Answer:
y = -x+3
Step-by-step explanation:
We have two points so we can find the slope
m =(y2-y1)/(x2-x1)
(1-2)/(2-1)
-1/1
The slope is -1
We can use the slope intercept form of the equation
y = mx+b where m is the slope and b is the y intercept
y = -x+b
Substitute a point into the equation to find b
2 = -1 +b
Add 1 to each side
2+1 =-1+1 +b
3 =b
y = -x+3
g The Enigma machine was used by Germany in World War II to send coded messages. It has gained fame because it was an excellent coding device for its day and because of the ultimately successful efforts of the British (with considerable aid from the Poles) to crack the Enigma code. The breaking of the code involved, among other things, some very good mathematics developed by Alan Turing and others. One part of the machine consisted of three rotors, each containing the letters A through Z. To read an encrypted message, it was necessary to determine the initial settings of the three rotors (e.g., PDX or JJN). This is only the beginning of the problem of deciphering the Enigma code. Other parts of the machine allowed for many more initial settings. How many different initial settings of the three rotors are there
Answer:
17576
Step-by-step explanation:
Each of the three rotors contained the letters A through Z.
For the first rotor: There are 26 Possible Initial Settings
(A,B,...Z)
For the second rotor: There are 26 possible initial combination with the first rotor likewise.
For the third rotor:There are also 26 possible combinations with the first and second rotors.
Therefore:
Number of Possible Initial Setting of the three rotor=26*26*26=17576
what is 943 divide by 4
Answer:
235.75
Step-by-step explanation:
Answer:
Math answers to fraction 943 divided by 4 can be calculated as follows.
943/4 math problems division = 235.75. Therefore 235.75 to 2 decimal places= 235.75
943/4 divided by 2 » (943/4) ÷ 2 » 235.75 ÷ 2 = 117.875 .
Step-by-step explanation:
Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $60, the optimal value of the objective function would not change.
1.True
2.False
Suppose x has a distribution with a mean of 50 and a standard deviation of 4. Random samples of size n = 64 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the distribution. x has distribution with mean μx = and standard deviation σx = . (b) Find the z value corresponding to x = 51. z = (c) Find P(x < 51). (Round your answer to four decimal places.) P(x < 51) =
Answer:
(a) [tex]\bar x\sim N(\mu_{\bar x}=50,\ \sigma_{\bar x}=0.5)[/tex]
(b) The z-score for the sample mean [tex]\bar x[/tex] = 51 is 2.
(c) The value of [tex]P(\bar X < 51)[/tex] is 0.9773.
Step-by-step explanation:
The random variable X has mean, μ = 50 and standard deviation, σ = 4.
A random sample of size n = 64 is selected.
(a)
According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.
Then, the mean of the distribution of sample mean is given by,
[tex]\mu_{\bar x}=\mu[/tex]
And the standard deviation of the distribution of sample mean is given by,
[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}[/tex]
The sample of X selected is, n = 64 > 30.
So, the Central limit theorem can be applied to approximate the distribution of sample mean ([tex]\bar x[/tex]).
[tex]\bar x\sim N(\mu_{\bar x}=50,\ \sigma_{\bar x}=0.5)[/tex]
(b)
The z-score for the sample mean [tex]\bar x[/tex] is given as follows:
[tex]z=\frac{\bar x-\mu_{\bar x}}{\sigma_{\bar x}}[/tex]
Compute the z-score for [tex]\bar x[/tex] = 51 as follows:
[tex]z=\frac{\bar x-\mu_{\bar x}}{\sigma_{\bar x}}[/tex]
[tex]=\frac{51-50}{0.5}\\[/tex]
[tex]=2[/tex]
Thus, the z-score for the sample mean [tex]\bar x[/tex] = 51 is 2.
(c)
Compute the value of [tex]P(\bar X < 51)[/tex] as follows:
[tex]P(\bar X < 51)=P(\frac{\bar x-\mu_{\bar x}}{\sigma_{\bar x}}<\frac{51-50}{0.5})[/tex]
[tex]=P(Z<2)\\=0.97725\\\approx 0.9773[/tex]
*Use a z-table for the probability.
