IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. Part (a) Give the distribution of X. X ~ , Part (b) Find the probability that the person has an IQ greater than 105. Write the probability statement. P What is the probability? (Round your answer to four decimal places.) Sketch the graph. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Part (c) Mensa is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the Mensa organization. Write the probability statement. P(X > x) = What is the minimum IQ? (Round your answer to the nearest whole number.) x = Sketch the graph. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Part (d) The middle 40% of IQs fall between what two values? Write the probability statement. P(x1 < X < x2) = State the two values. (Round your answers to the nearest whole number.) x1 = x2 = Sketch the graph. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot
Answers
Answer:
a) The normal distribution function for the IQ of a randomly selected individual is presented in the first attached image to this solution.
b) The probability that the person has an IQ greater than 105 = P(X > 105) = 0.3707
The sketch of this probability is presented in the second attached image to this solution.
c) The minimum IQ needed to qualify for the Mensa organization = 130.81. Hence, P(X > 131) = 2%
The sketched image of this probability is presented also in the second attached image to the solution. The shaded region is the required probability.
d) The middle 40% of IQs fall between 92 and 108 IQ respectively.
P(x1 < X < x2) = P(92 < X < 108) = 0.40
The sketched image of this probability is presented in the third attached image to the solution. The shaded region is the required probability.
Step-by-step explanation:
The IQ of an individual is given as a normal distribution withh
Mean = μ = 100
Standard deviation = σ = 15
If X is a random variable which represents the IQ of an individual
a) The distribution of X will then be given as the same as that of a normal distribution.
f(x) = (1/σ√2π) {e^ - [(x - μ)²/2σ²]}
The normal distribution probability density function is more clearly presented in the first attached image to this question
b) Probability that the person has an IQ greater than 105.
To find this probability, we will use the normal probability tables
We first normalize/standardize 105.
The standardized score of any value is that value minus the mean divided by the standard deviation.
z = (x - μ)/σ = (105 - 100)/15 = 0.33
P(x > 105) = P(z > 0.33)
Checking the tables
P(x > 105) = P(z > 0.33) = 1 - P(z ≤ 0.33) = 1 - 0.6293 = 0.3707
The sketch of this probability is presented in the second attached image to this question. The shaded region is the required probability.
c) Mensa is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the Mensa organization.
We need to find x for P(X > x) = 2% = 0.02
Let the corresponding z-score for this probability be z'
P(X > x) = P(z > z') = 0.02
P(z > z') = P(z ≤ z') = 1 - 0.02 = 0.98
From the normal distribution table, z' = 2.054
z = (x - μ)/σ
2.054 = (x - 100)/15
x = 2.054×15 + 100 = 130.81 = 131 to the nearest whole number.
The sketched image of this probability is presented also in the second attached image to the solution. The shaded region is the required probability.
d) The middle 40% of IQs fall between what two values?
P(x1 < X < x2) = 0.40
Since the normal distribution is symmetric about the mean, the lower limit of this IQ range will be greater than the lower 30% region of the IQ distribution and the upper limit too is lesser than upper 30% region of the distribution.
P(X < x1) = 0.30
P(X > x2) = 0.30, P(X ≤ x2) = 1 - 0.30 = 0.70
Let the z-scores of x1 and x2 be z1 and z2 respectively.
P(X < x1) = P(z < z1) = 0.30
P(X ≤ x2) = P(z ≤ z2) = 0.70
From the normal distribution tables,
z1 = -0.524
z2 = 0.524
z1 = (x1 - μ)/σ
-0.524 = (x1 - 100)/15
x1 = -0.524×15 + 100 = 92.14 = 92 to the nearest whole number.
z2 = (x2 - μ)/σ
0.524 = (x2 - 100)/15
x2 = 0.524×15 + 100 = 107.86. = 108 to the nearest whole number.
The sketched image of this probability is presented in the third attached image to the solution. The shaded region is the required probability.
Hope this Helps!!!
Related Questions
9) brainliest & 10 + points!
Answers
Answer:
no supplement
Step-by-step explanation:
Supplementary angles add to 180 degrees,
This angle is larger than 180 degrees by itself, so it has no supplement
Let x be a variable, and let n be an arbitrary constant. What is the derivative of x^n?
