Final Exam Review Problems Math 13 Statistics Summer 2013

Transcription

1 Final Exam Review Problems Math 13 Statistics Summer 2013 These problems are due on the day of the final exam. Name: (Please PRINT) Problem 1: (a) Find the following for this data set {9, 1, 5, 3, 6, 8, 8, 4, 3, 2, 1, 1, 8, 9, 7} Mean Median: Mode: Range: (b) Find standard deviation without using calculator for this data set: {4, -6, 5, -7} s s x x n 1 2 n x x 2 n n1 2

2 Problem 2: A couple wants to have three babies for the next three years, only one baby per year (either boy or girl) and no other possibility. Create a sample space, i.e. collection of all simple events: Find the probability that they will have two girls and a boy. Find the probability that they will have at least one girl. Find the probability that they will have no more than two boys. Find the probability that they will have no girls. Find the probability that they will have between one and three girls. Problem 3: Quality Control: As a quality control manager in a clothing company you randomly select 5 shirts from a collection of 2000 shirts that just came to your company from Bangladesh. You will reject all the shirts of if you find at least one faulty shirt. It is assumed that there are 20 faulty shirts in the lot of 2000 shirts. Find the probability of accepting all the shirts in this lot. Problem 4: On the day of an important exam such as the SAT, you keep a backup mechanical pencil in the event one fails so that you may use the other. Given that there is a 95% chance that a mechanical pencil would work, what are your chances that you would not have to get a third mechanical pencil at the test center at the expense of your precious exam time?

3 Problem 5: Permutation and combination: What are your chances of winning the Mega Millions Lottery? In Mega Millions you pick 5 numbers from 1 to 56 without replacement and 1 number from 1 to 46. Problem 6:Assume that you are investing $10,000 in one bond. There are two types of bonds available. The first bond gives you a 7% return with a default rate of 3% and the second bond gives you a return of 9% with a default rate of 5%. Which one these bonds would you consider for investing your $10,000 assuming that you want to maximize your profit.

4 Problem 7: A new drug named CURAIDS that is 60% effective in extending the average life of an AIDS patient by twenty years. Five randomly selected AIDS patients from Africa are treated with this new drug. Answer the following questions based on the above information. (a) Show that the above situation satisfies all four criteria for the Binomial probability distribution. (b) (10 points) Fill in the probabilities in the following table. Show your calculations. x P(x) (c) What is the probability that no more than 4 patients are cured? Use results from part (b), do not do the calculations again.

5 (d) Find the probability that more than 2 patients or less than or equal to 5 patients are cured. Use results from part (b), do not do the calculations again. (e) Find the probability that at least 4 patients are cured. Use results from part (b), do not do the calculations again. (f) Find probability that less than 2 or more than 3 patients are cured. Use results from part (b), do not do the calculations again. Problem 8: In a city named Dhaka there were 125 drug related crimes over one year period. Find the probability that on a given day there will be exactly 3 drug related crimes in that city. UsePoisson distribution. Explain the requirements for Poisson distribution. x e Px ( ) x!

6 Problem 9: According to the U.N. report, the average yearly income in Bangladesh is about $500/year. It is also estimated that the population standard deviation is $100. Find the following probabilities: a) If you randomly select a person find the probability that she/he would make between $400 and $600. b) If you randomly select 30 persons find the probability that their average yearly income would be between $400 and $600. Problem 10: (Normal Distribution)Assume that you are a restaurant owner. The customer waiting time at your restaurant is normally distributed with an average waiting time of 12 minutes and a standard deviation of 3 minutes. You want to reward (with a free burger) 2% of the customers who wait the longest amount of time. So what should you tell your customers about minimum waiting period before they can get a free burger?

7 Problem 11: In a survey of 200 Hartnell students, it was found that 60 students said that English was not their first language. Create a 95% confidence interval for true proportion of Hartnell students whose first language is not English. Assumptions: Margin of error: Graph: Confidence interval: Explanation of confidence interval:

8 Problem 12: Population proportion A Popular TV show named ROOTS that addressed the black history and culture in the U.S. was very popular in the 1980s. You are curious if the majority of the population nowadays have heard of this show or know about it. Your research indicated that 241 people knew about this show out of 495 people you surveyed. Create a 95% confidence interval for the true population proportion. Assumptions: Graph: Margin of error: Confidence interval and explanation:

