# Statistical Methods-Tests of Significance (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 5 - 9 of 17

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## Question number: 5

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2010

### Describe in Detail

For 2X2 table

A | b |

C | d |

prove that Chi-square test of independence gives

### Explanation

Let the contingent table is

Class | A | α | Total |

B | a | b | a + b |

β | c | d | c + d |

Total | a + c | b + d | a + b + c + d =N |

Here A denotes the presence of any attributes and α denotes the absence of attributes A

Since the marginal frequencies is fixed, therefore

Probability for a individual belong

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## Question number: 6

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2014

### Describe in Detail

Let X follow a binomial distribution B (n, P). Explain the test procedure for

H _{0}: P = P _{0} against H _{1}: P > P _{0}

when the sample size is (i) small, and (ii) large. It is desired to use sample proportion p as an estimator of the population proportion P, with probability 0·95 or higher, that p is within 0·05 of P. How large should sample size (n) be?

### Explanation

Let X follow a binomial distribution B (n, P) with mean nP and variance nP (1-P). In testing the hypothesis

H _{0}: P = P _{0} against H _{1}: P > P _{0}

The null hypothesis can be tested by z-test for assuming the sample size n is large (Central limit theorem). For binomial distribution, the test statistic is

The test criter

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## Question number: 7

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2015

### Describe in Detail

If X follows binomial b (n _{1}, p _{1}) distribution and Y follows binomial b (n _{2}, p _{2}), provide an appropriate test at level α for H _{0}: p _{1} =p _{2} against H _{1}: p _{1} > p _{2}.

### Explanation

The hypothesis is equivalent to testing the null hypothesis that p _{1} −p _{2} = 0 against the alternative is p _{1} -p _{2} > 0. The statistic on which we base our decision is the random variable . Suppose the samples of sizes n _{1} and n _{2} are selected from binomial populations are independent.

For large sample size, the estimator was approximately

… (57 more words) …

## Question number: 8

» Statistical Methods » Tests of Significance » F-Test

Appeared in Year: 2012

### Describe in Detail

Indicate how you would test the hypothesis that the means of k independent normal populations are identical, clearly mentioning the null and the alternative hypotheses, the assumptions made, the test statistic used, and the critical region.

### Explanation

Let there are k independent normal population with different sample size n _{1}, n _{2}, …, n _{k}. The test of null hypothesis is that the means of k normal population is same and the alternative hypothesis is any two mean are not same.

Vs H _{1}: at least two mean are not equal

Under the assumptions that the

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## Question number: 9

» Statistical Methods » Tests of Significance » F-Test

Appeared in Year: 2013

### Describe in Detail

Obtain 100 (1 -α) % confidence interval for the ratio of population variances by using two independent random samples from N (µ _{1}, σ _{1}^{2}) and N (µ _{2}, σ _{2}^{2}) under the assumption that the population means are (i) known and (ii) unknown.

### Explanation

Let X _{1}, X _{2}, …, X _{n} and Y _{1}, Y _{2}, …Y _{m} are the samples taken from independent N (µ _{1}, σ _{1}^{2}) and N (µ _{2}, σ _{2}^{2}).

In this question the hypothesis for testing is

Let s _{1}^{2} and s _{2}^{2} be the estimates variances of σ _{1}^{2} and σ _{2}^{2} based on sample sizes n and m.

and

Then

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