Thus, the value of [tex]P(\bar X < 51)[/tex] is 0.9773.
The x distribution is normally distributed with a mean of 50 and a standard deviation of 4. The z-value corresponding to x = 51 is 0.25. The probability of x being less than 51 is approximately 0.5987.
Explanation:a) The x distribution is normally distributed with a mean of 50 and a standard deviation of 4.
The mean of the distribution is μx = 50 and the standard deviation is σx = 4.
b) To find the z-value corresponding to x = 51, we can use the formula z = (x - μ) / σ. Plugging in the values, we get z = (51 - 50) / 4 = 0.25.
c) To find P(x < 51), we can use the standard normal distribution table or a calculator to find the corresponding cumulative probability. The value is approximately 0.5987, rounded to four decimal places.
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A dead body was found within a closed room of a house where the temperature was a constant 70∘F. At the time of discovery t 1 , the core temperature of the body was determined to be 85^\circ85 ∘ F. One hour later (time t=t 1 +1,) a second measurement showed that the core temperature of the body was 80∘F. The core temperature was 98.6^\circ98.6 ∘ F at the time of death (time t=0.) Determine how many hours elapsed before the body was found.
Answer: 1.59 hours elapsed before the body was found
Step-by-step explanation: Please see the attachments below
A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hr. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data support the use of a one-sample t test. The relevant hypotheses are H0: µ = 10 versus Ha: µ < 10.(a) If t = -2.4 and = .05 is selected, what conclusion is appropriate?a. Rejectb. Fail to reject(b) If t = -1.83 and = .01 is selected, what conclusion is appropriate?a. Rejectb. Fail to reject(c) If t = 0.57, what conclusion is appropriate?a.Rejectb. Fail to reject
Answer:
(a) We reject our null hypothesis.
(b) We fail to reject our null hypothesis.
(c) We fail to reject our null hypothesis.
Step-by-step explanation:
We are given that a certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 hr.
A random sample of 18 pens is selected.
Let [tex]\mu[/tex] = true average writing lifetime under controlled conditions
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 10 hr {means that the true average writing lifetime under controlled conditions is at least 10 hr}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 10 hr {means that the true average writing lifetime under controlled conditions is less than 10 hr}
The test statistics that is used here is one-sample t test statistics;
T.S. = [tex]\frac{\bar X -\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]
where, [tex]\bar X[/tex] = sample mean
s = sample standard deviation
n = sample size of pens = 18
n - 1 = degree of freedom = 18 -1 = 17
Now, the decision rule based on the critical value of t is given by;
If the value of test statistics is more than the critical value of t at 17 degree of freedom for left-tailed test, then we will not reject our null hypothesis as it will not fall in the rejection region.If the value of test statistics is less than the critical value of t at 17 degree of freedom for left-tailed test, then we will reject our null hypothesis as it will fall in the rejection region.(a) Here, test statistics, t = -2.4 and level of significance is 0.05.
Now, at 0.05 significance level, the t table gives critical value of -1.74 at 17 degree of freedom.
Here, clearly the value of test statistics is less than the critical value of t as -2.4 < -1.74, so we reject our null hypothesis.
(b) Here, test statistics, t = -1.83 and level of significance is 0.01.
Now, at 0.051 significance level, the t table gives critical value of -2.567 at 17 degree of freedom.
Here, clearly the value of test statistics is more than the critical value of t as -2.567 < -1.83, so we fail to reject our null hypothesis.
(c) Here, test statistics, t = 0.57 and level of significance is not given so we assume it to be 0.05.
Now, at 0.05 significance level, the t table gives critical value of -1.74 at 17 degree of freedom.
Here, clearly the value of test statistics is more than the critical value of t as -1.74 < 0.57, so we fail to reject our null hypothesis.
