PROBLEM 1

The mayor of a town informs an investor who would like to build a shopping mall there that average of the annual spending(expenses) (in food and in clothes) by home(foyer) is 15 000 $ there. Let us suppose that in this city kind (genre), we can admit that the distribution (casting) of the annual spending (expenses) of homes (foyers) is normal with a $2 000 standard deviation, as a previous survey (investigation) established her (it).

1. In a sample of homes (foyers) of this city of size (cutting) n = 15, we find that X = 14 000 $. By using the critical values, at the beginning of 5 %, verify if the information given by the mayor to the investor is exact or not.
2. The investor is worried in no way by the possibility that average of the annual spending (expenses) of homes (foyers) is less raised (brought up) than 15 000 $. Appropriate repeat the new useless hypothesis Ho and the piece of news (short story) against hypothesis H1 and make this test at the beginning of 5 % by granting (tuning) the benefit of the doubt to the opinion of the mayor.

3) The investor would hesitate to invest (surround) if the average of the spending (expenses) of homes (foyers) was at the most 13 500 $ instead of 15 000 $, as claims him (it) the mayor. Establish the probability to make the error of first sort (species) and calculate the probability to make the error of second sort (species) in this one-sided test.

4) Let us suppose that the mayor said to the investor that average of the annual spending (expenses) of homes (foyers) was at least 15 000 $. If the sample of size (cutting) n = 15 homes (foyers), we find X = 14 000 $ and = 2 000 $ there and supposing that the distribution (casting) of the spending (expenses) is normal, test the assertion of the mayor at the beginning of the 5 % meaning.

5) Let us suppose that the standard deviation of the population is not known and that the distribution (casting) of the annual spending (expenses) cannot be supposed normal. The taking of a sample of size (cutting) n = 30 gave X = 14 000 $, as previously, and a standard deviation = 2 000 $. Make out a will at the beginning of 5 % of the no hypothesis according to whom (which) the average of the annual spending (expenses) of homes (foyers) is at least $15 000.

6) In a sample chosen at random by size (cutting) n = 10 of the homes (foyers) of the city, we find a standard deviation of s = $1,500. We suppose that the distribution (casting) of the annual spending (expenses) of homes (foyers) is normal. Before pulling (firing) this sample, we claimed that the standard deviation of the population was most \(\sigma =2,000\). By means of the results (profits) obtained in the sample, test this hypothesis at the beginning of 1 % meaning.

7) Let us suppose that the hypothesis no is \({{H}_{0}}:\mu =15,000\) and that the hypothesis alternatives \({{H}_{1}}:\mu =13,500\). We also suppose that the critical value of \(\bar{X}\) is 14,147.28 when the size (cutting) of the sample is of n = 15 and when the standard deviation is 516. 80. Which would be then the power of this test if the value of the average of the population was?

(A) \(\mu =13,000\) ?

(B) \(\mu =14,000\) ?

C) Compare these two answers found there a) and b) and explain the difference if necessary.

8) Let us suppose that the investor wants to test the hypothesis \({{H}_{0}}:\mu \ge 15000\) at the beginning of 5 %. Let us say that he (it) considered the situation as completely risked if average of the annual spending (expenses) of homes (foyers) was lower or equal to 13 500 $. Given that he is hardly afraid of making the error to build a shopping mall in a community environment (middle) where his survival would be problematic, he (it) fixes the probability to make the error of second sort (species) \(\beta =0.01\). If we suppose that the standard deviation of the annual spending(expenses) of homes(foyers) is of \(\sigma =2000\) , determine the size(cutting) of the sample required to conform to the values of specified by the investor if we do not suppose that the population is normally distributed.

PROBLEM 2

During the annual meeting of Rhunéistes (Association of the walkers of the Rhune) which was held in Venka Udako, a representative of the Federation of the hike publishing(editing) Topo-Guides GR10 (one of the roads to climb Rhune) was invited. He (it) was engaged (surrendered) in a small survey (investigation) to know the time (weather) of rise (ascent) of a walker. The results (profits) obtained according to the time (weather) of rise (ascent) (minutes) are recorded (deposited) in the following picture (board):

For example, two rises (ascents) were made in 50 minutes or more but within 60.

1) By what punctual value can we estimate (esteem) the average and the variance of the time (weather) of elaborated?

2) In what interval is situated the average of the time of rise (ascent) with a reliable level of 0, 98?

3) What should have been the size (cutting) of the questioned sample if we had wished an interval of 8 minutes of amplitude and supposing that this new sample has the same characteristics as the precedent?

4) Test the following hypothesis: the time of rise (ascent) follows a normal law at the beginning of 5 %.

5) By supposing the normality of time, test the hypothesis \({{H}_{0}}:{{\sigma }^{2}}=225\) against the hypothesis \({{H}_{a}}:{{\sigma }^{2}}>225\) with a risk of 10 %.

6) By making his survey (investigation), he also asked the walkers if the GR10 was their road preferred to climb Rhune. People asked, 32 answered "yes".

1. Estimate (esteem), with a level of 95 % confidence (trust), the proportion of the walkers preferring the road of the GR10 to climb Rhune.
2. Test the following hypothesis: with a risk of 5 %, more half of the walkers prefer to borrow (take) the GR10.

7) While leading his investigation, he discreetly raised (found) the approximate age of people asked. From the following picture (board), he would like to know if there is a relation between the age of the walker and the preference of the route. The error on the conclusion that he (it) will set has to be 5 %.

8) According to inquiries made by the publisher (editor) of the Guide of the GR10, we know that 60 % of the walkers buy the guide. On 50 people asked by our investigator, what is the probability that more half is in ownership of this guide?

PROBLEM 3

The person in charge of the training(formation) of a big company read an article in a specialist magazine about a new method of learning(apprenticeship) and he wants to experiment him(it) with the technicians. This new method consists of a combination (overall) of sessions of programmed learning, sessions of laboratory and traditional lectures. Up to here, the used method contained only lectures. For his(its) experiment, he(it) trains(forms) 12 couples of technicians, every couple being established(constituted) by two technicians of the same experience(experiment) and of the same skill (competence). One of the technicians of every couple will do the training course according to the traditional method and the other one, according to the new method. Finally, an examination estimating at once (at the same time) the acquired knowledge and the fitness to use (get) it will determine the quality of the learning (apprenticeships).

As the purpose of the person in charge is not to abandon(to give up) at all costs the method of learning(apprenticeship) already ready(already in position), he wants to test her(it) by comparing him(it) with the piece of news(short story).

1. Formulate both hypotheses of the test, and make him (it) at the beginning of 5 %.
2. What is the conclusion which will pull (fire) the person in charge of the result (profit) of this test with regard to both methods of learning (apprenticeship), former (ancient) and the piece of news (short story)?

PROBLEM 4

We want to compare the average life expectancy (cycle) of the tires Goodway with that of tires Firerock. Let us suppose that these life expectancies (cycles) are normally distributed, and that their variances \(\sigma _{1}^{2}\) and \(\sigma _{2}^{2}\) are unknown. We want to verify the equality of the variances. To this end, we pull(fire) independently a random sample of 10 tires Goodrich and a random sample of 16 tires Firerock, and we obtain S1 = 6 000 km and S2 = 4 000 km. From these results (profits) can we assert that there is an equality of the variances at the level \(\alpha =0.02\) ?

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