Answers
Answer:
nx^(n-1)
Step-by-step explanation:
Find measure of arc or angle indicated
Answers
A shipment of 60 inexpensive digital watches, including 9 that are defective,is sent to a department store.The receiving department select's 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found defective.What is the probability that the shipment will be rejected?
Answers
Answer: 0.627 or 62.7 %
Step-by-step explanation:
The probability that shipment will be rejected P(rejected) = 1- probability that shipment will be accepted.
P(rejected)= 1-P(accepted)
P(accepted) is equal to probability when all 10 watches are not defective.
The probability that 1st one randomly selected watches are not defective is 51/60 (51 watches are not defective and 9 are defective)
The probability that 2-nd one randomly selected watches are not defective is 50/59 ( because the total number of the watches now is 1 unit less 60-1=59, and the total number of not defective watches is 1 unit less 51-1=50 units)
The probability that 3rd one randomly selected watches are not defective is 49/58 (49 watches are not defective total number of watches is 58)
Similarly P(4th)= 48/57 P(5th)=47/56 P(6th)=46/55 P(7th)=45/54
P(8th)=44/53 P(9th)=43/52 P(10th)=42/51
So P(accepted)= P(1st)*P(2nd)*P(3rd)*P(4th)*P(5th)*P(6th)*P(7th)*P(8th)*P(9th)*P(10th)=
=51*50*49*48*47*46*45*44*43*42/(60*59*58*57*56*55*54*53*52*51)=
= approx= 0.373
So P(rejected)=1-0.373=0.627
A composition of reflections over parallel lines is the same as a __________. A. translation B. rotation C. glide reflection D. double rotation
Answers
Answer: A
Step-by-step explanation:
Answer: A.
Step-by-step explanation:
A composition of reflections over two parallel lines is equivalent to a translation.
What do you go by for the pattern 360,60,10 would it be add 60, divided by 6 ,multiply by 6 or subtract 300?
Answers
Answer:
divided by 6
Step-by-step explanation:
Given pattern
360,60,10
would it be add 60
lets
add 60 to each term
360 +60 = 420
but next term is 60, hence it incorrect choice
divided by 6
lets divide each term by 6
360/6 = 60 which is the next term in the series as well
60/6 = 10 which is also the next term in the series as well
hence divided by 6 is the correct option.
multiply by 6
multiply by 6 to each term
360 *60 = 21600
but next term is 60, hence it incorrect choice
subtract 60 from each term
360 -300 = 60 which is the next term in the series
60 -300 = -240 which is not same the next term in the series that is 10
hence this is incorrect choice
finding angle measures between intersecting lines
Answers
Answer:
[tex]\boxed{<x = 17 degrees}[/tex]
Step-by-step explanation:
<EGB = 73 degrees (Vertically Opposite angles are congruent)
=> <x = 90-<EGB (Complementary angles)
=> <x = 90-73
=> <x = 17 degrees
Multiply: (−2x2 + 9x − 3) * (7x2 − 4x + 2)
Answers
Answer:
[tex]-14x^4+71x^3-61x^2+30x-6[/tex]
Step-by-step explanation:
All we are doing is distributing each number of the 1st equation to the 2nd equation to get our answer. Once we do so, we combine like terms and we get our answer.
Graph the system of linear inequalities. Please help!
Answers
Answer: Third option is correct
Step-by-step explanation: The y-intercepts match the constants (b in y=mx+b format) The y-values shaded above the line y≥2x+4 are correct. The y-values shaded in the region below the y≤2x -2 are also correct.
What is the value of x?
O 40
O 70
O 140
O 160
Answers
what is the question? if u give the question , I might help and edit this :)
Answer:
i think it's 70
Step-by-step explanation:
sorry if its wrong let me know plz
Solve the system of linear equations x - 3y = -3 x + 3y = 9
Answers
x-3y= -3x +3y=9
collect like terms
-3x will go to x and it will turn to
x+3x And -3y will go to +3y and it will turn to
×+3x=+3y+3y=9
solving time.
x+3x is 4x
3y +3y =6y
4x+6y =9
remove the 4x for now.
6y divided by 9.
