1.(3,5 points) After the introduction of a new wage policy, the Human Resources department of So Random, Inc. wants to ···
Note: No questions during the exam! You are allowed to use your statistical tables and calculator only.
After the introduction of a new wage policy, the Human Resources department of So Random, Inc. wants to estimate the proportion of workers that are satisfied with the new salaries. Since So Random, Inc. employs a large number of people, a random sample was obtained from the pool of employees. A 99% confidence interval for the proportion of satisfied workers was calculated, based on this sample of 100 individuals: ]0,3712; 0,6288[
(1,0) a) Find the error margin associated with the interval above.
(1,0) b) How many workers are satisfied in the collected sample?
(1,5) c) What should be the sample size if So Random, Inc. wanted to halve the error margin of the interval, keeping the confidence level constant?
So Random, Inc. also wants to analyze the average junior level salaries of its sales force. One of the items under analysis is the starting salary of its sales workers, which was obtained for 31 individuals, and produced the following results:
(2,0) a) Based on the information above, find a and b, and interpret the result.
(1,5) b) After an additional sampling effort, starting salaries were obtained for an additional 69 workers, totaling 100 valid observations. Curiously, the new sample showed the same mean and variance as the previous one. Does the additional sampling effort yield a different conclusion? How and why? Give a detailed explanation, without performing additional calculations.
A local public entity of a given country states that the average monthly household income in a certain region is greater than 120 m.u. Notwithstanding, some of the municipalities in that region have countered that this household income is overestimated.
The central government, concerned with a possible overestimation, decided to verify the claim by collecting a sample of 36 households in that region. The sum of all collected household incomes amounted to 4320 m.u.
Municipalities and central government agree on one fact: the true dispersion of monthly household income is 6 m.u.
(2,0) a) For α = 0,05, is it plausible to state that the local public entity is correct?
(2,0) b) Based on the decision made in a), what is the probability of making an error if the true average monthly household income is 123 m.u.
(1,5) c) Compute the test power if true average monthly household income is 121,645 m.u.
(1,0) d) Sketch the power function.
Several indicators were calculated based on data obtained from the PISA2002 study. One of this indicators is “Interest for Mathematics”, measured on a scale from 0 to 100. An analysis performed on a subset of the original sample produced the following results:
(1,0) a) What is intended with this analysis? State the null and alternative hypothesis in this context.
(1,0) b) What should be concluded (a=0,05)?
(1,5) c) Do we have reasons to believe that Interest for Mathematics is above the scale’s midpoint? Explain.
This group is made up of 6 multiple choice questions. Each correct answer is worth 0,5 points. Each incorrect answer is worth 0,15 negative points. You need not justify your answers. 12 minutes should be sufficient time to complete this group.
Which of the following statements on Confidence Intervals is false?
□ a) A confidence interval is an interval between two values, calculated from a sample, that will likely contain the true value of the parameter to be estimated.
□b) An approximate f ormula for a Confidence Interval for a location parameter is of the type Estimate ± Error margin.
□c) If a confidence interval for a sample proportion is ]0,2 ; 0,4[, then the true proportion of successes will be between 20 and 40%.
□ d) A random 99% confidence interval has greater probability of producing specific confidence intervals that contain the true parameter than a random 95% confidence interval.