For a sample of sample size n:{X 1
,…,X n
}, which is randomly chosen from a population with normal distribution N(μ,σ 2
), the sample mean is X
ˉ
= n
∑ i=1
n
X i
∼N(μ, n
σ 2
) and the sample variance is S 2
= n−1
∑ i=1
n
(X i
− X
ˉ
) 2
∼σ 2
χ n−1
2
/(n−1). The sample mean and sample variance are independent. Prove that the test statistic of one-sample t-test under H 0
:μ=μ 0
follows a t-distribution with df=n−1, i.e., S/ n
X
ˉ
−μ 0
∼t n−1
.
Answers
we have proved that the test statistic of a one-sample t-test under the null hypothesis H0: μ = μ0 follows a t-distribution with df = n - 1.
To prove that the test statistic of a one-sample t-test under the null hypothesis H0: μ = μ0 follows a t-distribution with df = n - 1, we need to show that the random variable (S/√n)(X - μ0) follows a t-distribution with df = n - 1, where S is the sample standard deviation, X is the sample mean, and n is the sample size.
First, let's rewrite the test statistic:
T = (X- μ0)/(S/√n)
To prove that T follows a t-distribution with df = n - 1, we need to show that T has the same probability density function (pdf) as a t-distribution with df = n - 1.
To do this, we can use the properties of the sample mean and sample variance:
1. The sample mean X follows a normal distribution with mean μ and standard deviation σ/√n.
2. The sample variance S^2 follows a chi-square distribution with df = n - 1.
Using these properties, we can express T as:
T = (X - μ0)/(S/√n)
= (X - μ0)/[σ/√n * (S/σ)]
= (X - μ0)/[σ/√n * √(S^2/σ^2)]
= (X - μ0)/[σ/√n * √(χ^2/(n - 1))]
= (X- μ0)/[σ/√(n/(n - 1)) * √(χ^2/(n - 1))]
= (X - μ0)/[S/√(n - 1)] * √(n - 1)/σ
Here, the term (X - μ0)/(S/√(n - 1)) follows a standard normal distribution, and the term √(n - 1)/σ is a constant.
Therefore, T can be expressed as the product of a standard normal random variable and a constant. This implies that T follows a t-distribution with df = n - 1.
Hence, we have proved that the test statistic of a one-sample t-test under the null hypothesis H0: μ = μ0 follows a t-distribution with df = n - 1.
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Related Questions
Find x when y=2 for y=6x+8
Answers
x
=
y
6
+
4
3
What do u get when u Simplify (5b)(-3a).
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9.) What is the containers dimensions
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Step-by-step explanation:
please mark me as a brainlist...
find the slope of the skate ramp that has a vertical change of 7 and horizontal change of 35
Answers
Answer:
1/5
Step-by-step explanation:
The difference of 3 times a number (n) and 10 is 5. What is the number?
Answers
If the difference of 3 times a number(n) and 10 is 5 , then the number is n = 5 .
In the question ,
it is given that
the number is = n
and also the statement is given as " the difference of 3 times of a number "n" and 10 is = 5 "
which means
3×n - 10 = 5 ...because times means × and difference means " - " .
3n - 10 = 5
on adding 10 to both sides of the equation and solving further ,
we get ,
3n = 5 + 10
3n = 15
n = 15/3
n = 5
Therefore , If the difference of 3 times a number(n) and 10 is 5 , then the number is n = 5 .
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Can anyone help me with my homework
Answers
Answer:
Below in bold.
Step-by-step explanation:
When x = -3, y = -(-3) - 2 = 3-2 = 1.
When x = 0, y = -0 - 2 = -2.
When x = 4, y = -4 - 2 = -6.
To draw the graph plot the 3 points and you should be able to draw a straight line through them.
Answer:
The first y value with -3, is 1.
The second y value with 0, is -2.
The third y value with 4, is -6.
A graph is shown below.
Step-by-step explanation:
It states that the rule is y = -x-2. So all we do is plug in the values shown in the table.
y = -(-3)-2
y = 1
The first point of intersection is at (-3,1)
y = -(0)-2
y = -2
The second point of intersection is at (0,-2)
y = -(4)-2
y = -6
The third point of intersection is at (4,-6)
So now, we plot the points, and draw a line connecting them all.
Someone help quickly!
Look at the picture!
Answers
Give the name (monomial,
binomial, trinomial etc.) and
degree of the polynomial.
x + 2
Answers
Answer:
Step-by-step explanation:
x+2
degree is 1
\(x^{1} +2\)
and has two terms so is a binomial
Discharge in a rectangular channel is Q=30 m3/s, the longitudinal channel bottom slope is Ie=1/400. fricition coefficiem is n=0.02, and channel bottom width is B=5 m. a) Calculate critical flow depth (y,) at which the given discharge is passed at minimum specific energy level. b) Determine critical longitudinal channel bottom slope (t.) to pass the given discharge at critical depth and decide on the flow regime by comparing Ie to Iw c) Determine uniform flow depth (yn) and the coressponding flow regime ( N5, the Froude number).