3 in 6 is 2 and 3 in 9 is 3.
therefore 4x =2=9
4/2=2
2x=9.
4and half is the answer
Answer:
x = 3
y = 2
Step-by-step explanation:
x - 3y = -3
x + 3y = 9
Solve for x in the second equation.
x = 9 - 3y
Pu x as 9 - 3y in the first equation and solve for y.
9 - 3y - 3y = -3
9 - 6y = -3
-6y = -12
y = 2
Put y as 2 in the second equation and solve for x.
x = 9 - 3(2)
x = 9 - 6
x = 3
The board of directors of Midwest Foods has declared a dividend of $3,500,000. The company has 300,000 shares of preferred stock that pay $2.85 per share and 2,500,000 shares of common stock. After finding the amount of dividends due the preferred shareholders, calculate the dividend per share of common stock.
Answers
Answer:
$1,06
Step-by-step explanation:
Calculation for dividend per share of common stock for board of directors of Midwest Foods
First step is to find the amount of dividends due to the preferred shareholders
Using this formula
Total Dividend =Dividend- (Preferred stock *Per share of preferred stock )
Let plug in the formula
Total Dividend =$3,500,000-($300,000*$2.85)
Total Dividend =$3,500,000-$855,000
Total Dividend =$2,645,000
The second step is to find Dividend per share of common stock
Using this formula
Dividend per share of common stock=Total dividend/Shares of common stock
Let plug in the formula
Dividend per share of common stock=$2,645,000/$2,500,000
Dividend per share of common stock=$1.06
Therefore the dividend per share of common stock for board of directors of Midwest Foods will be $1.06
two lines, 3y-2x=21 and 4y+5x=5, intersect at the point Q. find the coordinates of Q.
Answers
Answer:
Q = (- 3, 5 )
Step-by-step explanation:
Given the 2 equations
3y - 2x = 21 → (1)
4y + 5x = 5 → (2)
Multiplying (1) by 5 and (2) by 2 and adding will eliminate the x- term.
15y - 10x = 105 → (3)
8y + 10x = 10 → (4)
Add (3) and (4) term by term to eliminate x
23y = 115 ( divide both sides by 23 )
y = 5
Substitute y = 5 into either of the 2 equations and solve for x
Substituting into (2)
4(5) + 5x = 5
20 + 5x = 5 ( subtract 20 from both sides )
5x = - 15 ( divide both sides by 5 )
x = - 3
Solution is (- 3, 5 )
A linear function has a slope of -7/9 and ay-intercept of 3. How does this function compare to the linear function that is
represented by the equation y+11=-7/9(x-18)?
It has the same slope and the same y-intercept.
It has the same slope and a different y-intercept.
It has the same y-intercept and a different slope.
O It has a different slope and a different y-intercept.
Answers
Answer:
same slope and same y intercept
Step-by-step explanation:
y + 11 = -7/9 (x - 18)
y + 11 = -7/9x + 14
y = -7/9x + 3
What is the slope of the line shown?
Slope=-1/2
Slope =7
Slope=2
Slope=-2
Answers
Answer:
-2
Step-by-step explanation:
To find the slope: Rise/Run
You go 2 steps down for every step you go right.
Your rise is -2 and your run is 1.
So your slope is -2.
Answer:
-2
Step-by-step explanation:
Get two points that intersect that line.
(1, 5) and (2, 3)
Find the rise and run.
As we can see to get from (1,5) to (2,3), we have to go to the right by 1 and go down by 2 (in this case the movement is -2 steps).
rise/run
-2/1 = -2
How many feet of chain fence are necessary to enclosed a dog pen that is square and has a area of 64 sq feet
Answers
Answer:
32 feet
Step-by-step explanation:
area of square is given by side^2
Perimeter of square is given by 4*side
_______________________________
Given
area of square = 64 sq feet
side^2 = 64
side^2 = 8^2
side = 8
thus side = 8 feet
_______________________________________
The dog pen is fenced with chain, hence chain will be fence at the edge of square and at the perimeter.
Thus, length of chain required will be same as the Perimeter of square.
Perimeter of given dog pen with side length 8 feet = 4*8 = 32 feet.