4x-6 + 2x = 18
What’s the answer
Answer:
x=4
Step-by-step explanation:
4x-6 + 2x = 18
Combine like terms
6x -6 = 18
Add 6 to each side
6x-6+6 =18+6
6x = 24
Divide each side by 6
6x/6 = 24/6
x =4
Arlin has 9 dollars and 37 cents. Lauren has 6 dollars and 63 cents. How much money does Arlin need to give Lauren so that each of them has the same amount of money?
Answer:
Arlin has to give lauren 1.37
Step-by-step explanation:
9.37 + 6.63 / 2 = 16 / 2 = 8
9.37 - 8 = 1.37
Answer:
$1.37
Step-by-step explanation:
I would start by adding up the total money between them. $9.37 + $6.63 = $16.00. They want the same amount of money, so divide the total by two.
$16.00/2 = $8.00.
Now take the difference between how much Arlin had ($9.37) and how much she has now ($8.00).
$9.37 - $8.00 = $1.37
Gabriellas school is selling tickets to a fall musical. On the dirst day of ticket sales the school sold 10 senior citizen tickets and 14 student tickets for a total of $212. Tje school took in$232 on the second day by selling 12 senior citizen tickets and 14 student tickets. What is the price each of one senior citizen tickets and one student ticket?
Jamie places fifteen 1 inch cubes in the bottom of a box. She adds 4 more layers of the same number of cubes to completely fill the box. What is the volume of the box?
Answer:
60
Step-by-step explanation:
15cubes assuming 3rows of 5 cubes in one layer so volume of first layer is 3x5x1 =15
4 layers means 4 times or 4 inch in height, either way 4x15 = 60
Answer:
75
Step-by-step explanation:
he amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.3 minutes and standard deviation 1.4 minutes. Suppose that a random sample of n equals 47 customers is observed. Find the probability that the average time waiting in line for these customers is
Complete question:
He amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.3 minutes and standard deviation 1.4 minutes. Suppose that a random sample of n equals 47 customers is observed. Find the probability that the average time waiting in line for these customers is
a) less than 8 minutes
b) between 8 and 9 minutes
c) less than 7.5 minutes
Answer:
a) 0.0708
b) 0.9291
c) 0.0000
Step-by-step explanation:
Given:
n = 47
u = 8.3 mins
s.d = 1.4 mins
a) Less than 8 minutes:
[tex]P(X<8) = P \frac{X'-u}{s.d/ \sqrt{n}} < \frac{8-8.3}{1.4/ \sqrt{47}}][/tex]
P(X' < 8) = P(Z< - 1.47)
Using the normal distribution table:
NORMSDIST(-1.47)
= 0.0708
b) between 8 and 9 minutes:
P(8< X' <9) =[tex] [\frac{8-8.3}{1.4/ \sqrt{47}}< \frac{X'-u}{s.d/ \sqrt{n}} < \frac{9-8.3}{1.4/ \sqrt{47}}][/tex]
= P(-1.47 <Z< 6.366)
= P( Z< 6.366) - P(Z< -1.47)
Using normal distribution table,
[tex] NORMSDIST(6.366)-NORMSDIST(-1.47) [/tex]
0.9999 - 0.0708
= 0.9291
c) Less than 7.5 minutes:
P(X'<7.5) = [tex] P [Z< \frac{7.5-8.3}{1.4/ \sqrt{47}}] [/tex]
P(X' < 7.5) = P(Z< -3.92)
NORMSDIST (-3.92)
= 0.0000
5-2+12÷4
use the order of operations
Step-by-step explanation:
= 5- 2 + 12 /4
= 5 -2 + 3
= 8- 2
= 6
Answer:
6
Explanation:
What you do is you take 12 divided by 4 and you get 3. The equation is now 5-2+3, you subtract 2 from 5 and get 3. Now you have 3 plus 3 which gets you 6.
What is the volume of this rectangular prism? 2 cm 1/4 cm 2 cm
Answer:
1 cm cubed
Step-by-step explanation:
The volume of a rectangular prism is found by the equation: [tex]V=lwh[/tex] , where [tex]l[/tex] is the length, w is the width, and h is the height.