9 Problem 13: t-distribution: Following table represents the number of hours 10 different Hartnell College students work per week It is known that the distribution of the number of hours a HartnellCollege student works has approximately bell shape. Create a 95% confidence interval for the true mean of the number of hours per week a HartnellCollege student works. Assumptions: Calculations: x E x E E t /2 s n Graphs: Confidence interval and explanation:

10 Problem 14: Assume that the distribution of average yearly salary for Hartnell College graduates is a bell shaped curve. A random sample of 13 Hartnell College graduates has a mean salary of $35,000 and a standard deviation of $5,000. Create a 95% confidence interval for the population standard deviation. Assumptions: Calculations: n s n s R L Graph: Confidence interval and explanation:

11 Problem 15: It is known that in Bangladesh a person makes on the average $75 a month with a standard deviation of $9. You, as a researcher, are interested in determining if the monthly average income per person has increased. Therefore, you take a random sample of 100 people and find that the sample average is $79. If you desire a significance level of 0.05, then state your conclusion based on the calculations you make. Assumptions: Null and Alternative hypotheses: Calculations: z x n Graph and critical values: Conclusions:

12 Problem 16: A Popular TV show named ROOTS that addressed the black history and culture in the U.S. was very popular in the 1980s. You are curious if the majority (more than 50%) of the population nowadays have heard of this show or know about it. Your research indicated that 241 people knew about this show out of 495 people you surveyed. What is your conclusion? Significance level is Assumptions: Null and Alternative hypotheses: Calculations: z ˆp p pq n Graph and critical values: Conclusion:

13 Problem 17: Hypothesis Testing: 200 field workers were surveyed and their average yearly income was $13,700. The population standard deviation for the income distribution of field workers is assumed to be $4,000. Use the sample data, with 0.05 significance level, and test the claim that average income for the population of field workers is different from $14,000 per year. Requirements: Null and Alternative Hypotheses: Test Statistic: z x * x n Graph and critical regions: Conclusion:

14 Problem 18: Hypothesis test: (Matched pair) An exercise program is claimed to be effective in reducing weight. The following represents the weights of 8 people before and after the exercise program. Is there sufficient evidence to support the claim that there is a difference in weights before and after the program? Use a 0.05 significance level. Before After Assumptions: Null and alternative hypotheses: Calculations: * d d t and degrees of freedom = n 1 s n Graphs and critical points: Conclusion:

15 Problem 19: Two population proportions inferences: According to the PEW Research Center for the People & the Press in 2000 approximately 50% of the people surveyed said that the immigrants strengthen the U.S. with their hard work and talents whereas in 2006 approximately 41% responded similarly. Let us assume that each year the survey was conducted on 2,000 randomly selected adults in the U.S. Based on this information would you conclude that there is a progressively negative attitude towards immigrants in the U.S.? (Use significance level of 0.05, i.e. 95% confidence level) Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: z * pˆ pˆ p p pq pq n n 1 2 x x x x where pˆ and pˆ and p and q 1 p n1 n2 n1 n2

16 Graphs and critical points: Conclusion: Create a confidence interval: pˆ pˆ E p p pˆ pˆ E where margin of error: E z /2 pˆ qˆ pˆ qˆ n n

17 Problem 20: A 50-year long study by the British researchers shows that on the average smokers live 10 years less than the nonsmokers do. You being the curious cat collected some data from a reliable source and found that 621 smokers had average life span of 77 while 831 nonsmokers had an average life of 71. You also have the information that the sample standard deviation for life expectancy for both the smokers and nonsmokers is 10. Based on this information would you reject the null hypothesis that there is no difference in the average life expectancy between the smokers and nonsmokers? Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: t * x x s n s n Graphs and critical points: Conclusion:

18 Problem 21: F-distribution: A new drug was tested on treatment population and placebo population. Test the claim that variances for two populations differ. Significance level is Treatment group: sample size = 17, mean = 23.84, standard deviation = 2.31 Placebo group: sample size = 36, mean = 21.97, standard deviation = 2.01 Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: F s 2 * 1 2 s2 Graph and critical values: Conclusion:

19 Problem 22: Following represents the amount of tips paid for different bills at a restaurant. Bill (in $) Tips (in $) (a) Use hypothesis testing to determine whether there is linear correlation between the bill and the tip paid? * r t 2 1 r Degree of freedom (n 2) n 2 (b) Find the equation of the regression line for the above set of data. (c) Predict tip for $95. (d) Predict tip for $72 END OF FINAL EXAM REVIEW

