# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 128 - 134 of 164

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## Question 128

Appeared in Year: *2012*

### Describe in Detail

Essay▾Solve the equation f (x) = 0 by using a suitable interpolation formula on the following values:

x | 3 | 4 | 5 | 6 |

f (x) | -2.8 | -1.2 | -0.3 | 1.8 |

### Explanation

To solve the equation f (x) = 0, using Lagrange՚s inverse interpolation method

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## Question 129

Appeared in Year: *2015*

### Describe in Detail

Essay▾Prove that for r = 1,2, … , n

### Explanation

We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.

The incomplete gamma function is

Then

provided that *r* is an integer. Thus recall that Γ (*r*) = (*r* − 1) ! for *r* integer.

Rewriting this expression we arrive at the desired expression

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## Question 130

Appeared in Year: *2011*

### Describe in Detail

Essay▾Let the temperature before and after administration of aspirin be

Patient | Before | After |

1 | 100·0 | 98·1 |

2 | 102·1 | 97·2 |

3 | 100·6 | 98·6 |

4 | 100·1 | 99·1 |

5 | 101·5 | 97·6 |

6 | 102 | 98·6 |

7 | 99·9 | 98·2 |

8 | 102·7 | 98·1 |

9 | 100.40 | 98·2 |

10 | 100·8 | 97.1 |

Test by the sign test; whether aspirin is effective in reducing temperature. What is the p-value of the calculated statistic?

### Explanation

Let M_{1} and M_{2} is the median temperature of the patients. The hypothesis for test is

Fir this first find the difference of median

Also find the difference of temperature of before and after

d_{i} 2.1 4.9 2.0 1.0 3.9 3.4 1.7 4.6 2.4 3.7

Then find the sign of difference between d_{i} and d_{0} .

-+ -- ++ -+ -+

The number of positive sign is r = 5 and negative sign i…

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## Question 131

### Explanation

We have to prove that

By definition of Kurtosis

Let x_{1} , x_{2} , … , x_{n} are n observations in a set have frequency f_{1} , f_{2,} … , f_{n} and the mean of observation is , then

Assume

which is always true because this a variance which is always positive

So,

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## Question 132

Appeared in Year: *2010*

### Describe in Detail

Essay▾How large a sample must be taken in order that the probability will be at least 0·95 that X_{n} will be within 0·5 of µ (µ is unknown and σ = l) .

### Explanation

Here, you would know the standard deviation of a population but not know the mean of that population. So, you want to estimate the mean μ to within a given margin of error in a given confidence interval. The required sample size is

Margin of error E = 0.5,

1-α = 0.95 ⇾ α = 0.05

σ = 1 and z_{α/2} = 1.96

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## Question 133

Appeared in Year: *2013*

### Describe in Detail

Essay▾Find the weighted arithmetic mean of the first ‘n’ natural numbers, the weights being the corresponding numbers.

### Explanation

If x_{1} , x_{2} , … , x_{n} are the values having weights w_{1} , w_{2} , … , w_{n} respectively, the weight mean is

Here X is first ‘n’ natural numbers that is 1,2, 3, … , n

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## Question 134

Appeared in Year: *2015*

### Describe in Detail

Essay▾Fit the exponential curve y = a + bx to the following data

x: 0 2 4

y: 5.01 10 31.62

### Explanation

his is not exponential equation, it is a straight line y = a + bx

S. no | x | y | Xy | x ^{2} |

1 | 0 | 5.01 | 0 | 0 |

2 | 2 | 10 | 20 | 4 |

3 | 4 | 31.62 | 126.48 | 16 |

Sum | 6 | 46.63 | 146.48 | 20 |

The normal equations is

Using the table , putting these values in normal equations. We get

Solving these two equations, we get the value of a and b

So, required line is y = 2.188 + 6.677x