Here, our dimensions are 2 by 1/4 by 2. So: [tex]l=2,w=1/4,h=2[/tex].
Substituting these into the equation, we have:
[tex]V=2*(1/4)*2=1[/tex]
Thus, the volume is 1 cm cubed.
Hope this helps!
Answer:
1
Step-by-step explanation:
2*2=4
4*1/4=1
multiplying 1/4 is the same as dividing by 4
a line contains the points (-3 -2) and (7,2) determine whether the slope of this line is positive or negative
Answer:
Positive. From those two points the line would slant upwards. Going left to right it would be going up, therefore it's a positive slope
Step-by-step explanation:
The slope of the given lines with two points is positive.
How to find the slope?Slope of a line or straight object is the ratio of how much amount of rise occurs in correspondence to the increment in the run.
Thus, we get:
Slope = rise/ run
y-y₁ = m(x-x₁)
We are given that;
The points =(-3 -2) and (7,2)
Now,
For this line, let’s use (-3, -2) and (7, 2) as the two points. Plugging these values into the formula, we get:
m = (2 - (-2)) / (7 - (-3)) = 4 / 10 = 0.4
Therefore, the slope of this line is 0.4.
A positive slope means that the line goes up from left to right3. A negative slope means that the line goes down from left to right3.
Since 0.4 is a positive number,
Therefore, the slope of this line will be positive.
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ten times a number increased by 150
Hey there!
"A number" is referred to an unknown number so we can say it is labled as
[tex]x[/tex]
"Increased" means you're going up/ adding
ten = 10
150 stays the same
"Ten times a number" =
[tex] \bf{10x}[/tex]
"Increased by 150" =
[tex] \bf{ + 150}[/tex]
Thus your answer should look like this:
[tex] \bf{10x + 150}[/tex]
Good luck on your assignment and enjoy your day!
~
[tex] \frak{loveyourselffirst }[/tex]
To solve this problem, we can use the algebraic expression 10x + 150, where 'x' represents the number.
Explanation:To solve the problem, we can translate the given phrase into an algebraic expression. Let's assume the number is represented by the variable 'x'. 'Ten times a number increased by 150' can be written as 10x + 150. This expression represents ten times the number 'x' plus 150.
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shayna had $22 to spend on six notebooks. After buying them she had $10. How much did each notebook cost ? solving equations: application
equation and a solution
Answer:
Each notebook costs $2
Step-by-step explanation:
We have to find the amount she spent on each notebook.
22-10=12
We know she spent $12 on six notebooks
We need to divide to find the answer
12/6=2
Each notebook costs $12
Answer:
$2
Step-by-step explanation:
First subtract 10 from 22 to get the price she spent on notebooks which is $12.
Then divide 12 by 6 to get the price she spent on each which is, $2
6. A cone is 10 inches tall and ha s a radius of 3 inches. What is the cone’s volume? A. 31.4 cubic inches B. 94.2 cubic inches C. 282.6 cubic inches D. 847.8 cubic inches
Final answer:
To find the volume of a cone with a radius of 3 inches and a height of 10 inches, use the formula V = [tex]\frac{1}{3}[/tex] * π * r² * h. Substitute the values and calculate to find a volume of B) 94.26 cubic inches.
Explanation:
The volume of the cone can be calculated using the formula V = [tex]\frac{1}{3}[/tex] * π * r² * h.
Substitute the values for the radius (3 inches) and height (10 inches) into the formula to find the volume:
V = [tex]\frac{1}{3}[/tex] * 3.142 * 3² * 10
V = [tex]\frac{1}{3}[/tex] * 3.142 * 9 * 10
V = 94.26 cubic inches
Therefore, the cone's volume is 94.26 cubic inches.
Determine whether the samples are independent or dependent. To test the effectiveness of a drug comma cholesterol levels are measured in 200 men and 200 women after the treatment. Choose the correct answer below. A. The samples are independent because there is a natural pairing between the two samples. B. The samples are dependent because there is a natural pairing between the two samples. C. The samples are independent because there is not a natural pairing between the two samples.