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21 Midterm 1 Review Problems Math 13 Statistics Summer 2013 These problems are due on the day of the midterm. Name: (Please PRINT) Problem 1: Identify different levels of measurements for (a) through (d) (a) Heights of buildings in the city of Salinas Nominal Ordinal Interval Ratio (b) Temperature on different days of the year in Salinas Nominal Ordinal Interval Ratio (c) Possible letter grades you may receive in the class Nominal Ordinal Interval Ratio (d) Names of 10 students from the class Nominal Ordinal Interval Ratio (e) Identify different types of sampling and data: At a buffet they have different types of foods; these are American food, Asian food, Indian food, and Italian food. You eat two items randomly from each category of food. Convenience Systematic Stratified Clustering (f) Determine if the following example represents discrete or continuous data: Candy store sells candies only in the following amount: 0.2 lbs, 0.4 lbs, 0.6 lbs Continuous Discrete

22 (g) Many students are getting A s in Mo s Statistics class. Tom says Mo is a great teacher, while Jessica says Mo is an easy grader. This situation is called: Problem 2: Your grade in the class consists of 2 midterms 15% each, homework 10%, project 5%, attendance 10%, and final 30%. Find the weighted mean if your scores are as follows in the class. Midterm 1: 85 Midterm 2: 90 Final: 80 Homework: 95 Project: 100 Attendance: 75 Weighted Mean: x w x w Problem 3: (a) Find the following for this data set {9, 1, 5, 3, 6, 8, 8, 4, 3, 2, 1, 1, 8, 9, 7} Mean Median: Mode: Range: (b) Find standard deviation without using calculator for this data set: {4, -6, 5, -7} s s x x n 1 2 n x x 2 n n1 2

23 Problem 4: For the given set of data create a histogram: {9, 1, 5, 3, 6, 8, 8, 4, 3, 2, 1, 1, 8, 9, 7}. To find class frequencies, find the number of digits in each class. Step 1: Find the class boundaries and frequencies Classes Frequency Class boundaries Step 2: Create a histogram for the above dataset

24 Problem 5: Assume that you take a random sample of 200 people from Salinas and find that their average income is $48,000 per year with a standard deviation of $9000: (a) What can you say about the number of people who make between $25,500 and $70,500 out of the sample of 200 people? (b) What can you say about the number of people who make more than $70,500 out of the sample of 200 people?

25 (c) Now assume that you are given the additional information that the average yearly earning in Salinas is Bell shaped. Now what can you say about the number of people who make more than $57,000 per year? Draw graph and label it. (d) If the data distribution is bell shaped, then approximately how many people make between $39,000 and $66,000? Draw graph and label it. Problem 6:A couple wants to have three babies for the next three years, only one baby per year (either boy or girl) and no other possibility. Create a sample space, i.e. collection of all simple events: Find the probability that they will have two girls and a boy. Find the probability that they will have at least one girl. Find the probability that they will have no more than two boys. Find the probability that they will have no girls. Find the probability that they will have between one and three girls.

26 Problem 7: Determine whether the following events are independent or dependent: a) Being African American or any person of color raised in a the poverty stricken part of a big city and going to college for higher education b) Growing up in beautiful Pebble beach community and going to college for higher education c) A randomly selected student from CA community colleges being successful in four year college and a randomly selected student from MA community colleges being successful in four year college Problem 8: Disjoint events? a) A person being born in the U.S. and the same person being born in Mexico b) Father having a college degree and the son having a college degree from the same college as the father did c) Sunshine and drizzle Problem 9: Quality Control: As a quality control manager in a clothing company you randomly select 5 shirts from a collection of 2000 shirts that just came to your company from Bangladesh. You will reject all the shirts of if you find at least one faulty shirt. It is assumed that there are 20 faulty shirts in the lot of 2000 shirts. Find the probability of accepting all the shirts in this lot.

27 Problem 10: On the day of an important exam such as the SAT, you keep a backup mechanical pencil in the event one fails so that you may use the other. Given that there is a 95% chance that a mechanical pencil would work, what are your chances that you would not have to get a third mechanical pencil at the test center at the expense of your precious exam time? Problem 11: Two events A and B are disjoint if P(A and B) = 0. Are the events A and Bdisjoint? Use the formula P(A or B) = P(A) + P(B) P(A and B) and find the value of P(A and B)given thatp(a) = 0.6, P(B) = 0.3, P(A or B) = 0.7 Problem 12:The average income for the city of Salinas is normally distributed with a mean of $48,000 and a standard deviation of $9,000. Find the z-scores associated with the following incomes: Income for Mr. Chris Mickens is $16,000 per year Income for Ms. Sandra is $90,000 per year Are the above data outliers? Explain why. Problem 13:You have the option of buying one car from 5 different types of cars and you may pick one insurance from 4 different choices. What are total number of ways you can have the car and insurance combination?

