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alexgriva [62]
3 years ago
13

A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The chi-square val

ues to be used for this interval estimation are
a. 16.151 and 40.113.

b. 14.573 and 43.195.

c. 15.308 and 44.461.

d. 11.808 and 49.645.
Mathematics
1 answer:
Natali5045456 [20]3 years ago
7 0

Answer:

\chi^2_{\alpha/2}=43.195

\chi^2_{1- \alpha/2}=14.573

b. 14.573 and 43.195.

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 mean or variance 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.

The Chi square distribution is the distribution of the sum of squared standard normal deviates .

2) Solution to the problem

The confidence interval for the population variance is given by the following formula:

\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}

We need on this case to calculate the critical values. First we need to calculate the degrees of freedom given by:

df=n-1=28-1=27

Since the Confidence is 0.95 or 95%, the value of \alpha=1-0.95=0.05 and \alpha/2 =0.025, and we can use excel, a calculator or a table for the Chis square distribution with 27 degrees of freedom to find the critical values. We need a value that accumulates 0.025 of the area on the left tail and 0.025 of the area on the right tail.  

The excel commands would be: "=CHISQ.INV(0.025,27)" "=CHISQ.INV(0.975,27)". so for this case the critical values are:

\chi^2_{\alpha/2}=43.195

\chi^2_{1- \alpha/2}=14.573

b. 14.573 and 43.195.

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Step-by-step explanation:

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In 1990 the average family income was about $ 39 , 000 , and in 2010 it was about $ 70 , 768 . Let x = 0 represent 1990, x = 1 r
iren2701 [21]

Answer:

<em />f(x) = 1588.4x + 39000<em />

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Step-by-step explanation:

Given

In 1990; Income= $39000

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Solving (a): An equation in form of f(x) = ax + b

First, we need to determine the slope, a

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Taking y as income and x as year index.

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Substitute these values in the above formula

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Next, is to determine the formula using:

y - y_1 = a(x - x_1)

<em>Considering :When x = 0; y = 39000, we have</em>

<em />y - 39000 = 1588.4(x - 0)<em />

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<em />y = 1588.4x + 39000<em />

<em />

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<em />f(x) = 1588.4x + 39000<em />

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So:

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3 0
3 years ago
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Answer:

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(a)We want to show that E[X + Y ] = E[X] + E[Y ].

When we have two random variables instead of one, we consider their joint distribution function.

For a function f(X,Y) of discrete variables X and Y, we can define

E[f(X,Y)]=\sum_{x,y}f(x,y)\cdot P(X=x, Y=y).

Since f(X,Y)=X+Y

E[X+Y]=\sum_{x,y}(x+y)P(X=x,Y=y)\\=\sum_{x,y}xP(X=x,Y=y)+\sum_{x,y}yP(X=x,Y=y).

Let us look at the first of these sums.

\sum_{x,y}xP(X=x,Y=y)\\=\sum_{x}x\sum_{y}P(X=x,Y=y)\\\text{Taking Marginal distribution of x}\\=\sum_{x}xP(X=x)=E[X].

Similarly,

\sum_{x,y}yP(X=x,Y=y)\\=\sum_{y}y\sum_{x}P(X=x,Y=y)\\\text{Taking Marginal distribution of y}\\=\sum_{y}yP(Y=y)=E[Y].

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Since X and Y are independent, Cov(X,Y)=0

=Var(X)+Var(Y)

Therefore as required:

Var(X+Y)=Var(X)+Var(Y)

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