Answers
In a rectangular channel with a given discharge, longitudinal slope, friction coefficient, and channel width, we need to determine the critical flow depth, critical slope, and uniform flow depth. These calculations will help us understand the flow regime and hydraulic characteristics of the channel.
a) To calculate the critical flow depth (yc) at which the given discharge is passed at the minimum specific energy level, we can use the specific energy equation. By equating the specific energy equation to its critical value, we can solve for yc.
b) To determine the critical longitudinal channel bottom slope (Io) to pass the given discharge at critical depth, we can use Manning's equation. By rearranging the equation and solving for Io, we can find the critical slope. Comparing Io to the longitudinal slope (Ie), we can determine the flow regime.
c) To determine the uniform flow depth (yn) and the corresponding flow regime, we can use Manning's equation again. By substituting the given values into the equation and solving for yn, we can find the depth of uniform flow. The flow regime can be determined by calculating the Froude number (Fr).
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Which point on the coordinate plane is a reflection of point A across the y-axis
Answers
On wednesday 72% of the customers who bought gas at a gas station mad additional purchases. There were 250 customers who bought gas. how many of these 250 customers made additional purchases?
Answers
180
Explanation:
72% of 250 equals
0.72 x 250
Hope this helps!
21. The table shows four quantities. Each quantity
is assigned a numeric label.
3
√₂
1.3
78
52
1
●
2
3
4
5
Order the quantities from least to greatest.
● Next, write the sequence of numeric labels
in the same order as their corresponding
quantities.
This sequence of numeric labels is your
answer.
Answers
The quantities from least to greatest are:
3√₂1.3785212345Therefore, the sequence of numeric labels is: B, A, C, D.
How to determine sequence of numeric labels?To find the sequence of numeric labels, these are the calculations for ordering the quantities from least to greatest:
Identify the smallest quantity.Identify the largest quantity.Order the remaining quantities in between the smallest and largest quantities.Assign the numeric labels to the quantities in the order that you have determined.For the following quantities:
3√₂
3√₂ = 3 × √₂
= 3 × 1.41421356237
= 4.24264068712
1.378 = 1.378
521 = 521
2345 = 2345
Determine the sequence of numeric labels by following these steps:
The smallest quantity is 1.378.
The largest quantity is 2345.
The remaining quantities are 3√₂ and 521.
Order these quantities in between 1.378 and 2345.
The sequence of numeric labels is B, A, C, D.
1.378 is the smallest quantity, followed by 3√₂, 521, and 2345. Therefore, the sequence of numeric labels is B, A, C, D.
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The whittendale family purchases a new refrigerator on a no interest for one year plan. The cost is $1,385. There is no down payment If they make a monthly payment of x dollars until the last month, express their last month's payment algebraically.
Answers
Integrate f(x,y)=(x + y + 3)^−2 over the triangle with vertices (0,0),(4,0), and (0,8).
Use symbolic notation and fractions where needed.
Answers
To integrate the function f(x, y) = (x + y + 3)^-2 over the given triangle with vertices (0,0), (4,0), and (0,8), we can set up the integral using symbolic notation and fractions.
The integral can be written as ∫∫R (x + y + 3)^-2 dA, where R represents the region of integration.
To evaluate this integral, we need to determine the limits of integration for x and y based on the triangle's vertices. Since the triangle is defined by the points (0,0), (4,0), and (0,8), we can set the limits as follows:
For x: 0 ≤ x ≤ 4
For y: 0 ≤ y ≤ 8 - (2/4)x
Now, we can rewrite the integral as ∫[0,4]∫[0,8-(2/4)x] (x + y + 3)^-2 dy dx.
Evaluating this integral will yield the exact value of the integral over the given triangle region.
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please help me i dont understand
Answers
Answer:
alternate interior, 37 degrees
step-by-step explanation:
these are alternate interior angles and alternate interior angles are always equal :)
Cans of corn are on sale at 10 cans for $4. What's the cost for 30 cans?
Answers
Answer:
12
Step-by-step explanation:
30 is 3x larger than 10
4 x 3 = 12
Sabah has $450 to pay for college textbooks. She expects to pay about $75 per book. Her friend told her that 4 of them can be checked out of the library for free.
Complete the equation below to find the total number of books that Sabah can get for her classes. Use b to represent the total books.
Answers
Answer: I believe it would be 10 books?
Step-by-step explanation: because 450/75=6 and since 4 are free, then 6+4=10 b
What is the image of the point (2,3) after the transformation Ry-axis?
Answers
Answer:
The image of the point (2, 3) after the reflection across the y-axis will be: (-2, 3)
Step-by-step explanation:
We know that when a point, let say P(x, y), is reflected across the y-axis, the sign of x-coordinate is reversed but y-coordinate remains the same.
