# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 113 - 115 of 165

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## Question number: 113

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2012

### Describe in Detail

125out of 285 college-going male students in 1995 from a city were found to be smokers. Another sample of 325 such students from the same city in 2012 included 95 smokers. Examine at 5 % level of significance if smoking habit among college-going students is on the decrease in this city.

### Explanation

The given information can be written as in a table:

Year | Smoker | Non-smoker | Total |

1995 | 125 | 160 | 285 |

2012 | 95 | 230 | 325 |

Total | 220 | 390 | 610 |

Here, the null hypothesis is that the proportion of smoker in this two year is same and the alternative is there is decrease in the smoker in 2012. Le

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## Question number: 114

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

### Describe in Detail

If f _{x} (x) be the probability density function of a N (µ, σ ^{2}) distribution, then show that

where , and φ (x) and Φ (x) are the probability density function and distribution function of the standard normal distribution respectively.

### Explanation

Given that f _{X} (x) be the probability density function follows N (µ, σ ^{2}).

Then

………. (1)

Let assume that =z, then differentiate

The limit is also change, the lower limit is and upper limit is

Equation (1) can be written as

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## Question number: 115

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2012

### Describe in Detail

Ten short-distance runners were put to a rigorous training for two months. Times taken by them to clear 100 metres before and after the training were as follows:

Sl. no. of runner | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Time (in sec) before training | 10.6 | 10.9 | 10.1 | 10.5 | 11.0 | 11.2 | 10.7 | 10.2 | 10.9 | 10.6 |

Time (in sec) after training | 10.1 | 10.7 | 9.9 | 10.0 | 11.1 | 10.9 | 10.6 | 10.3 | 10.5 | 10.8 |

Use Wilcoxon’s paired sample signed rank test to examine at 1 % level if the training was at all effective. (The critical value of Wilcoxon’s statistic at 1 % level of significance for n = 10 is 5)

### Explanation

In Wilcoxon’s paired sample signed rank test, the null hypothesis that we are sampling two continuous symmetric populations with for the paired-sample case, we rank the differences of the paired observations without regard to sign and proceed as in the single-sample case

Let and represent the median time of all runner put to a r

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