Exam Practice II 1) Ashley calculates a 95% confidence interval for the proportion of students at A&M who are foreign. Her confidence interval was (0.102, 0.435). Which of the following statements about Ashley's Confidence Interval are true? (Check all that apply) _______ A) There is a 95% probability that the true proportion is between 0.102 and 0.435 _______ B) There is a 95% probability that Ashley's interval captures the true proportion _______ C) Ashley's results are accurate with 95% probability _______ D) If Ashley had made 100 confidence intervals 95 of them would capture the true proportion _______ E) Between 10.2% and 43.5% of the students at Texas A&M are foreign _______ F) Ashley is 95% confident that her interval captures the true proportion _______ G) The chance of meeting a foreign student is between 0.102 and 0.435 _______ H) If Ashley made infinitely many intervals, then 95% of them would capture the true proportion _______ I) There is a 95% chance the next interval will contain the true proportion 2) The table below lists some of the animals at the Houston Zoo. If an animal escapes then zoo officials have the following probabilities of which animal it was. Animal P(Escaped ) Elephant 0.01 Zebra 0.10 Cobra 0.19 Tarantula 0.35 Eagle 0.08 Falcon Censored
If we know that one of animals above escaped, what is the probability it is a mammal given that we know it is not a bird? 3) The weight of a Humvee is N (1428, 322) in pounds, and the local bridge can only support 5760 pounds. If four randomly chosen Humvees are scheduled to cross the bridge at the same time, what is the probability that the bridge will collapse? (Assume it will collapse if more than 5760 pounds are on it, and it will not collapse if less than 5760 pounds are on it) 4) Adam works at a job where people come in with work. The amount of money Adam will be paid for each job, and the probability of that job are given in the following table: Job payment Probability $10 0.85 $100 0.08 $400 0.06 $1,500 0.01
When the next person walks into Adam's Place, how much money can Adam expect to get from them? 5) Let p42 ~ N (0.7, 0.0712). What is P (p42 < 0.5)? A. 0.5 B. 0 C. 0.9976 D. 0.0024 E. 0.2
For questions 6 through 8, refer to the following example: The number of children in a family is distributed as follows. X 0 1 2 P(X) .35 .25 .22 6) What is the mean and median of X? A. Mean = 1.25 Median = 1 B. Mean = 2.89 Median = 1 C. Mean = 1.25 Median = 2 D. Mean = 2.89 Median = 2 E. Not enough information 7) What is P (1 X < 3)? A. 0.22 B. 0.38 C. 0.47 D. 0.63 E. None of the above 8) What is P (Three random families have at least two children)? A. 0.0058 B. 0.0640 C. 0.4000 D. 1.2000 E. None of the above
3 .16
4 .02
9) The level of nitrogen oxides (NOX) in the exhaust of a particular car model varies. A company has 200 cars of this model in its fleet. We're interested in obtaining a confidence interval for the mean NOX emission level. The data doesn't appear to come from a normal distribution but it's not too far from normality and no outliers are present. Assuming a sample size of 200 is large enough, why can we use the confidence interval ordinarily reserved for normal data? A. Statistics is a fairly flexible science. You can get away with a lot more than you think. B. The Central Limit Theorem says it's justified. C. It's justified since we're using a typical level of confidence, 95%. D. The confidence interval for the normal distribution is used when the data is numerical, especially when the variable is continuous such as measuring the diameter of a piston. E. We can't use the normal confidence interval here. 10) Suppose a sample proportion based on samples of size 50, p50 ~ N (0.75, 0.06122). What is the smallest proportion in the TOP 5%? A. 0.753 B. 0.649 C. 0.872 D. 0.870 E. 0.851
11) Which of the following theorems and/or rules is NOT stated correctly? A. If the population distribution is normal then the distribution of the sample mean is approximately normal. B. X-Y = X - Y If X and Y are independent. C. X-Y = X - Y D. All of the above are not stately correctly. E. Only two of the above are not stately correctly. Use the following problem to answer the next two questions. Two machines are used to fill 50 pounds bags of dog food. Sample information for these two machines is given below Machine 1 Machine 2 n: 81 64 Sample means (lbs.): 51 48 Variance of sample means: 16 12 12). Suppose that machine 1 is denoted by the r.v. X81 and machine 2 by W64. What is the standard deviation of X81 - 2W64? A. 8 B. 32 C. 64 D. 40 E. 832 13) Suppose that we want to test the hypothesis H0: X =W vs. Ha: X W with an = 0.05. We found that the p-value = 0.003. Which of the following statements is TRUE? A. Machine 1 has a mean significantly larger than machine 2. B. There is not enough evidence to reject the hypothesis that machine 1 has a larger mean than machine 2. C. We fail to reject the hypothesis that machine 1 has smaller mean than machine 2. D. None of the above is TRUE. E. Two of the above are TRUE. 14) Which of the following best describes the relationship between a (1 - )100% confidence interval for and a 2-sided test of hypotheses = for some value, 0? A. There is no relationship between confidence intervals and hypothesis tests. B. If the hypothesized value, 0, falls within the confidence interval, we would reject the null. C. If the hypothesized value, 0, falls within the confidence interval, we would fail to reject the null. D. If the confidence interval contains 0, we would reject the null. E. If the confidence interval contains 0, we would fail to reject the null. 15) Determine whether each of the following statements regarding increasing the value of the power is true or false. A. The power will increase when we increase the significance level . B. The power will increase when we increase the sample size. C. The power will increase when we consider an alternative value that is farther from the null value (the value of the parameter of interest under the null hypothesis).