Thus, 32 feet of chain fence is required.
listed below are costs in dollars of round trip flights between two cities. All flights involve one stop and a two week stay. Find a coefficient of variation for each of the two sets of data, then compare the variation.
30 days in advance: 250 286 305 256 288 282 254
1 day in advance: 454 619 557 912 619 1049 562
The coefficient of variation for the prices of tickets purchased 30 days in advance is ____% (round to the three decimal places as needed)
Answers
Answer:
coefficient of variation = 7.108%
Step-by-step explanation:
From the given information:
The objective is to determine the coefficient of variation for the prices of tickets purchased 30 days in advance is ____%
The mean [tex]\overline x[/tex] = [tex]\dfrac{250+286+305+256+288+282+254}{7}[/tex]
The mean [tex]\overline x[/tex] = [tex]\dfrac{1921}{7}[/tex]
The mean [tex]\overline x[/tex] = 274.4285714
The standard deviation also can be computed as follows:
[tex]\sigma =\sqrt{ \dfrac{\sum (x_i-\mu)^2}{N}}[/tex]
[tex]\sigma =\sqrt{ \dfrac{ (250-274.43)^2+(286-274.43)^2+(305-274.43)^2+...+(254-274.43)^2}{7}}[/tex][tex]\sigma =19.507[/tex]
Finally; the coefficient of variation can be calculated with the formula:
coefficient of variation = [tex]\dfrac{\sigma}{\overline x}[/tex]
coefficient of variation = [tex]\dfrac{19.507}{274.43}[/tex]
coefficient of variation = 0.07108
coefficient of variation = 7.108%
c. Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $12 and $18
Answers
Answer:
0.0066
Step-by-step explanation:
The x has a distribution that is approximately normal. For normal distribution the probability of x will be,
u = 12 and 18
P (12 [tex]\leq[/tex] [tex]x[/tex] [tex]\leq[/tex] 18)
P (- 2.3 [tex]\leq[/tex] [tex]x[/tex] [tex]\leq[/tex] 2.85 )
P (0.9912 - 0.9978)
= 0.0066
Mari Seni works exactly 40 hours in a 5-day work week. She worked 6 1/2 hours on Monday, 8 3/4 hours on Tuesday, 6 5/6 hours on Wednesday, and 10 1/4 hours on Thursday. How many hours must Mari work on Friday.
Answers
Answer:
gdk
Step-by-step explanation:
The population of a town is 9,000, and it grows at a rate of 7% per year. What will the population be in 6 years?
Answers
Answer:
12,780
Step-by-step explanation:
Initial population = 9000
grows 7% of 9000= 630 people in a year
after 6 yrs, number of added people = 630× 6=3780 ...... totally, population = 9000+ 3780
= 12,780
The population of the town after 6 years will be 13506.
Concept:As the population grows at r% per year and if the Current population is P, then After 'x' years, the population will be [tex]P_x = P(1 + \frac{r}{100} )^x[/tex]How to solve the given question?Initial Population, P = 9000Rate of increase in population, r = 7% per yearPeriod , x = 6 years∴ The population after 6 years,[tex]P_x = P(1 + \frac{r}{100} )^x[/tex]∴ [tex]P_6 = 9000(1 + \frac{7}{100})^6\\[/tex]
∴[tex]P_6 = 13506 \ peoples[/tex]
Thus, the population of the town after 6 years will be 13506.
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g python Suppose the instructor of the course is convinced that the mean engagement of students who become knowledgeable in the material (i.e., the eng1 population) is 0.75. Formulate null and alternative hypotheses for a statistical test that seeks to challenge this belief. What are the null and alternative hypotheses, and what type of test can be used
Answers
Answer:
Step-by-step explanation:
The null and alternative hypothesis that would seek to challenge this belief would be
Null hypothesis: u = 0.75
Alternative hypothesis: u =/ 0.75
The type of test to be used would be a one sample z test where one sample is drawn from the population and the mean engagement of students who become knowledgeable in the material is measured and tested in an experiment against the null hypothesis.
Determine the absolute maximum and minimum of f(x)= 2 cosx+ sin 2x
Answers
Answer:
2.598 and -2.598.
