviation of p^ is expected to change, because it depends on the sample and its size.
d. The standard error of p^ will change, because it depends on the sample.
e. The sampling distribution of p^, including its shape, mean, and standard deviation, will not change, because it is estimated with the data of the previous sample and it is supposed to be a property of the population and the sample size. Although the new information can be used to review the sample mean and standard deviation.
You play two games against the same opponent. The probability you win the first game is 0.7. If you win the first game, the probability you also win the second is 0.5. If you lose the first game, the probability that you win the second is 0.3.(a) Are the two games independent?(b) What's the probability you lose both games?
Answer:
(a) No
(b) 0.21 or 21%
Step-by-step explanation:
(a) Since the outcome of the first game influences in the probability of winning the second game, the two games are not independent.
(b) The probability of losing both games is given by the product of the probability of losing the first game and the probability of losing the second game given that you have lost the first:
[tex]P = (1-0.7)*(1-0.3)\\P=0.21=21\%[/tex]
The probability you lose both games is 21%
Given Information:
Probability of wining 1st game = p₁ = 0.7
Probability of wining 2nd game given 1st game won = p₂|p₁ = 0.5
Probability of wining 2nd game given 1st game lost = p₂|q₁ = 0.3
Required Information:
(a) Are the two games independent = ?
(b) Probability of losing both games = ?
Answer:
(a) Are the two games independent = No
(b) Probability of losing both games = 0.21
Step-by-step explanation:
(a) Independent Events:
Two events are said to be independent when the success of one event is not affected by the success or failure of another event.
In this case, the probability of 2nd game depends on the success or failure of the 1st game, therefore, the two games are not independent.
(b) Probability of losing both games
The probability of losing the both games is the product of the probabilities of losing each game.
Probability of losing 1st game = 1 - Probability of wining 1st game
Probability of losing 1st game = 1 - 0.7 = 0.30
Probability of losing 2nd game = 1 - Probability of wining 2nd game given 1st game lost
Probability of losing 2nd game = 1 - 0.3 = 0.70
Please note that since we are finding the probability of losing both games that's why we used the condition of 1st game lost
Probability of losing both games = Probability of losing 1st game*Probability of losing 2nd game
Probability of losing both games = 0.30*0.70
Probability of losing both games = 0.21
A local board of education conducted a survey of residents in the community concerning a property tax levy on the coming local ballot. They randomly selected 850 residents in the community and contacted them by telephone. Of the 850 residents surveyed, 410 supported the property tax levy. Let p p represent the proportion of residents in the community that support the property tax levy. A 90% confidence interval for p p is_______________.
A.0.4489 to 0.5159.
B. 0.4542 to 0.5105.
C.0.4487 to 0.5161.
D.0.4463 to 0.5185.
Answer:
[tex]0.482 - 1.64 \sqrt{\frac{0.482(1-0.482)}{850}}=0.454[/tex]
[tex]0.482 + 1.64 \sqrt{\frac{0.482(1-0.482)}{850}}=0.510[/tex]
And the 90% confidence interval would be given (0.454;0.510).
B. 0.4542 to 0.5105.
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
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".
Solution to the problem
The estimated proportion of residents in the community that support the property tax levis is given by:
[tex] \hat p =\frac{x}{n}= \frac{410}{850}= 0.482[/tex]
The confidence interval for a proportion is given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 90% confidence interval the value of [tex]\alpha=1-0.9=0.1[/tex] and [tex]\alpha/2=0.05[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.64[/tex]
And replacing into the confidence interval formula we got:
[tex]0.482 - 1.64 \sqrt{\frac{0.482(1-0.482)}{850}}=0.4542[/tex]
[tex]0.482 + 1.64 \sqrt{\frac{0.482(1-0.482)}{850}}=0.5105[/tex]
And the 90% confidence interval would be given (0.4542;0.5105)
B. 0.4542 to 0.5105.
Answer:
90% confidence interval for p is [0.4542 , 0.5105] .
Step-by-step explanation:
We are given that a local board of education conducted a survey of residents in the community concerning a property tax levy on the coming local ballot. Of the 850 residents surveyed, 410 supported the property tax levy.
Let p = proportion of residents in the community that support the property tax levy
[tex]\hat p[/tex] = proportion of residents in the community that support the property tax levy in a survey of 850 residents = [tex]\frac{410}{850}[/tex] = [tex]\frac{41}{85}[/tex]
The pivotal quantity that will be used here population proportion p is;
P.Q. = [tex]\frac{\hat p - p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
So, 90% confidence interval for p is given by;
P(-1.6449 < N(0,1) < 1.6449) = 0.90 {At 10% significance level the z table give
critical value of 1.6449)
P(-1.6449 < [tex]\frac{\hat p - p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < 1.6449) = 0.90
P( [tex]-1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] < [tex]{\hat p - p}[/tex] < [tex]1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] ) = 0.90
P( [tex]\hat p -1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] < p < [tex]\hat p +1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] ) = 0.90
90% confidence interval for p = [ [tex]\hat p -1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] , [tex]\hat p +1.6449 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }[/tex] ]
= [ [tex]\frac{41}{85} -1.6449 \times {\sqrt{\frac{\frac{41}{85} (1-\frac{41}{85} )}{850} }[/tex] , [tex]\frac{41}{85} +1.6449 \times {\sqrt{\frac{\frac{41}{85} (1-\frac{41}{85} )}{850} }[/tex] ]
= [ 0.4542 , 0.5105 ]
Therefore, 90% confidence interval for p is [0.4542 , 0.5105] .
It takes approximately 4 medium apples to make 3 servings of homemade
apple sauce. The cafeteria purchased a bushel of apples (approximately 126
medium apples) at a discount. How many whole servings of applesauce can they
make with the bushel? (Round to the nearest whole number.)
Answer: 95 servings of applesauce
Step-by-step explanation:
It takes approximately 4 medium apples to make 3 servings of homemade apple sauce. It means that the number of servings of homemade apple sauce that can be made from 1 medium apple is
3/4 = 0.75 servings
Approximately 126 medium apples were purchased. Therefore, the number of servings of applesauce that they can make with the bushel is
126 × 0.75 = 95 servings of applesauce rounded to the nearest whole number.
A researcher is interested in studying crime rates in each state. To do so, he records the number of violent crimes that occur per every 1000 people living in the state. The collection of crime rates from all 50 states represents the:a. datab. scorec. variable
Answer:
A. Data
Step-by-step explanation: Data is a term used to describe facts, information or statistics that are collected together in order for it to be used as a reference or for analysis.
An effective data collection is one of the most important aspects in research,experiments or statistics as it helps to guarantee a reliable and effective outcome.
Data collection should be done in such a way that it helps to solve the problem which is being studied or handled.
An insurance company is reviewing its current policy rates. When originally setting the rates, they believed that the average claim amount was $1,800. They are concerned that the true mean is actually higher than this because they could potentially lose a lot of money. They randomly select 40 claims and calculate a sample mean of $1,950. Assuming that the standard deviation of claims is $500, and set α= 0.05; α= 0.1, test to see if the insurance company should be concerned.
Answer:
Reject There is sufficient evidence to support the claim that true mean is actually higher than the claim amount $1800.
Step-by-step explanation:
Based on the decision rule, the test statistic is lies in the rejection region. So reject the null hypothesis at 5% level of significance.
There is sufficient evidence to support the claim that the true mean is actually higher than the claim amount $1800.
Solution is attached below