28 Problem 14:An access code has 6 characters. First four are digits and the last two are alphabets which are case sensitive. A thief trying to break this code has a probability of success: Problem 15:A student committee consists of 13 members. They need to elect a president, a vice president, and a treasurer. How many different ways this can be accomplished? Problem 16:Age discrimination: Among 13 managers the company laid off 3 oldest managers. Do you think there was discrimination involved in the process based on your calculations?

29 Problem 17: Permutation and combination:what are your chances of winning the Mega Millions Lottery? In Mega Millions you pick 5 numbers from 1 to 56 without replacement and 1 number from 1 to 46. Problem 18: In a hiring process at a company 6 top managers were hired by taking into consideration the diversity of pool of applicants drawn from a community which had at least 80% minority populations. At the end of the hiring process no minority manager was hired from a pool of 30 applicants which proportionately represented the population of the community. Do you have enough statistical/mathematical justification to question the integrity of the hiring process given that at least 80% of the population in that community is minority?

30 END OF MIDTERM 1 REVIEW Review Problems Math 13 Statistics Name: (Please PRINT) Problem 1: Identify different levels of measurements for (a) through (d) (a) Heights of buildings in the city of Salinas Nominal Ordinal Interval Ratio (b) Temperature on different days of the year in Salinas Nominal Ordinal Interval Ratio (c) Possible letter grades you may receive in the class Nominal Ordinal Interval Ratio (d) Names of 10 students from the class Nominal Ordinal Interval Ratio (e) Identify different types of sampling and data: At a buffet they have different types of foods; these are American food, Asian food, Indian food, and Italian food. You eat two items randomly from each category of food. Convenience Systematic Stratified Clustering (f) Determine if the following example represents discrete or continuous data: Candy store sells candies only in the following amount: 0.2 lbs, 0.4 lbs, 0.6 lbs Continuous Discrete (g) Many students are getting A s in Mo s Statistics class. Tom says Mo is a great teacher, while Jessica says Mo is an easy grader. This situation is called:

31 Problem 2: Your grade in the class consists of 2 midterms 15% each, homework 10%, project 5%, attendance 10%, and final 30%. Find the weighted mean if your scores are as follows in the class. Midterm 1: 85 Midterm 2: 90 Final: 80 Homework: 95 Project: 100 Attendance: 75 Weighted Mean: x w x w Problem 3: (a) Find the following for this data set {9, 1, 5, 3, 6, 8, 8, 4, 3, 2, 1, 1, 8, 9, 7} Mean Median: Mode: Range: (b) Find standard deviation without using calculator for this data set: {4, -6, 5, -7} s s x x n 1 2 n x x 2 n n1 2

32 Problem 4: For the given set of data create a histogram: {9, 1, 5, 3, 6, 8, 8, 4, 3, 2, 1, 1, 8, 9, 7}. To find class frequencies, find the number of digits in each class. Step 1: Find the class boundaries and frequencies Classes Frequency Class boundaries Step 2: Create a histogram for the above dataset

33 Problem 5: Assume that you take a random sample of 200 people from Salinas and find that their average income is $48,000 per year with a standard deviation of $9000: (a) What can you say about the number of people who make between $25,500 and $70,500 out of the sample of 200 people? (b) What can you say about the number of people who make more than $70,500 out of the sample of 200 people?

34 (c) Now assume that you are given the additional information that the average yearly earning in Salinas is Bell shaped. Now what can you say about the number of people who make more than $57,000 per year? Draw graph and label it. (d) If the data distribution is bell shaped, then approximately how many people make between $39,000 and $66,000? Draw graph and label it. Problem: A data set has a Bell shape with a mean of 23 and a standard deviation of 4. (a) Find the data point that is associated with az-score of (b) Is the data point you found in part (a) an outlier? Explain.