Answer:
C
Step-by-step explanation:
The samples are independent because there is not a natural pairing between the two samples.
Since Independent samples are samples that are selected randomly so that its observations do not depend on the values other observations also data set in which each data point in one sample is not paired to a data point in the second sample
One side of a square has a value of 3x+2, find the perimeter of the square
Answer:
P = 12x +8
Step-by-step explanation:
The perimeter of a square is given by
P = 4s where s is the side length
P = 4(3x+2)
Distribute
P = 12x +8
Answer:
[tex]12x+8[/tex]
Step-by-step explanation:
[tex]3x+2[/tex] for one side of a square, for a perimeter for the square we need 4 times the side length, so we need:
[tex]4(3x+2)=12x+8[/tex]
if i have 5 blue pens and 3 black pens, What fraction of the number of black pens is the number of blue pens?
Answer:
the answer to the question is 3/5
(1,4) (6,-1) what is the y intercept of the line and how did you find it?
Answer:
Step-by-step explanation:
(1,4) (6,-1)
1-6= -5
6--1= 7 its 7 because 6 minus negative 1 would be 7
-5/7 would be your slope
---------------------------------------------------
y=mx+b
y= -5/7x+b
-1= -5/7(6)+b
-1= 6[tex]\frac{-5}{7}[/tex]+b
you take those two numbers and subtract
5 [tex]\frac{-5}{7}[/tex] is y intercept
y=[tex]\frac{-5}{7}[/tex]x+5[tex]\frac{-5}{7}[/tex]
What is the area of a triangle with a base of 23 feet and a height of 6 feet
Answer:
A= 69
Step-by-step explanation:
A= h*b/2= (6*23)/2=69
Answer:
A = 69 [tex]ft^{2}[/tex]
Step-by-step explanation:
The formula utilised to determine the area of a triangle is:
A = [tex]\frac{1}{2}[/tex] * b * h
The base and height are given, and thus, can easily be substituted for in the formula to find the area.
A = [tex]\frac{1}{2}[/tex] * 23 * 6
A = 69 [tex]ft^{2}[/tex]
The smaller star has been enlarged which is a good estimated scale factor according to your evaluation of the coordinate changes
Answer:B 1.5
Step-by-step explanation:
I just did it
Answer:
B 1.5
Step-by-step explanation:
(Based on Q1 ~ Q3) According to the Bureau of the Census, 18.1% of the U.S. population lives in the Northeast, 21.9% inn the Midwest, 36.7% in the South, and 23.3% in the West.. In a random sample of 200 recent calls to a national 800-member hotline, 39 of the calls were from the Northeast, 55 from the Midwest, 60 from the South, and 46 from the West. At the 0.05 level, can we conclude that the geographical distribution of hotline callers could be the same as the U.S. population distribution?
Answer:
We can therefore conclude that the geographical distribution of hotline callers could be the same as the U.S population distribution.
Step-by-step explanation:
The null Hypothesis: Geographical distribution of hotline callers could be the same as the U.S. population distribution
Alternative hypothesis: Geographical distribution of hotline callers could not be the same as the U.S. population distribution
The populations considered are the Midwest, South, Northeast, and west.
