**Statistics****II – 2017/2018 **

**Midterm exam **

**Duration: 60+30 minutes**

统计Midterm代考 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. **

**1.(3,5 points) **

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?

**2.(3,5 points) 统计Midterm代考**

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**.

**3. (6,5 points) 统计Midterm代考**

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.

**4.(3,5 points) **

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.

**5.(3,0 points) ****统计Midterm代考**

**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. **

**1.**

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.

**2. ****统计Midterm代考**

Which of the following statements on the Standard Error of the mean is **true**?

□ a) The standard error of the mean is the sampling standard deviation df the sampling mean.

□ b) The standard error of the mean is always normally distributed.

□ c) The standard error of the mean may take values below 0.

□ d) The standard error of the mean is the square of variance.

**3.**

A 95% confidence interval for the mean of a population is computed from a random sample and found to be ]9 – 3; 9 + 3[. Which of the following is a correct statement?

□ a) there is a 95% probability that m is between 6 and 12.

□ b) 95% of values sampled are between 6 and 12.

□ c) If we took many, many additional random samples and from each computed a 95% confidence interval for , approximately 95% of these intervals would contain the true mean, μ .

□ d) all of the above are true

**4. 统计Midterm代考**

A hypothesis test is done in which the alternative hypothesis is that more than 10% of a population is left-handed. The p-value for the test is calculated to be 0.25. Which statement is correct?

□ a) We can conclude that more than 10% of the population is left-handed.

□ b) We can conclude that more than 25% of the population is left-handed.

□ c) We can conclude that exactly 25% of the population is left-handed.

□ d) We cannot conclude that more than 10% of the population is leet-handed.

**5.**

Which of the following statements is **false**?

□ a) An hypothesis test is built on the assumption that H0 is true.

□ b) When a p-value lower than 0,05 is produced by a statistical test, this implies that the possibilities of H1 being false are lower than 5%.

□ c) When a p-value lower than 0,05 is produced by a statistical test, this implies that, if H0 is true, the sample data had less than 5% chance of occurence, *a priori*.

□ d) When a p-value lower than 0,05 is produced by a statistical test, this provides strong evidence for H1 being true.

**6.**

In hypothesis testing, a Type II error occurs when

□ a) The null hypothesis is not rejected when the null hypothesis is true

□ b) The null hypothesis is rejected when the null hypothesis is true

□ c) The null hypothesis is not rejected when the alternative hypothesis is true

□ d) The null hypothesis is rejected when the alternative hypothesis is true

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