STATUse the information below to answer Questions 1 through 4..docx
- 1. STAT Use the information below to answer Questions 1 through 4. Given a sample size of 36, with sample mean 670.3 and sample standard deviation 114.9, we perform the following hypothesis test. Null Hypothesis Alternative Hypothesis 1. What is the test statistic? 2. At a 10% significance level (90% confidence level), what is the critical value in this test? Do we reject the null hypothesis? 3. What are the border values between acceptance and rejection of this hypothesis? 4. What is the power of this test if the assumed true mean were 710 instead of 700?. Questions 5 through 8 involve rolling of dice. 5. Given a fair, six-sided die, what is the probability of rolling the die twice and getting a “1” each time? 6. What is the probability of getting a “1” on the second roll when you get a “1” on the first roll? 7. The House managed to load the die in such a way that the faces “2” and “4” show up twice as frequently as all other faces. Meanwhile, all the other faces still show up with equal frequency. What is the probability of getting a “1” when rolling this loaded die? 8. Write the probability distribution for this loaded die, showing each outcome and its probability. Also plot a histogram to show the probability distribution.
- 2. Use the data in the table to answer Questions 9 through 11. x 3 1 4 4 5 y 1 -2 3 5 9 9. Determine SSxx, SSxy, and SSyy. 10. Find the equation of the regression line. What is the predicted value when 11. Is the correlation significant at 1% significance level (99% confidence level)? Why or why not? Use the data below to answer Questions 12 through 14. A group of students from three universities were asked to pick their favorite college sport to attend of their choice: The results, in number of students, are listed as follows: Football Basketball Soccer Maryland 60 70 20 Duke
- 3. 10 75 15 UCLA 35 65 25 Supposed a student is randomly selected from the group mentioned above. 12. What is the probability that the student is from UCLA or chooses football? 13. What is the probability that the student is from Duke, given that the student chooses basketball? 14. What is the probability that the student is from Maryland and chooses soccer? Use the information below to answer Questions 15 and 17. There are 3600 apples in a shipment. The weight of the apples in this shipment is normally distributed. It is found that it a mean weight of 14 ounces with a standard deviation of 2.5 ounces. 15. How many of apples have weights between 13 ounces and 15 ounces? 16. What is the probability that a randomly selected mango weighs less than 12.5 ounces? 17. A quality inspector randomly selected 100 apples from the shipment. a. What is the probability that the 100 randomly selected apples have a mean weight less than 12.5 ounces? b. Do you come up with the same result in Question 16? Why or why not? 18. A pharmaceutical company has developed a screening test for a rare disease that afflicted 2% of the population.
- 4. Unfortunately, the reliability of this test is only 80%, which means that 20% of the tested will get a false positive. If a subject is tested positive based on this test, what is the probability that he has the disease? Use the information below to answer Questions 19 and 20. Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way shown in the following table. Leading Digit 1 2 3 4 5 6 7 8 9 Distribution of Leading Digit (%) 30.1 17.6 12.5 9.7 7.9 6.7 5.8 5.1 4.6 The owner of a small business would like to audit its account payable over the past year because of a suspicion of fraudulent activities. He suspects that one of his managers is issuing checks to non-existing vendors in order to pocket the money. There have been 790 checks written out to vendors by this manager. The leading digits of these checks are listed as
- 5. follow: Leading Digits 50 15 12 74 426 170 11 23 9 19. Suppose you are hired as a forensic accountant by the owner of this small business, what statistical test would you employ to determine if there is fraud committed in the issuing of checks? What is the test statistic in this case? 20. What is the critical value for this test at the 5% significance level (95% confidence level)? Do the data provide sufficient evidence to conclude that there is fraud committed? Hypothesis Test versus Confidence Interval – Questions 21 through 22 Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. 21. Perform a hypothesis test of p1 = p2 with a 5% significance level (95% confidence level). 22. Obtain a 95% confidence interval estimate of p1 - p2. Do you come up with the same conclusion for Question 21? Why
- 6. or why not? 95% confidence interval estimate of p1 - p2is, Hardness of Gem – Questions 23 and 24 Listedbelow are measured hardness indices from three different collections of gemstones. Collection Hardness Indices A 9.3 9.3 9.3 8.6 8.7 9.3 9.3 -- --- --- --- --- --- 9.91 0.10 B 8.7 7.7 7.7 8.7 8.2 9.0
- 7. 7.4 7.0 --- --- --- --- --- 8.03 0.60 C 7.2 7.9 6.8 7.4 6.5 6.6 6.7 6.5 6.5 7.1 6.7 5.5 7.3 6.82 0.34 You are also given that . 23. What is the test statistic? 24. Use a 5% significance level (95% confidence level) to test the claim that the different collections have the same mean hardness. 25. A couple has 3 daughters. The wife is expecting another baby.
- 8. a. What is the probability that the new baby is a girl again? b. Suppose the new baby turns out to be a girl. What is the probability that a family with 4 children that are all girls? Use the data below to answer Questions 26 and 27. This is a summary of the midterm scores for two sections of STAT 230. The midterm questions and the grading criteria are different in these two sections. Section A Student Score Section B Student Score A 70 H 15 B 42 I 57 C
- 9. 53 J 48 D 61 K 90 E 22 L 85 F 85 M 73 G 59 N 49 ----- ------ O 39 26. What are the mean and standard deviation of the scores in
- 10. Section A? 27. We notice that Student F in Section A and Student L in Section B have the same numerical score. a. How do they stand relative to their own classes? b. Which student performed better? Explain your answer. 28. Composite sampling is a way to reduce laboratory testing costs. A public health department is testing for possible fecal contamination in public swimming pools. In this case, water samples from 5 public swimming pools are combined for one single test, and further testing is performed only if the combined sample shows fecal contamination. Based on past experience, there is a 3% chance of finding fecal contamination in a public swimming area. What is the probability that a combined sample from 5 public swimming pools has fecal contamination? 29. Peter, Paul, Mary, Andrew, John, and Martha are members of the pastoral council at a local church. They are to be seated at one side of a long conference table in a pastoral council meeting. a. How many possible ways can these 6 council members can be seated? b. How many possible sitting arrangements are there if only gender is considered in the process? 30. A banquet organizer knows that not all 600 invited guests will show up at an event. Based on past experience, only 80% of the invited guest for this special event will come. When expensive dishes are served, it would be prudent not to order
- 11. the full 600 plates because a good number of them will be wasted. On the other hand, the banquet organizer will try to stay within 7% probability that he would not have to rush to prepare the expensive dishes. How many of these expensive dishes would you order if you were organizing this banquet? i x 2 i s 7.99 x = 0 :700 H m = :700 a H m ¹ 4? x =