## Introduction to Mathematical Statistics |

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Page 166

Such a rule is called a test of the

hypothesis H1: 0 = 2. There is no limit to the number of rules or tests that can be

constructed. We shall compare two such tests, and to keep the exposition simple,

...

Such a rule is called a test of the

**null hypothesis**Ho: 6 = 1 against the alternativehypothesis H1: 0 = 2. There is no limit to the number of rules or tests that can be

constructed. We shall compare two such tests, and to keep the exposition simple,

...

Page 176

Find a best test of the

the alternative simple

Xn denote a random sample from a normal distribution n(z; 6, 100). Show that ...

Find a best test of the

**null**simple**hypothesis**Ho: 61 = 6'1 = 0, 6, = 6'2 = 1 againstthe alternative simple

**hypothesis**H1: 61 = 6"1 = 1, 0, = 6'2=4. 9.10. Let X1, X2, ...,Xn denote a random sample from a normal distribution n(z; 6, 100). Show that ...

Page 188

If the

X)+:(r. - ?). n + m – 2 has, in accordance with page 134, a t distribution with n + m

– 2 degrees of freedom. Thus the random variable defined by X*/("+") is n + m ...

If the

**null hypothesis**Ho: 61 = 0, is true, the random variable 70??? T #(x - ?) źw,X)+:(r. - ?). n + m – 2 has, in accordance with page 134, a t distribution with n + m

– 2 degrees of freedom. Thus the random variable defined by X*/("+") is n + m ...

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### Common terms and phrases

Accordingly best critical region binomial distribution cent confidence interval chi-square distribution composite hypothesis conditional p.d.f. confidence interval Consider continuous type degrees of freedom denote a random discrete type distribution function distribution having p.d.f. distribution with mean ExAMPLE Exercises f(zi function F(z hypothesis H1 independent random variables integral Jacobian joint p.d.f. joint sufficient statistics Let the random Let X1 limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent Mx(t My(t normal distribution n(z null hypothesis null simple hypothesis one-to-one transformation order statistics p.d.f. of Yi Poisson distribution positive integer Pr(a Pr(X Pr(Y probability density functions quadratic form random experiment random interval random sample random variables X1 respectively ſ ſ sample space Show significance level ſº stochastically independent random subset sufficient statistic Yi theorem tion type of random unbiased statistic values X1 and X2 Xn denote zero elsewhere