The number of categories, k = 4
Number of recent calls = 200
Let the number of estimated parameters that must be estimated, m = 0
The degree of freedom is given by the formula:
df = k - 1-m
df = 4 -1 - 0 = 3
Let the significance level be, α = 5% = 0.05
For α = 0.05, and df = 3,
from the chi square distribution table, the critical value = 7.815
Observed and expected frequencies of calls for each of the region:
Northeast
Observed frequency = 39
It contains 18.1% of the US Population
The probability = 0.181
Expected frequency of call = 0.181 * 200 = 36.2
Midwest
Observed frequency = 55
It contains 21.9% of the US Population
The probability = 0.219
Expected frequency of call = 0.219 * 200 =43.8
South
Observed frequency = 60
It contains 36.7% of the US Population
The probability = 0.367
Expected frequency of call = 0.367 * 200 = 73.4
West
Observed frequency = 46
It contains 23.3% of the US Population
The probability = 0.233
Expected frequency of call = 0.233 * 200 = 46
[tex]x^{2} = \sum \frac{(O_{i} - E_{i}) ^{2} }{E_{i} } , i = 1, 2,.........k[/tex]
Where [tex]O_{i} =[/tex] observed frequency
[tex]E_{i} =[/tex] Expected frequency
Calculate the test statistic value, x²
[tex]x^{2} = \frac{(39 - 36.2)^{2} }{36.2} + \frac{(55 - 43.8)^{2} }{43.8} + \frac{(60 - 73.4)^{2} }{73.4} + \frac{(46 - 46.6)^{2} }{46.6}[/tex]
[tex]x^{2} = 5.535[/tex]
Since the test statistic value, x²= 5.535 is less than the critical value = 7.815, the null hypothesis will not be rejected, i.e. it will be accepted. We can therefore conclude that the geographical distribution of hotline callers could be the same as the U.S population distribution.
Factor completely. − 3 x 2 + 6 x + 9 = −3x 2 +6x+9=minus, 3, x, squared, plus, 6, x, plus, 9, equals
Answer:
-3 (x-3) (x+1)
Step-by-step explanation:
− 3 x ^2 + 6 x + 9
Factor out -3
-3( x^2 -2x-3)
The terms inside the parentheses can be factored
What 2 numbers multiplies to -3 and adds to -2
-3*1 = -3
-3+1 =-2
-3 (x-3) (x+1)
Find 2.4% of $109. Show work.
Answer:
$2.62
Step-by-step explanation:
[tex]2.4\% \: of \: \$109 \: \\ \\ = \frac{2.4}{100} \times 109 \\ \\ = 0.024 \times 109 \\ \\ = \$2.616 \\ \\ \approx \: \$2.62[/tex]
An ethanol railroad tariff is a fee charged for shipments of ethanol on public railroads. An agricultural association publishes tariff rates for railroad-car shipments of ethanol. Assuming that the standard deviation of such tariff rates is $1250, determine the probability that the mean tariff rate of 350 randomly selected railroad-car shipments of ethanol will be within $110 of the mean tariff rate of all railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.
Answer:
The probability that the mean is less than 110
P(x⁻<110) =0.5
Step-by-step explanation:
Explanation:-
Given the standard deviation of the Population' σ' = 1250
Given sample size 'n' = 350
The standard error of the mean determined by
[tex]S.E = \frac{S.D}{\sqrt{n} }[/tex]
Standard error = [tex]\frac{1250}{\sqrt{350} } = 66.8153[/tex]
by using normal distribution [tex]z = \frac{x -mean}{S.E}[/tex]
[tex]z = \frac{x^{-} -110}{66.8}[/tex]
cross multiplication 66.8z = x⁻-110
x⁻ = 66.81Z+110
P(x⁻<110)=P(66.81Z+110<110)
= P(66.81Z < 110-110)
= P(66.81Z<0)
= P(Z<0)
= 0.5- A(z₁)
= 0.5 - A(0) (here z₁=0)
= 0.5 -0.00
=0.5
Conclusion:-
The probability that the mean is less than 110
P(x⁻<110) =0.5
A magician shuffles a standard deck of playing cards and allows an audience member to pull out a card, look at it, and replace it in the deck. Three additional people do the same. Find the probability that of the 4 cards drawn, at least 1 is a face card. (Round your answer to one decimal place.)
Answer:
0.23 or 3/13.
Step-by-step explanation:
There are 52 cards in a deck. There are four different suits, dividing the decks into 4 sets of 13. There are three face cards for each suit so 3/13. 3 divided by 13 is 0.23. Use fractions if you can because they are easier and more accurate.
To find the probability that at least one of the 4 cards drawn is a face card, calculate the probability of all cards not being face cards and subtract that from 1, resulting in approximately 64.9%.