35 Problem 6:A couple wants to have three babies for the next three years, only one baby per year (either boy or girl) and no other possibility. Create a sample space, i.e. collection of all simple events: Find the probability that they will have two girls and a boy. Find the probability that they will have at least one girl. Find the probability that they will have no more than two boys. Find the probability that they will have no girls. Find the probability that they will have between one and three girls. Problem 7: Determine whether the following events are independent or dependent: a) Being African American or any person of color raised in a the poverty stricken part of a big city and going to college for higher education b) Growing up in beautiful Pebble beach community and going to college for higher education c) A randomly selected student from CA community colleges being successful in four year college and a randomly selected student from MA community colleges being successful in four year college Problem 8: Disjoint events? a) A person being born in the U.S. and the same person being born in Mexico b) Father having a college degree and the son having a college degree from the same college as the father did c) Sunshine and drizzle

36 Problem 9: Quality Control: As a quality control manager in a clothing company you randomly select 5 shirts from a collection of 2000 shirts that just came to your company from Bangladesh. You will reject all the shirts of if you find at least one faulty shirt. It is assumed that there are 20 faulty shirts in the lot of 2000 shirts. Find the probability of accepting all the shirts in this lot. Problem 10: On the day of an important exam such as the SAT, you keep a backup mechanical pencil in the event one fails so that you may use the other. Given that there is a 95% chance that a mechanical pencil would work, what are your chances that you would not have to get a third mechanical pencil at the test center at the expense of your precious exam time? Problem 11: Two events A and B are disjoint if P(A and B) = 0. Are the events A and Bdisjoint? Use the formula P(A or B) = P(A) + P(B) P(A and B) and find the value of P(A and B)given thatp(a) = 0.6, P(B) = 0.3, P(A or B) = 0.7

37 Problem 12: PB A P A and B P A In a statistics class the following are the outcomes at the end of the semester: Passed Failed Students who expected to pass 35 5 Students who expected to fail 4 15 Find the probability that a randomly selected student passed, given that the student expected to fail. Problem 13:You have the option of buying one car from 5 different types of cars and you may pick one insurance from 4 different choices. What are total number of ways you can have the car and insurance combination? Problem 14:An access code has 6 characters. First four are digits and the last two are alphabets which are case sensitive. A thief trying to break this code has a probability of success:

38 Problem 15:A student committee consists of 13 members. They need to elect a president, a vice president, and a treasurer. How many different ways this can be accomplished? Problem 16:Age discrimination: Among 13 managers the company laid off 3 oldest managers. Do you think there was discrimination involved in the process based on your calculations? Problem 17: Permutation and combination:what are your chances of winning the Mega Millions Lottery? In Mega Millions you pick 5 numbers from 1 to 56 without replacement and 1 number from 1 to 46.

39 Problem 18: In a hiring process at a company 6 top managers were hired by taking into consideration the diversity of pool of applicants drawn from a community which had at least 80% minority populations. At the end of the hiring process no minority manager was hired from a pool of 30 applicants which proportionately represented the population of the community. Do you have enough statistical/mathematical justification to question the integrity of the hiring process given that at least 80% of the population in that community is minority? Problem 19:Assume that you are investing $10,000 in one bond. There are two types of bonds available. The first bond gives you a 7% return with a default rate of 3% and the second bond gives you a return of 9% with a default rate of 5%. Which one these bonds would you consider for investing your $10,000 assuming that you want to maximize your profit.

40 Problem 20:A new drug named CURAIDS that is 60% effective in extending the average life of an AIDS patient by twenty years. Five randomly selected AIDS patients from Africa are treated with this new drug. Answer the following questions based on the above information. (a) Show that the above situation satisfies all four criteria for the Binomial probability distribution. (b) (10 points) Fill in the probabilities in the following table. Show your calculations. x P(x) (c) What is the probability that no more than 4 patients are cured? Use results from part (b), do not do the calculations again.

41 (d) Find the probability that more than 2 patients or less than or equal to 5 patients are cured. Use results from part (b), do not do the calculations again. (e) Find the probability that at least 4 patients are cured. Use results from part (b), do not do the calculations again. (f) Find probability that less than 2 or more than 3 patients are cured. Use results from part (b), do not do the calculations again. Problem 21:In a city named Dhaka there were 125 drug related crimes over one year period. Find the probability that on a given day there will be exactly 3 drug related crimes in that city. UsePoisson distribution. Explain the requirements for Poisson distribution. x e Px ( ) x!

42 Problem 22: Normal approximation to Binomial Distribution: Assume that the probability of giving birth to a baby boy is 0.5; find the probability of giving birth to at least 180 boys when a survey was conducted on 300 pregnant women. Draw appropriate graphs and label the points of interest. Problem 23: Dr. Mohammed Yunus who won the Nobel Peace prize for promoting microfinancing/mini-loan (normally under $200 per person) for women around the world, especially in Bangladesh. In a survey of 1000 people who received micro-financing, 89% said that people who received micro-financing have benefitted from this innovative program. Given that there is a 50% chance of success for a micro-finance program, find probability that at least 890 people, i.e. 89% of 1000 people surveyed would say that micro-financing was beneficial.

