Subject: Mathematics    / Statistics
Question

In each exercise, you must include the formulas used, with the corresponding substitutions of values, and indicate with absolute clarity the answer to the question asked. Must calculate with adequate mathematical accuracy.

1. A new surgical procedure has been developed which is claimed to be successful 80% of the time. Suppose that eight (8) surgeries are performed in which this new procedure is used and that the results are independent of each other. Determine the probabilities of the following events:

A. That the eight (8) surgeries be successful.

B. That exactly five (5) of those surgeries are successful.

C. That less than three (3) of those surgeries are successful.

2. The time it takes to answer a school entrance exam is a aleatory variable that is assumed to be normal. On average, answering it completely takes people 70 minutes, with a standard deviation of 12 minutes. Determine how much time should be given to those who take the test so that 90% of the applicants can answer it in full.

3. The wages of the workers of a company have an average of $ 38,000 and a standard deviation of $ 2,500. The company took a sample of the salaries of 60 workers.

A. Within what limits would you expect to find that the sample mean falls with a 95% probability?

B. Determine the probability of observing a sample mean greater than $ 38,600.

C. What would you think if you observe a sample mean of $ 36,500? Explain.

4. Samples of 400 units were taken from each of two production lines, X and Z, of a manufacturing plant. The production line X samples had 40 defective units while the Z line samples had 80 defective units.

· Estimate the difference in the fraction of defective units produced by both lines with a confidence coefficient of 90%.

5. You want to determine the sample size needed to estimate the mean of a population with an error not greater than 1.6, with a probability of 95%. Based on previous experience you know that the standard deviation of that population is 12.7.

· Determine the minimum sample size to take.

6. An experiment was conducted to compare the average time in days required to recover from a common cold flu in people who consumed a daily dose of 4 mg of vitamin C versus those who were not given that vitamin supplement. 35 adults were randomly selected in each group and the following results were observed:

Without vitamin supplement

With 4 mg of vitamin C

Size of the sample

35

35

Sample mean

6.9

5.8

Sample standard deviation

2.9

1.2

A. Establish the null and alternative hypotheses for this situation and indicate whether it is a one-tailed or two-tailed test.

B. Carry out the statistic hypothesis test and state with absolute clarity what is your conclusion. Use a value of c. = 0.05

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