The problem can be approached by finding the probability that none of the 4 cards drawn is a face card and then subtracting that from 1 to find the probability that at least one is a face card. There are 12 face cards in a standard deck of 52 cards, leaving 40 non-face cards. When the audience members draw and replace the cards, each draw is independent of the previous draw.
First, calculate the probability of drawing a non-face card (P(NF)):
P(NF) = number of non-face cards / total number of cards = 40/52
Since the card is replaced each time, the probability remains the same for each of the four draws. Thus, the probability that all 4 cards are non-face cards is:
P(all four are NF) = [tex]P(NF)^4 = (40/52)^4[/tex]
Then subtract this probability from 1 to get the probability of at least one face card:
P(at least one face card) = 1 - P(all four are NF)
Calculation:
P(at least one face card) = [tex]1 - (40/52)^4 = 1 - (0.7692)^4[/tex]
P(at least one face card) ≈ 1 - 0.3515 ≈ 0.6485
Therefore, the probability that at least one of the 4 cards drawn is a face card is approximately 64.9% (rounded to one decimal place).
The Information Technology Department at a large university wishes to estimate the proportion of students living in the dormitories, p, who own a computer with a 99% confidence interval. What is the minimum required sample size the IT Department should use to estimate the proportion p with a margin of error no larger than 5 percentage points
Answer:
[tex]n=\frac{0.5(1-0.5)}{(\frac{0.05}{1.96})^2}=384.16[/tex]
And rounded up we have that n=385
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
In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. And the critical value would be given by:
[tex]z_{\alpha/2}=-2.58, z_{1-\alpha/2}=2.58[/tex]
The margin of error for the proportion interval is given by this formula:
[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex] (a)
And on this case we have that [tex]ME =\pm 0.05[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex] (b)
We can use as an estimator for p [tex]\hat p =0.5[/tex]. And replacing into equation (b) the values from part a we got:
[tex]n=\frac{0.5(1-0.5)}{(\frac{0.05}{1.96})^2}=384.16[/tex]
And rounded up we have that n=385
A company that makes shampoo wants to test whether the average amount of shampoo per bottle is 16 ounces. The standard deviation is known to be 0.20 ounces. Assuming that the hypothesis test is to be performed using 0.10 level of significance and a random sample of n = 64 bottles, how large could the sample mean be before they would reject the null hypothesis? Question 50 options: 16.2 ounces 16.041 ounces 15.8 ounces 16.049 ounces
Answer:
The correct option is 16.041 ounces.
Step-by-step explanation:
A single mean test can be used to determine whether the average amount of shampoo per bottle is 16 ounces.
The hypothesis can be defined as:
H₀: The average amount of shampoo per bottle is 16 ounces, i.e. μ = 16.
Hₐ: The average amount of shampoo per bottle is different from 16 ounces, i.e. μ ≠ 16.
The information provided is:
[tex]n=64\\\sigma=0.20\\\alpha =0.10[/tex]
We can compute a 90% confidence interval to determine whether the population mean is 16 ounces or not.
Since the population standard deviation is known we will compute the z-interval.
The critical value of z for 90% confidence interval is:
[tex]z_{0.05}=1.645[/tex]
*Use a z-table.
Compute the 90% confidence interval for population mean as follows:
[tex]CI=\bar x\pm z_{\alpha/2}\times\frac{\sigma}{\sqrt{n}}\\[/tex]
Since the sample size is quite large, according to the law of large numbers the on increasing the sample size, the mean of the sample approaches the whole population mean.
So, the 90% confidence interval estimate for sample mean is:
[tex]CI=\mu\pm z_{\alpha/2}\times\frac{\sigma}{\sqrt{n}}\\=16\pm 1.645\times \frac{0.20}{\sqrt{64}}\\=16\pm0.041125\\=(15.958875, 16.041125)\\\approx (15.959, 16.041)[/tex]
Thus, the correct option is 16.041 ounces.