In other words, the rule is:
P(x, y) → P'(-x, y)
Here, P'(x, y) is the image of the original point P(x, y) after the reflection across the y-axis.
Considering the point
A(2, 3)Let us reflect the point A(2, 3) across the y-axis.
We know the rule:
P(x, y) → P'(-x, y)
so
A(2, 3) → A'(-2, 3)
Thus, the image of the point (2, 3) after the reflection across the y-axis will be: (-2, 3)
Please help, I’ll give brainliest to the first person.
Answers
Answer:
option 1
Step-by-step explanation:
The equation cosx = square root 3/2 is an identity.
A. True
B. False
Answers
Answer:
false, because cos(x) has no definite answer, and equations aren't identities
Step-by-step explanation:
Entertainment Software Association would like to test if the standard deviation for the age of "gamers" (those that routinely play video games) is equal to 6 years. The correct set of hypotheses is: Group of answer choices
Answers
The calculate the chi-square statistic using the formula:
χ² = (n - 1) s² / σ²
How to find the correct set of hypotheses?The correct set of hypotheses to test whether the standard deviation for the age of "gamers" is equal to 6 years can be stated as:
Null Hypothesis: The standard deviation of age for gamers is equal to 6 years.Alternative Hypothesis: The standard deviation of age for gamers is not equal to 6 years.In symbols, the hypotheses can be expressed as:
H0: σ = 6
Ha: σ ≠ 6
Where σ represents the population standard deviation of age for gamers.
To test this hypothesis, we can use a statistical test called the chi-square test for variance.
We would need a sample of ages of gamers and calculate the sample variance, denoted as s².
We can then calculate the chi-square statistic using the formula:
χ² = (n - 1) s² / σ²
where n is the sample size.
Under the null hypothesis, the chi-square statistic follows a chi-square distribution with degrees of freedom equal to n - 1.
We can then calculate the p-value associated with the observed chi-square statistic and reject or fail to reject the null hypothesis based on a chosen significance level.
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A stack of textbooks weighs 38.25 pounds. Each textbook weighs 2.25 pounds. How many textbooks are in the stack?
Answers
Answer
17
Step-by-step explanation:
Divide 38.25 by 2.25
Answer:17
Step-by-step explanation:
38.25 divided by 2.25
HELP! The vertices of a rectangle are plotted.
What is the perimeter of the rectangle?
11 units
66 units
17 units
34 units
Answers
Explanation: The rectangle is 11 units by 6 units. Since counting the perimeter manually is pretty tedious, you can just add 11 and 6 and multiply its sum by two, since opposite sides of a rectangle are equal.
if its assumptions are met, the analysis of variance technique is appropriate when ____.
Answers
Answer:
comparing the means of three or more groups
Step-by-step explanation:
Two different meal combinations at a chicken restaurant have the same number of total
calories. The first meal has 8 chicken nuggets and a large order of fries, while the second
meal has 12 nuggets and a small order of fries. The large order of fries contains 288.5
calories and the small order of fries has 193.5 calories. Which equation and solution can
be used to determine n, the number of calories in each chicken nugget?
Answers
Answer: 8x+288.5=12x+193.5
Step-by-step explanation:
Let x=calories per nugget
First meal=second meal
8x+288.5=12x+193.5
Simplify: 3 square root 3 + square root 12
Answers
Answer:
Step-by-step explanation:
Number Square Root (√)
12 3.464
What decimal is equivalent to 1/8
Answers
Answer:
0.125
Step-by-step explanation:
Becuase math
A certain arithmetic sequence has this explicit formula for the nth term:
an = 11 + (n - 1)(4)
The same sequence has this recursive formula:
an = an-1 +__
What number belongs in the blank space in the recursive formula?
Answers
Answer:
4
Step-by-step explanation:
The n th term of an arithmetic sequence is
\(a_{n}\) = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
Given
\(a_{n}\) = 11 +(n - 1)4 , then
d = 4
The recursive formula allows a term in the sequence to be found by adding d to the previous term, thus
\(a_{n}\) = \(a_{n-1}\) + 4
Answer:
The answer is 4
Step-by-step explanation:
Tell whether the equation has one, zero, or infinitely many solutions.
5(x - 2) + 8 = 5x - 1
Answers
Answer:
zero (no solution)
Step-by-step explanation:
3(x+2)=5x-7
Work out value of x
Answers
Answer:
first distribute:
3x+6=5x-7
then combine the x's:
-2x+6=-7
get rid of the 6:
-2x=-13
divide the 2:
6.5
Step-by-step explanation:
Answer:
x = 13/2
Step-by-step explanation:
3(x+2) = 5x-7 3x+6 = 5x-73x-5x = -7-6-2x = -13x = -13/-2x = 13/2