# Statistics and Hypothesis Testing

1. A producer of orange juice buys all of his oranges from an orange Farmer in Las Marías. The quantity of juice squeezed from each of these oranges is approximately normal with an average of 4.70 ounces and a standard deviation of .40 ounces.

a) What is the probability that a randomly selected orange weighs between 4.50 and 4.60 ounces?

b) If a sample of 50 oranges is selected, what is the probability that the average of the sample is less than 4.60 ounces?

2. The manager of a local bank wants to estimate the average quantity that their depositors had on their savings accounts. A random sample of 20 clients indicated an average of $4,750 and a standard deviation of $1,200.

a) Establish a trust estimate interval of the average quantity of savings accounts with a 95% level of trust.

b) What assumption did you have to make to solve the first part of the problem?

3. A big shipment of air filters arrived in Western Auto. There will be a sample selected of the air filters to estimate the defective portion. The experience indicates that the proportion of defects is approximately 0.10. How big does the random sample must be to estimate the real proportion of defective air filters with an error of or less than .07 and with a .99% level of trust.

4. A market research indicated that 37% of all clients of a big pizza chain are university students.

a) What is the probability that a random sample of 625 clients of the chain have 225 or more people that are university students?

5. A survey took place amongst the members of the Book of the Month Club, to determine if they pass more time watching television than reading. Suppose that in a sample of 10 surveyed people, the weekly hours that they dedicate to watch television and the weekly hours dedicated to reading were obtained. (These times are all shown on the table below). With a significance level of 0.05, can we conclude that the members of the Book of the Month Club spend more time, in average, watching television instead of reading?

Surveyed Television Reading

1 10 6

2 14 16

3 16 8

4 18 10

5 15 10

6 14 8

7 10 14

8 12 14

9 4 7

10 8 8

6. A performance study of automobiles had the goal of proving the following hypothesis.

Hipothesis Conclusion

Ho: μ ≥ 25 m/gal The manufacturer's statement is supported

Ha: µ < 25 m/gal The manufacturer's statement is rejected;

The average performance is less than the stated one.

For σ = 3 and a level of significance of 0.02, what size of the sample is recommended, if the researcher wants to have 80% of the possibilities of detecting that μ is less than 25 miles per gallon when in reality is of 24?

7. A fast food restaurant plans a special offer which allows the clients to buy different special design glasses with known caricature characters. If more than 15% of the clients buy these glasses, they will be implemented in promotion. In a good preliminary test various establishments, 88 of 500 clients bought them. Should the special promotion be implemented? Realize a hypothesis test that backs up your decision. Use the level of significance of 0.01. Which is your recommendation? (10 points) Also calculate the value p.

8. The Evaluation Education Service carried out a study to research the differences between male and female students on the scholar aptitude test (PAE). The study identified a random sample of 562 female students and 852 male students which reached the same high grade on the mathematics section. This is, it was considered that the students, both male and female, had high and similar aptitudes in mathematics. The classification of oral expression of the PAE, for both samples, can be summarized as the following:

Female students Male students

x bar = 547 x bar = 525

s1 = 83 s2 = 78

The data, does this data support the conclusion that, given a population of female students and a population of male students with high mathematical aptitudes, the female students have a very high oral expression aptitude? Do the test with the significance level of 0.02. What is your conclusion.

9. The accounting department analyses the variance of the weekly unitary costs that two production departments report. A sample of 16 costs reports, of each of these departments, indicates variances of cost of 2.3 and 5.4, respectively. Is this sample enough to conclude that both production departments are different in relation to the variance of the unitary cost? Use the level of significance of 0.10.

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