Step-by-step explanation:
f(x) = 2 cos x + sin 2x
f'(x) = -2 sin x + 2 cos 2x = 0 for turning points.
cos 2x = 1 - 2 sin^2 x so we have
-2 sin x + 2 - 4 sin^2 x = 0
4sin^2 x + 2 sin x - 2 = 0
2(2 sin^2 x + sin x - 1) = 0
2(2sinx - 1)(sinx + 1) = 0
sin x = 0.5, -1 when f(x) is at a turning point.
x = π/6, -π/2, 5pi/6
The second derivative is 2 cos x + 2 * -2 sin 2x
= 2 cos x - 4 sin 2x
When x = π/6, this is negative , when x = -π/2 it is positive
so x = π/6 gives a maximum f(x) and x = -π/2 gives 0 so this is a point of inflection
When x = π/6 , f(x) = 2.598
When x = 5pi/6, f(x) = -2.598.
Does anyone know the slope of this line?
Answers
Answer:
3/4
Step-by-step explanation:
Use [tex]\frac{rise}{run}[/tex]. From the bottom red point, you have to go up 3 and left 4 to get to the top point. That's your answer.
To compare the production techniques used by foreign and local firms in Brazil, a random sample of 80 foreign firms and a random sample of 80 local firms are selected.This study uses_________ design
Answers
Answer:
An independent sample.
Step-by-step explanation:
In this scenario, to compare the production techniques used by foreign and local firms in Brazil, a random sample of 80 foreign firms and a random sample of 80 local firms are selected. We can safely conclude that this study uses an independent sample design.
An independent sample design can be defined as a research method that usually involves the use of multiple experimental groups (two or more). The samples or participants are only in one group and as such each group has no relationship with the other. This simply means that, the samples in a particular group is having no relationship with the other samples in another group.
Ultimately this implies, each samples are independent and satisfies only one condition of the independent sample design during the experiment to compare the production technique used by foreign and local firms in Brazil.
Hence, the researcher would use only two variables or conditions: a random sample of 80 foreign firms and a random sample of 80 local firms are selected.
A set of data is summarized by the stem and leaf plot below.
Steam Leaf
1 0 0 2 4 4 4 7 8 8 8 8 9
2 2 2 5 5 7 8
3 3 4 4 5 5 5 6 6 6 7 9 9
4 3 3 3 5 5 7 9
a. There are _____ values in the data set which are greater than or equal to 10 and less than or equal to 19.
b. There are _____ values in the data set which are greater than or equal to 30 and less than or equal to 39.
c. There are ____ values in the data set which are greater than or equal to 40 and less than or equal to 49.
Answers
Answer:
(a) There are 12 values in the data set which are greater than or equal to 10 and less than or equal to 19.
(b) There are 12 values in the data set which are greater than or equal to 30 and less than or equal to 39.
(c) There are 7 values in the data set which are greater than or equal to 40 and less than or equal to 49.
Step-by-step explanation:
We are given a set of data that is summarized by the stem and leaf plot below;
Steam Leaf
1 0 0 2 4 4 4 7 8 8 8 8 9
2 2 2 5 5 7 8
3 3 4 4 5 5 5 6 6 6 7 9 9
4 3 3 3 5 5 7 9
This shows that the data values are: 10, 10, 12, 14, 14, 14, 17, 18, 18, 18, 18, 19, 22, 22, 25, 25, 27, 28, 33, 34, 34, 35, 35, 35, 36, 36, 36, 37, 39, 39, 43, 43, 43, 45, 45, 47, 49.
(a) From the data above, the number of values in the data set which are greater than or equal to 10 and less than or equal to 19 is 12, i.e; 10, 10, 12, 14, 14, 14, 17, 18, 18, 18, 18, 19.
(b) From the data above, the number of values in the data set which are greater than or equal to 30 and less than or equal to 39 is 12, i.e; 33, 34, 34, 35, 35, 35, 36, 36, 36, 37, 39, 39.
(c) From the data above, the number of values in the data set which are greater than or equal to 40 and less than or equal to 49 is 7, i.e; 43, 43, 43, 45, 45, 47, 49.
