# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 67 - 71 of 165

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## Question number: 67

» 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: 68

» Numerical Analysis » Inverse Interpolation

Appeared in Year: 2015

### Describe in Detail

Compute the value of by Simpson’s 1/3 ^{rd} rule. Given that ln4.0 = 1.39, ln4.2 = 1.43, ln4.4 = 1.48, ln4.6 = 1.53, ln4.8 = 1.57, ln5.0 = 1.61, ln5.2 = 1.65

### Explanation

So, the seven ordinates of the integrand using Simpson’s rule is

Where h = 1/6 (b-a) and y _{k} =f (a + kh) for k = 0,1, 2,3, 4,5, 6

Here, b = 5.2, a = 4, h = 0.2, f (x) = lnx

Using Simpson’s rule is

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## Question number: 69

» Numerical Analysis » Inverse Interpolation

Appeared in Year: 2014

### Describe in Detail

Given the following values of the function y = f (x); evaluate f (4) and also find x for which f (x) = 25.

f (1) = 10, f (2) = 15, f (3) = 42.

### Explanation

Given that y = f (x), So

x 1 2 3 4

y 10 15 42?

To find the missing value, we use binomial expansion method. Here 3 values are known, we would take third order finite difference zero. Thus,

Second question is that if we have given that value of y = f (x) =25, then find x using Lagrange’s

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## Question number: 70

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

### Describe in Detail

Let (X, Y) be jointly distributed with p. d. f.

Find marginal probability density function of X and Y.

### Explanation

The marginal probability density function of X is

The range is 0 < x < 1

The marginal probability density function of Y is

The range is 0 < y < 1

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## Question number: 71

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2010

### Describe in Detail

The observed-values of a function are respectively 168,120,72 and 63 at the four positions 3,7, 9 and 10 of the independent variable. What is the best estimate you can give for the value of the function at the position 6 of the independent variable?

### Explanation

Using Lagrange’s formula

Given that a = 3, b = 7, c = 9, d = 10 and f (a) =168, f (b) =120, f (c) =72, f (d) =63

Find x = 6, the values of f (x) =?

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