D. The power will increase when we increase the standard deviation. 16) 90% CI: (2.3265, 16.4735) 95% CI: (0.972, 17.828) 99% CI: (-1.694, 20.494) Using the information above, what is the correct range of the p-value if I wanted to test H0: = 0 vs. Ha: 6= 0? A. p-value > 0.10 B. 0.10 > p-value > 0.05 C. 0.05 > p-value > 0.01 D. p-value< 0.01 E. You need a test statistic value to determine the p-value 17) A small company consists of 25 employees. As a service to the employees, the company arranges for each of the employees to have a complete physical exam for free. Among other things, the weight of each employee is measured. The mean weight is found to be 165 pounds. The standard deviation of the weight measurements is 20 pounds. It is believed that a mean weight of 160 pounds would be expected for this group. To see if there is evidence that the mean weight of the population of all employees of the company is significantly larger than 160, the hypotheses H0: = 160 versus Ha: > 160 are tested. You obtain a P-value of about 0.106. Determine whether each of the following statements is true or false. A. At the 5% significance level, you have proved that H0 is true. B. You have failed to obtain any evidence for Ha. C. At the 5% significance level, you have failed to prove that H0 is true and a larger sample size will be needed to do so. D. The calculation was in error; a difference of 5 pounds is very significant. 18) Answer each of the following questions with yes, no, or can't tell. A. Does the sample mean lie in the 95% confidence interval? B. Does the population mean lie in the 95% confidence interval? C. If we were to use a 92% confidence level, would the confidence interval from the same data produce an interval wider than the 95% confidence interval? D. With a smaller sample size, all other things being the same, would the 95% confidence interval be wider? 19) A sprinkler system is being installed in a newly renovated building on campus. The average activation time is supposed to be at most 20 seconds. A series of 12 fire alarm/sprinkler system tests result in an average activation time of 21.5 seconds. Do these data indicate that the design specifications have not been met? Assume that activation times for this system are Normally distributed with = 3 seconds. Set up and perform the test to show if the design specifications have been met. Show all your work including what the hypotheses are. 20) In the game of Statistical Luck you roll two dice, and if the sum is 7 or 11 you lose. What is the probability of winning at this game if you only play once?
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F, H P (Mammal | not bird) = P (Mammal and not bird)/P (not bird) =P (Elephant or Zebra)/P (Elephant or Zebra or Cobra or Tarantula) = [.01+.10] / [.01+.10+.19+.35] =.11/.65 = 11/65 P (bridge collapsing) = P (h1+h2+h3+h4>5760) h1 +h2 +h3 +h4 ~ N (1428+1428+1428+1428, 322 + 322 + 322 + 322) = N (5712, 642) z* = (5760 5712) / 64 = .75 P(Z > z*) = 1- P(Z < z*) = 1 - .7734 = 0.2266 E ($) = 10*.85 + 100*.08 + 400*.06 + 1500*.01 = $55.50 D P (p42 < 0.5) = P(Z < (0.5-0.7)/ 0.0712) = P(Z < -2.82) = 0.0024 A C B B ^ ^ E Find p such that P (Z > ( p -.75)/.0612) = 0.05 B A Since they are independent: Var(X81-2W64)=Var(X81)+Var(-2W64)=Var(X81)+(-2)2Var(W64)=Var(X81)+4Var(W64) A C A) True, B) True, C) True, D) False C A) False, B) False, C) False, D) False A) Yes, B) Can't tell, C) No, D) Yes
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13) 14) 15) 16) 17) 18) 19) Step 1 & 2: Ho:20 vs. Ha: >0 Step 3: = 0.05 Step 4a: activation times are normally distributed and simple random sample (SRS) Step 4: z* = (21.5-20)/(3/12) = 1.732 Step 5: p-value = P(Z > z*) = P(Z > 1.732) = 1- P(Z < 1.732) = 1- 0.9584 = 0.0416 Step 6 We reject the null hypothesis since p-value < 0.05 = Step 7 We conclude that there is significant evidence at = 0.05 significance level that the sprinkler system needs to be adjusted or is not up to design specifications. 20)
P(winning) = 1 P(losing) then P(losing)= P(rolling 7 OR rolling 11) = P(rolling 7) + P(rolling 11) = P(roll (1,6) or (6,1) or (2,5) or (5,2) or (3,4) or (4,3)) + P(roll (6,5) or (5,6)) =[ 1/6*1/6 + 1/6*1/6 + 1/6*1/6 + 1/6*1/6 + 1/6*1/6 + 1/6*1/6 ] + [1/6*1/6 + 1/6*1/6 ] = 6* 1/36 + 2*1/36 =8/36 =2/9 therefore P(winning) = 1- 2/9 = 7/9