Practice_Problems_forFinalF23

Practice Problems for The Final Exam MATH 161A Fall 2023 Professor Gottlieb 1 Multiple Choice Directions: Clearly circle the best answer 1) Two events A and B are mutually exclusive if A) P ( A ∪ B ) = P ( A ) × P ( B ) B) P ( A ∪ B ) = P ( A ) + P ( B ) C) P ( A ∩ B ) = P ( A ) × P ( B ) D) P ( A | B ) = P ( A ) 2) The sample mean is an example of a A) Parameter B) Statistic C) Experimental Unit D) Population 3) Suppose you have a simple random sample X 1 , X 2 , · · · , X n from an exponential distribution with λ = 3. Which of the following statements is true about X ? A) The standard error of X = 1 3 √ n B) E ( X ) = 1 3 n C) X = 1 3 D) None of the above 4) Suppose P( A )=0.6 and P( B )=0.3 and that events A and B are independent. Then P ( A ∪ B ) is A) 0.90 B) 0.72 C) 0.18 D) Not enough information 5) How many ways are there for me to arrange 4 of my 7 Harry Potter books on a shelf? A) ( 7 7 ) B) P 7 4 C) P 4 7 D) ( 7 4 ) 6) Of 10 students in a class, 4 will be selected to serve on a committee. How many committees could be formed? A) ( 4 10 ) B) P 10 4 C) ( 10 4 ) D) 10!

7) Suppose that the number of customers entering a bike shop per hour can be modeled as a Pois- son variable. The average number of customers per hour is 5. What is the (rounded) probability that exactly 2 customers enter the bike shop in an hour? A) 0.084 B) 0.360 C) 0.034 D) 0.140 8) Suppose that a 95% confidence interval for a population mean μ is computed for a sample of size n = 100. Then we can conclude A) That the confidence interval contains μ with probability 0.95. B) That 95% of the observations in the sample are contained in the confidence interval. C) That for 95% of samples of the same size the population mean is contained in the confidence interval. D) That for 95% of samples of the same size the sample mean is contained in the confidence interval. 9) Let X be a continuous random variable with probability density function f ( x ) and cumulative distribution function F ( x ). Then for any two numbers a and b with a < b which of the following statements is generally true? A) P ( a ≤ X ≤ b ) = F ( a ) - F ( b ) . B) F ( X ) = ( x - a ) / ( b - a ) . C) P ( X > a ) = 1 - F ( a ) . D) F ( x ) = R ∞ -∞ xf ( x ) dx 10) After constructing a confidence interval estimate for a population mean, you believe that the interval is useless because it is too wide. In order to correct this problem, you need to: A) increase the population size. B) increase the sample mean. C) increase the confidence coefficient. D) increase the sample size. 11) Which of the following p -values would lead us to reject the null hypothesis if the level of sig- nificance is α = 0 . 05 A) 0.025. B) 0.06. C) 0.10. D) 0.20. 12) Which of the following exemplifies a Type I error of incorrectly rejecting a true null hypothe- sis A) Interrupting the production process in order to adjust machinery that needs no adjusting. B) Switching to a new supplier of raw materials although the performance of the old was satisfactory. C) Condemning a firm for ignoring polluting standards even though they have done no such thing. D) All of the above.

19) A 99% confidence interval estimate for a population mean μ is determined to be (85.58, 96.62). If the confidence level is reduced to 90%, the confidence interval for μ A) will become wider. B) remains the same. C) becomes narrower. D) None of the above. 20) Which of the following is not a part of the formula for constructing a confidence interval esti- mate of the population proportion? A) A point estimate of the population proportion. B) The standard error of the point estimator. C) The confidence coefficient. D) The value of the population proportion. 2 Short Answer 1. The following probability table is a breakdown on age and gender of soap opera stars: Consider the following three events: Age (years) Gender < 35 35 - 44 45 - 54 55 - 64 > 64 Male 0.20 0.15 0.10 0.08 0.03 Female 0.26 0.12 0.03 0.02 0.01 A : One randomly selected soap opera star is female. B : One randomly selected soap opera star is 35-44 years old. C : One randomly selected soap opera star is less than 35 years old. (a) Find P ( A ), P ( B ), and P ( C ) (b) Find P ( A ∩ B ) (c) Find P ( A ∪ B ) (d) Find P ( A | C ) (e) Are A and C independent? 2. Say I have a wallet that contains either a $2 bill or a $20 bill (with equal probability), but I don’t know which one. I add a $2 bill. Later, I reach into my wallet and randomly remove a bill (without looking). Its a $2 bill. There is one bill remaining in the wallet. What are the chances that it is $2 bill?

3. The unemployment rate in a city is 12.0%. A random sample of five people from the labor force was drawn. Let X =number of unemployed people in the sample. (a) What is the probability that the sample contains exactly 2 unemployed people? (b) What is the probability that the sample contains at least one unemployed person? (c) Find the mean of X (d) Find the standard deviation of X 4. Suppose a random variable X has cumulative distribution function (CDF) F ( x ) =    1 - 1 x 5 x ≥ 1 0 x < 1 (a) Find f ( x ), the probability density function (PDF). (b) Verify that the PDF you found in (a) is a valid probability density function. (c) Find E ( X 2 ) . 5. Assume that the amount of time T required to process an order at the drive-through window of a fast-food restaurant is an exponential random variable with a mean of 2.5 minutes. (a) What is the probability that it takes more than 4 minutes to process an order? (b) Ten cars arrive at the drive-through. What is the expected time needed to process all ten orders, assuming the cars are independent? 6. The average lifetime of a Samsung HDTV is 8.5 years and the standard deviation is 3.1 years. A random sample of 45 Samsung HDTV is selected and the lifetimes of each television is recorded. (a) What is the sampling distribution of the sample mean? Why? (b) Find the probability that the sample mean is between 7.5 and 9 years (c) Find the 90th percentile of the sample mean 7. State DMV records indicate that of all vehicles undergoing emissions testing during the previous year, 70% passed on the first try. A random sample of 200 cars tested in a particular county during the current year yields 124 that passed on the initial test. Does this suggest that he true proportion for this county during the current year differs from the previous statewide proportion? Find an appropriate confidence interval to make your conclusion. 8. The Bud Light label claims that each bottle only contains 100 calories. Suppose that 55 Bud Light bottles are randomly selected and their calorie content is measured. The average calorie count in the sample is 106 with a standard deviation of 17 calories. Does this data provide sufficient evidence that the true average calorie content of Bud Light bottles is different than 100 calories? (a) State the null and alternative hypotheses (b) Test the hypothesis at 0.01 level of significance. Please clearly state the rejection region and the test conclusion. (c) Find the p-value. (d) Interpret your result.