Paper, Order, or Assignment Requirements
Question 1
The diagram below is a graph of density function (frequency function) of normal distribution. We are going to assume the case to “ estimate the interval with 95% confidence coefficient”. Using the bell graph below, please explain what these phrases mean: “confidence coefficient”, “”confidence limit”, and “significance (ex. Significant difference)” Then, please show where the 5% probability corresponds to in the bell graph below. Please draw in the bell graph.
Question 2
Assume that the diagram you see below is the graph of density function (frequency function) of normal distribution N (50, 102) with standard deviation 10 and average 50. We want to know what value will be included in the top 3% (x has to be >= to what value to be included in top 3%). Please explain how I will be able to solve this question (find the value that will be the top 3%) Then, using excel, please actually calculate and solve the problem. Please draw in the bell graph below.
Question 3
We know that a random valuable (probability valuable) X has normal distribution of average μ and variance σ². Please show this of this in a standardized equation.
We know that random valuable (probability valuable) X1, …, Xn will follow normal distribution N (μ, σ²) with identically and independently distributed. In this condition, we will define sample mean of X1, …, Xn as
x̅ = X1+…+Xn
———————–
n
What will happen to the distribution of x̅ ? Also, please write the equation of standardized x̅.
We know that when random valuable (probability valuable) that has normal distribution is standardized, it will be normal distribution. Then, what is its mean and variance?
What kind of distribution of random valuable (probability valuable) can we think as standardization itself? Can we conclude that any random valuable (probability valuable) will follow the nature of “The distribution after standardization is the same type of distribution as that from before”？
Question 4
We want to estimate the interval with the population mean and variance data that we have. We will assume that the population follows the normal distribution. Please answer the question below. You are welcome to use the knowledge you used up until this question. However, you have to mention the distribution for assumption, theory of statistics and “percent point (known as alpha point).
We are thinking of estimating the population mean. We know the population variance. Please explain in an easily understandable way, how to perform interval estimation.
We are thinking of estimating the population mean. However, we do not know the population variance. . Please explain in an easily understandable way, how to perform interval estimation.
We want to estimate the population mean with interval estimation with confidence coefficient 95% . Please explain in an easily understandable way, what to do.
Question 5
The mathematical definition of t distribution is below
Y
t = ——- (1)
√ Z/k
Here, we will assume that Y follows the normal distribution N(0, 1), and Z is independent random valuable that will follow the degree of freedom k’s Chi-squared distribution. With this fact, when we assume X1 , …, Xn will follow independent and identical normal distribution N (μ, σ²),
√ n (x̅ – μ)
—————– (2)
s
please prove that it will follow the t distribution of degree of freedom n – 1 mathematically . (This)
√ n (x̅ – μ)
*Hint : We know that ——————– follows normal distribution N (o, 1), and
σ
(n – 1)s2
———- follows chi squared distribution of degree of freedom n – 1. Apply these facts
σ2
into equation (1). When you rearrange, you should get the (2) equation.
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