Answer:5,9
Step-by-step explanation:
Poly(3-hydroxybutyrate) (PHB), a semicrystalline polymer that is fully biodegradable and biocompatible, is obtained from renewable resources. From a sustain-ability perspective, PHB offers many attractive proper-ties though it is more expensive to produce than standard plastics. The accompanying data on melting point (°C) for each of 12 specimens of the polymer using a differential scanning calorimeter appeared in the article "The Melting Behaviour of Poly(3-1-1ydroxybutyrate) by DSC. Reproducibility Study" (Polymer Testing, 2013: 215-220).
180.5 181.7 180.9 181.6 182.6 181.6
181.3 182.1 182.1 180.3 181.7 180.5
Compute the following:
a. The sample range
b. The sample variance S2 from the definition (Hint: First subtract 180 from each observation.]
c. The sample standard deviation
d. S2 using the shortcut method
Answers
Answer:
(a) 2.3
(b) 0.5245
(c) 0.7242
(d) 0.5245
Step-by-step explanation:
The data provided is:
S = {180.5, 181.7, 180.9, 181.6, 182.6, 181.6, 181.3, 182.1, 182.1, 180.3, 181.7, 180.5}
(a)
The formula to compute the sample range is:
[tex]\text{Sample Range}=\text{max.}_{x}-\text{min.}_{x}[/tex]
The data set arranged in ascending order is:
S' = {180.3 , 180.5 , 180.5 , 180.9 , 181.3 , 181.6 , 181.6 , 181.7 , 181.7 ,, 182.1 , 182.1 , 182.6}
The minimum value is, 180.3 and the maximum value is, 182.6.
Compute the sample range as follows:
[tex]\text{Sample Range}=\text{max.}_{x}-\text{min.}_{x}[/tex]
[tex]=182.6-180.3\\=2.3[/tex]
Thus, the sample range is 2.3.
(b)
Compute the sample variance as follows:
[tex]S^{2}=\frac{1}{n-1}\sum(x_{i}-\bar x)^{2}[/tex]
[tex]=\frac{1}{12-1}\times [(180.5-181.41)^{2}+(181.7-181.41)^{2}+...+(180.5-181.41)^{2}]\\\\=\frac{1}{11}\times 5.7692\\\\=0.524473\\\\\approx 0.5245[/tex]
Thus, the sample variance is 0.5245.
(c)
Compute the sample standard deviation as follows:
[tex]s=\sqrt{S^{2}}[/tex]
[tex]=\sqrt{0.5245}\\\\=0.7242[/tex]
Thus, the sample standard deviation is 0.7242.
(d)
Compute the sample variance using the shortcut method as follows:
[tex]S^{2}=\frac{1}{n-1}\cdot [\sum x_{i}^{2}-n(\bar x)^{2}][/tex]
[tex]=\frac{1}{12-1}\cdot [394913.57-(12\times (181.41)^{2}]\\\\=\frac{1}{11}\times [394913.57-394907.80]\\\\=\frac{5.77}{11}\\\\=0.5245[/tex]
Thus, the sample variance is 0.5245.
Suppose you were told that a 98% confidence interval for the population mean of mpg of a hybrid car was (22, 38). Determine the point estimate for this population mean.
Answers
Answer:
The point estimate for the mean mpg of hybrid cars is 30 mpg.
Step-by-step explanation:
A confidence interval has two bounds, a lower bound and an upper bound.
A confidence interval is symmetric, which means that the point estimate used is the mid point between these two bounds, that is, the mean of the two bounds.
In this problem, we have that:
Lower bound: 22
Upper bound: 38
Point Estimate:
(22 + 38)/2 = 30
The point estimate for the mean mpg of hybrid cars is 30 mpg.
round to the nearest hundredth: 2.0625
Answers
Answer: 2.06
Step-by-step explanation:
Remember if the number is greater than 5 round up, if it is less than 5 don't round up.
After round off the number 2.0625 to its nearest hundredth, it is 2.06.
When we round to the nearest hundredth, we follow the rule that says:
If the digit in the thousandths place is less than 5 we round down the hundredths place.If the digit in the thousandths place is greater than 5, we round up the hundredths place.The given number is 2.0625
In this number, the thousandth digit is 2 that is less than 5.
So we round down the hundredth place. The hundredth digit is 6 which remain 6 after rounding down.
So, the rounded number will be 2.06.
