Kamil Ələsgərov Ehtimal nəzəriyyəsi və riyazi statistika



Yüklə 27,19 Kb.
tarix07.11.2017
ölçüsü27,19 Kb.
#8848

Kamil Ələsgərov

Ehtimal nəzəriyyəsi və riyazi statistika

  1. Define and describe population, sample, parameter and statistic.

  2. Open the following expression:

  3. Define the scales of measurement for all of the following variables: eye color, number of children per family, GDP, electricity consumption, water pressure.

  4. Give an example of 5 continuous and discrete variables

  5. Give a proper definition of a random and deterministic variables with some examples.

  6. You have 10 observations of some variable. Build a summary (frequency) table: 1, 1, 3, 1, 2, 5, 5, 4, 2, 5

  7. Describe the methods used to summarize continuous numerical data.

  8. Describe the methods used to summarize discrete numerical data.

  9. Which graphical forms are used for visualizing the qualitative data?

  10. If the data is described in histogram what is the most likely type of the variable

  11. P(A) = 0.3, P(B) = 0.4. If the two event are statistically independent what is the value for joint probability.

  12. P(A) = 0.2, P(B) = 0.6. If the two events are statistically independent what is the value of P(A or B)?

  13. P(A) = 0.1, P(B) = 0.5. If the two events are mutually exclusive what is the value of P(A or B)?

  14. The probability distribution of a variable (collectively exhausting outcomes) is given as below. Find the value of b.

    X

    P(X)

    1

    b

    3

    2b

    5

    3b

    7

    4b

  15. Probability of graduating from university in the UK is 0.6. The joint probability of being a foreign student and graduating is 0.2. If the person graduated, what is the probability that s/he is a foreign student?

  16. The probabilities for collectively exhaustive outcomes are given as below. Find the ratio of cumulative probability of 3 to the cumulative probability of 1.

X

P(X)

0

0.5

1

0.4

2

0.07

3

0.03



  1. P(A) = 0.5 and P(B) = 0.2. If the two are statistically independent, find the probability of P(A or B).

  2. Skewness of the variable is 3 whereas the kurtosis is 5. Comment on symmetricity and tail probability.

  3. Calculate mean and variance for the following sample data: 2, 3, 4, 3, 3, 1, 1, 3, 4, 0

  4. The means for two variables, X and Y, are 5 and 6 respectively. Find the mean the following combination: 5X + Y

  5. The portfolio variance is 0.04. There are 2 assets in the portfolio with equal weights. The covariance of the assets is 0.03. If the first asset has the variance of 0.04, find the variance of the second.

  6. E(X) = 5 and var(X) = 3. Find E(X2)

  7. The Bernoulli variable has p = 0.5. What is the standard deviation of the variable?

  8. The Bernoulli variable has p = 0.5. If you have a Binomial variable which is 20 times repeated Bernoulli experiment, what are the mean and variance of that variable?

  9. Number of visitors every hour at some supermarket follows Poisson distribution with lambda parameter equal to 10. What are mean and standard deviation of the variable?

  10. On average 10% of the mobile phones tend to be defect. If you bought 10 mobile phones what is the probability that 2 of them will be defect?

  11. Assume there are 3 cars. The probabilities of cars having a defect are 0.1, 0.2 and 0.3 respectively. Build probability distribution of defects for those three cars.

  12. On average there are 5 car accidents a day. What is the probability that next day there will be less that 2 car accidents?

  13. There are 7 balls in the box and 3 of them are green. If I take 5 of them without returning back what is the probability that none of them will be green?

  14. Assume you have 4 friends (Nizami, Baxtiyar, Tofig and Zaur) and a car. You have an agreement that, every time you go somewhere in your car they should sit in a different order. What is the (joint) probability that Zaur will be in the front and Tofiq will be in the middle back seat?

  15. The level of water in the electric station dam (reservoir) follows normal distribution with the average of 30 m and variance of 49. If the level drops below 5 m the electric generators will not work. What is the probability of that?

