# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 113 - 115 of 164

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## Question 113

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Appeared in Year: 2012

### Describe in Detail

Essay▾

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. Let P1 and P2 are the proportion of smoker in year 1995 and 2012 with sample size n1 an&#8230;

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## Question 114

Appeared in Year: 2014

### Describe in Detail

Essay▾

If fx (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 fX (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

The first integral is the distribution function of standard normal distribution

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## Question 115

Appeared in Year: 2012

### Describe in Detail

Essay▾

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)beforetraining 10.6 10.9 10.1 10.5 11 11.2 10.7 10.2 10.9 10.6 Time (in sec)aftertraining 10.1 10.7 9.9 10 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 rigorous training b&#8230;

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