Hence, when we round 2.0625 to its nearest hundredth, it will be 2.06.
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To decide whether two different types of steel have the same true average fracture toughness values, n specimens of each type are tested, yielding the following results.
Type Sample Average Sample SD
1 60.7 1.0
2 60.5 1.0
Required:
a. Calculate the P-value for the appropriate two-sample z test, assuming that the data was based on n = 100. (Round your answer to four decimal places.)
b. Calculate the P-value for the appropriate two-sample z test, assuming that the data was based on n = 500. (Round your answer to four decimal places.)
c. Is the small P-value for n = 500 indicative of a difference that has practical significance
Answers
Answer:
a. P-value = 0.1589
b. P-value = 0.0016
Step-by-step explanation:
a. This is a hypothesis test for the difference between populations means.
The claim is that the two types of steel have different true average fracture toughness values.
Then, the null and alternative hypothesis are:
[tex]H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0[/tex]
The significance level is α=0.05.
The sample 1, of size n1=100 has a mean of 60.7 and a standard deviation of 1. The sample 2, of size n2=100 has a mean of 60.5 and a standard deviation of 1.
The difference between sample means is Md=0.2.
[tex]M_d=M_1-M_2=60.7-60.5=0.2[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2+\sigma_2^2}{n}}=\sqrt{\dfrac{1^2+1^2}{100}}\\\\\\s_{M_d}=\sqrt{\dfrac{2}{100}}=\sqrt{0.02}=0.1414[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{0.2-0}{0.1414}=\dfrac{0.2}{0.1414}=1.4142[/tex]
The degrees of freedom for this test are:
[tex]df=n_1+n_2-2=100+100-2=198[/tex]
This test is a two-tailed test, with 198 degrees of freedom and t=1.4142, so the P-value for this test is calculated as (using a t-table):
[tex]P-value=2\cdot P(t>1.4142)=0.1589[/tex]
As the P-value (0.1589) is greater than the significance level (0.05), the effect is not significant.
The null hypothesis failed to be rejected.
There is not enough evidence to support the claim that the two types of steel have different true average fracture toughness values.
b. As the sample size changes, the standard error and the degress of freedom change.
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{M_d}=\sqrt{\dfrac{\sigma_1^2+\sigma_2^2}{n}}=\sqrt{\dfrac{1^2+1^2}{500}}\\\\\\s_{M_d}=\sqrt{\dfrac{2}{500}}=\sqrt{0.004}=0.0632[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{0.2-0}{0.0632}=\dfrac{0.2}{0.0632}=3.1623[/tex]
The degrees of freedom for this test are:
[tex]df=n_1+n_2-2=500+500-2=998[/tex]
This test is a two-tailed test, with 998 degrees of freedom and t=3.1623, so the P-value for this test is calculated as (using a t-table):
[tex]P-value=2\cdot P(t>3.1623)=0.0016[/tex]
As the P-value (0.0016) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the two types of steel have different true average fracture toughness values.
A sequence starts 1, 4, 9, 16 ... The nth term is n^2. Use this fact to find the nth term of the following sequences: a) 2, 5, 10, 17 b) 2, 8, 28, 32
Answers
Answer:
n²+1
2n²
Step-by-step explanation:
We can see that 2, 5, 10, and 17 becomes 1, 4, 9, and 16 when we subtract 1.
So it adds 1. The nth will be n² + 1.
2, 8, 18, and 32 is the double of n². 1, 4, 9, and 16 are their half.
So the nth term becomes 2n².
The value of the equation is
a) The nth term of the equation is Aₙ = n² + 1
b) The nth term of the equation is Bₙ = 2n²
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
a)
Let the nth term of the equation be represented as Aₙ
Now , the sequence is
P = { 2 , 5 , 10 , 17 , .. }
when n = 1 , P₁ = 2
when n = 2 , P₂ = 5
So , the equation for the nth term is Aₙ = n² + 1
b)
Let the nth term of the equation be represented as Bₙ
Now , the sequence is
Q = { 2 , 8 , 18 , 32 , .. }
when n = 1 , P₁ = 2
when n = 2 , P₂ = 8
So , the equation for the nth term is Bₙ = 2n²
Hence , the equations are solved
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