## Help with Statistics

Hello everyone, I need help with these examples from statistics.
They are basic but because I never really studied this subject I need a little help with the procedure and understanding.

I would be thankful for any help.

17.1. In a small lottery, there are 100 tickets, 15 from them winning. If you have 5 tickets, What is the probability that at least 2 from your tickets will win?

17.2.Lifetime (in hours) of a device is described by a r. variable with Normal distribution with parameters µ = 200 hours, variance σ2 = 400. What is the mean lifetime?
Further compute, with the help Excel (or of table of N(0,1) distribution function)

1. probability that device survives 180 hours;

2. such time that 90% of devices survive it (i. e. compute the 10% quantile of the distribution);
Write the density function of this normal distribution.

17.3.Probability of occurrence of a success in one trial is 0,3. What is the probability that from 100 trials the number of successes will be between 20 and 40?
(use binomial distribution and also approximation with Central Limit Theorem, compare the results).

17.4.

Analyze the following grouped data - table of incomes (ascertained by survey): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data.
 INCOME (class intervals), in 1000 CzK Number of people (observed frequency) from 24 to 26 3 26 28 4 28 30 10 30 32 16 32 34 14 34 36 5 36 38 2
From observed relative frequencies, estimate the probability (proportion) of incomes smaler than 30 000 CzK.
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17.5. Consider a machine park with 5 machines, let variable X denote the (preventive) service (hours per week) and variable Y the number of failures per week (averaged):
 X: 2 4 6 8 10 12 Y: 1.7 1.3 1 0.8 0.6 0.5
Analyse the dependence of Y on X:
a)Find the correlation coefficient r (x,y).
b)Find the parameters of the regression line y = a + b*x
c)Plot the regression line and the data.
d)Compute also the residual variance s^2.
e)Assume that x hours of preventive service cost x^2 times 100 CzK while the repair of one failure costs (in average) 10000 CzK.
Estimate the optimal length of service time per
week (find x for which the average cost tis minimal).