  16. X ~ N(3, 16). Cumulative probability of certain X is 20%. Find that value of X

  17. X follows normal distribution and has population mean of 10. P(X > 15) is 2.5%. Find the variance of the variable.

  18. X follows normal distribution. P(X > 5) = 30%, P(X < 2) = 10%. Find mean and variance of X.

  19. Y ~ N(5, 8) and Z ~ N(0, 1). Find the distribution of Y + 10Z

  20. Number of traffic accidents in a portfolio of insured car follows binomial distribution. If you have 4000 cars in the portfolio and general probability of accident is 0.2, what is the probability that 900 or more cars will get into accident?

  21. The temperature has uniform distribution and it changes between 10 and 50 degrees Celcius. Find out mean and variance of the temperature.

  22. Variable has uniform distribution with mean and variance equal to 5 and 4 respectively. If you take a sample of 20 observations from the population, what would be the distribution of the sample means?

  23. Population proportion of smokers is 0.25. What is the sampling distribution of sample proportion if the sample size is 50?

  24. 95% confidence interval for population mean using a sample of 400 observations is (12;14). Assuming you do not know the population variance what is the sample mean?

  25. 90% confidence interval for population mean using a sample of 400 observations is (13;16). Assuming you do not know the population variance what is the sample standard deviation?

  26. 95% confidence interval for population mean is equal to (10; 12). If population variance was used for the estimation and it is equal to 15, what is the number of observations in the sample?

  27. Population proportion of blue-eyed people is 10%. If in the sample you had 50 people what would be the sampling distribution of sample proportions?

  28. Assume you are looking into sampling distribution of sample mean where you do not have population variance, but you know population mean. If you have 30 observations, which distributions you would use to check validity of the sample and what would be your critical value with 95% confidence level?

  29. Apply CLT and find distribution of Xave + Yave where Xave is the average of 40 observations from binomial with n = 100 and p = 0.2, and Yave is the average of 50 observations from normal with mean of 5 and variance of 25.

  30. Assume you know the population mean of daily forex returns which is 0. You do not know the population variance. You have taken a sample of 49 observations from the population and sample mean and variance are 0.005 (to make it clear - 0.5%) and 0.0004 respectively. Is this sample representative based on the mean value?

  31. The sample has a skewness of 0.2 and kurtosis of 3.5. Assuming you had 20 observations in the sample, can we claim that this sample comes from a population with normal distribution?

  32. Sample variance of 15 observations was 30. If the population variance is 25, can we claim that this sample is a good approximate for population when it comes to variance?

  33. We took a sample of tap water from 400 places in Baku and the average amount of heavy metals in the water was 220 ppm (particles per million). The sample standard deviation was 50 ppm. Find p-value associated with the hypothesis that the population average amount of heavy metals is equal to 215 ppm.

  34. Assume you are comparing math test results of Chinese and Americans. For them to be equal, the population means and variances should be equal. Sample variance of 25 Chinese students’ math scores was 42 and sample variance of 15 American students’ math scores was 52. Can we claim that the population variances are equal with 90% confidence?

  35. Population variance and sample variance for the number of daily equipment failures in some large factory are 25 and 36 respectively. If the sample had 40 observations and sample mean was 70, find 90% interval for population mean.

  36. Sample proportion of smokers was 15% in a sample of 200 people. Find 95% confidence interval for population proportion of smokers.

  37. The number of items sold in auctions in New-York every day follows binomial distribution with variance of 5. If you observed the sales for November and December and the daily average number of sales was 3.5, build 95% confidence interval for the population mean.

  38. Sample variance of 8 observations is 9. Find 95% confidence interval for population variance.

  39. The population variance of the Cola amount in the bottle is 0.0016 liter-squared. In a sample of 36 observations you found that the sample average was 0.99 liter and sample variance was 0.0009 liter-squared. Find 99% confidence interval for the mean of Cola amount.

  40. Test the hypothesis that average consumer eats less than 7kg of meat every months given the average and standard deviation of 50 people in April were 3kg and 4kg respectively using 90% confidence level.

