# Statistical Methods-Measures of Location [ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)]: Questions 1 - 7 of 7

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## Question 1

Appeared in Year: *2010*

### Describe in Detail

Essay▾A cyclist pedals from his house to his college at a speed of 10 km per hour and back from the college to his house at 15 km per hour. Find the average speed.

### Explanation

Given that a cyclist pedals from his house to his college at a speed of 10 km per hour and back from the college to his house at 15 km per hour. Assume that the distance between the house to college is d. Here the average speed is finding by harmonic mean because it is suitable when the values are pertaining to the rate of change per unit time such…

… (9 more words) …

## Question 2

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let F_{n} (x) be the distribution function define by

Is the lim F_{n} (x) a distribution function? If not, why?

### Explanation

The function F_{n} (x) is a distribution function if

By the definition , but here this is not equal to one. This says that F_{n} (x) is not a distribution function.

## Question 3

Appeared in Year: *2011*

### Describe in Detail

Essay▾Let Z be a random variable with p. d. f. f (z) . Let z_{α} be its upper α ^{th} quantile. Show that if X is

a random variable with p. d. f. then σz_{α} + µ is the upper α ^{th} quantile of X.

### Explanation

Let Z be a random variable with p. d. f. f (z) . Given that the quantile of z is defined as

X is a random variable with p. d. f. , then quantile is

Assume

Then the limit is

Lower limit is - and upper limit is

If z = then the integral is same for given z and the quantile is

So, the quantile of X is .

## Question 4

Appeared in Year: *2010*

### Describe in Detail

Essay▾Show that for any discrete distribution, standard deviation is not less than mean deviation from mean.

### Explanation

Here, we show that standard deviation is greater than mean deviation from mean.

S. D. ⩾ mean deviation from mean

(S. D.) ^{2} ⩾ (mean deviation from mean) ^{2}

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 essentially true as f_{1} , f_{2} , … are not negative. The equality ho…

… (16 more words) …

## Question 5

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let X have the probability density function

Find the mean, geometric mean and harmonic mean.

### Explanation

Then mean is

The geometric mean is

The geometric mean is

The harmonic mean is

## Question 6

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 7

Appeared in Year: *2013*

### Describe in Detail

Essay▾For the following frequency distribution on 229 values:

Class | Frequency |

0 - 10 | 12 |

10 - 20 | 30 |

20 - 30 | x |

30 - 40 | 65 |

40 - 50 | y |

50 - 60 | 25 |

60 - 70 | 18 |

the median is found to be 46. Find the values of x and y.

### Explanation

The cumulative frequency is

Class | Frequency | Cumulative frequency |

0 - 10 | 12 | 12 |

10 - 20 | 30 | 42 |

20 - 30 | x | 42 + x |

30 - 40 | 65 | 107 + x |

40 - 50 | y | 107 + x + y |

50 - 60 | 25 | 132 + x + y |

60 - 70 | 18 | 150 + x + y |

Median is 46 which lies in the class 40 - 50. So, median class is 40 - 50.

Here n = 229, l = lower limit of median class = 40

cf = cumulative frequency of the above median class = 1…

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