  41. For the last 50 days the number of defects in a production line of cars has had the sample average of 15. Assume the population variance is 25. Find out p-value associated with the hypothesis that population mean number of defects is below 14.

  42. The number of workplace incidents for the last 60 months has the average of 2 and variance of 2.25. Test the hypothesis that on average the place has more than 3 incidents a month.

  43. The daily average number of packages sent by the post office was the 2000 in April. Assume population variance of 100. Find the p-value for the hypothesis that post office sends 2010 packages a day.

  44. Branch manager claims that the annual percentage of fraudulent loans is below 5%. Assuming that for the last 30 years under the same manager’s rule the percentage has been 7%, test the manager’s claim.

  45. For the last 40 days on average 1 million people used Baku Metro every day. The standard deviation was 200 000. Using 99% confidence level, test the hypothesis that the population average of Metro users is 1.1 million.

  46. Sample standard deviation of TOEFL test scores for 22 Azerbaijani students was 8. Test the hypothesis that population variance above 70.

  47. You are interested in the impact of new training course on English test scores of students. You train 8 students. The difference of scores before and after the training are 2, 3, 2, 4, 2, 1, 3, 3. Did the training work with 95% confidence level?

  48. The population variances of math test scores are 25 and 36 for women and men respectively. If the average of test scores were 78 for 40 women and 71 for 50 men, can we claim that the population averages are equal?

  49. Assume you are trying to compare the population means of two groups. You statistically test and see that the population variances are equal. In the first sample you have 30 observations and sample variance is 20. In the second sample you have 40 observations and sample variance is 25. Calculate pooled variance.

  50. Pooled variance for two samples is 40. You have 20 observations in the first and 15 observations in the second sample. If the sample averages are 90 and 94 respectively, can we claim that the population means are equal with 5% significance?

  51. You are trying to prove that ability to jump higher is genetic (sorry, could not think of something useful). Assume you take 5 twins which were raised in different places and measure how high (in terms of meters) they can jump. The results are shown in the table below. Is the hypothesis in the first sentence valid with 90% confidence?

1st sibling

2nd sibling

1.5

2

1.1

1.3

1.2

1.8

1

1.6

2

2.1



  1. Usually the people in Azerbaijan think that if you have the foreign education you get higher salaries. We took two samples: 1) a sample of 30 locally educated people 2) a sample of 20 foreign educated people. The first sample has average salary of 1000 AZN and standard deviation of 200 AZN. The second sample has the average salary of 1500 AZN and standard deviation of 500 AZN. If we assume that the population variances for those groups are equal, is there significant salary difference with 90% confidence level?

  2. You want to apply pooled-variance t-test for comparison of two sample means. The first sample has 10 observations, the average of 10 and standard deviation of 5. The second sample has 15 observations, the average of 12 and the standard deviation of 6. Is it right to apply pooled-variance test here?

  3. There are two groups of students: 1) state university students 2) private university students. Assume the population variance for their GMAT scores is not the same. You take a sample of 100 students from the first and 225 students from the second group. The sample means and standard deviations are: for group (1) 500 and 100, for group (2) 550 and 45. Are their results statistically different?

  4. The regression analysis yields the following result: Y = 5 + 3X + e. What is the value of Y when X is 4?

  5. The regression and standard errors of the coefficients are estimated as follows. Test the hypothesis that slope coefficient is equal to 0 (assume critical equal to 2).





  1. The regression and standard errors of the coefficients are estimated as follows. Test the hypothesis that intercept is equal to 0.





  1. Given the following regression, explain the meaning of intercept: Exchange rate = 1.3 – 0.007*Trade_balance + e

  2. Given the following regression, explain the meaning of slope coefficient (Assume the trade balance is measured in billions of dollars): Exchange rate = 1.4 – 0.02*Trade_balance + e

Yüklə 27,19 Kb.

Dostları ilə paylaş:




Verilənlər bazası müəlliflik hüququ ilə müdafiə olunur ©genderi.org 2024
rəhbərliyinə müraciət

    Ana səhifə