43 Problem 24: According to the U.N. report, the average yearly income in Bangladesh is about $500/year. It is also estimated that the population standard deviation is $100. Find the following probabilities: a) If you randomly select a person find the probability that she/he would make between $400 and $600. b) If you randomly select 30 persons find the probability that their average yearly income would be between $400 and $600. Problem 25: (Normal Distribution)It is known that the average life for a DVD player is 5.4 years and a standard deviation of 0.94 years. If you want to provide a warranty so that only 2% of the DVD players will be replaced before the warranty expires, what is the time length of the warranty? Assume normal distribution.

44 Problem 26: (Normal Distribution)Assume that you are a restaurant owner. The customer waiting time at your restaurant is normally distributed with an average waiting time of 12 minutes and a standard deviation of 3 minutes. You want to reward (with a free burger) 2% of the customers who wait the longest amount of time. So what should you tell your customers about minimum waiting period before they can get a free burger? Problem 27:Amounts of nicotine in cigarettes (in general) have a mean of grams and a standard deviation of 0.33 grams. The manufacturers claim that they have reduced the amounts of nicotine in their cigarettes. You do a survey and find that 50 cigarettes have mean nicotine of grams. Assume that the mean and the standard deviation of nicotine in cigarettes have not changed, based on this information find the probability of randomly selecting 50 cigarettes with a mean of or less. Also based on your calculations, would you agree with the manufacturer s claim?

45 Problem 28: In a survey of 200 Hartnell students, it was found that 60 students said that English was not their first language. Create a 95% confidence interval for true proportion of Hartnell students whose first language is not English. Assumptions: Margin of error: Graph: Confidence interval: Explanation of confidence interval:

46 Problem 29: Population proportion A Popular TV show named ROOTS that addressed the black history and culture in the U.S. was very popular in the 1980s. You are curious if the majority of the population nowadays have heard of this show or know about it. Your research indicated that 241 people knew about this show out of 495 people you surveyed. Create a 95% confidence interval for the true population proportion. Assumptions: Graph: Margin of error: Confidence interval and explanation:

47 Problem 30: t-distribution: Following table represents the number of hours 10 different Hartnell College students work per week It is known that the distribution of the number of hours a HartnellCollege student works has approximately bell shape. Create a 95% confidence interval for the true mean of the number of hours per week a HartnellCollege student works. Assumptions: Calculations: x E x E E t /2 s n Graphs: Confidence interval and explanation:

48 Problem 31: Assume that the distribution of average yearly salary for Hartnell College graduates is a bell shaped curve. A random sample of 13 Hartnell College graduates has a mean salary of $35,000 and a standard deviation of $5,000. Create a 95% confidence interval for the population standard deviation. Assumptions: Calculations: n s n s R L Graph: Confidence interval and explanation:

49 Problem 32:It is known that in Bangladesh a person makes on the average $75 a month with a standard deviation of $9. You, as a researcher, are interested in determining if the monthly average income per person has increased. Therefore, you take a random sample of 100 people and find that the sample average is $79. If you desire a significance level of 0.05, then state your conclusion based on the calculations you make. Assumptions: Null and Alternative hypotheses: Calculations: z x n Graph and critical values: P-value: Conclusions:

50 Problem 33:A Popular TV show named ROOTS that addressed the black history and culture in the U.S. was very popular in the 1980s. You are curious if the majority (more than 50%) of the population nowadays have heard of this show or know about it. Your research indicated that 241 people knew about this show out of 495 people you surveyed. What is your conclusion? Significance level is Assumptions: Null and Alternative hypotheses: Calculations: z ˆp p pq n Graph and critical values: P-value: Conclusion:

51 Problem 34:Hypothesis Testing: 200 field workers were surveyed and their average yearly income was $13,700. The population standard deviation for the income distribution of field workers is assumed to be $4,000. Use the sample data, with 0.05 significance level, and test the claim that average income for the population of field workers is different from $14,000 per year. Requirements: Null and Alternative Hypotheses: Test Statistic: z x * x n Graph and critical regions: P-value: Conclusion:

52 Problem 35:Hypothesis test: (Matched pair) An exercise program is claimed to be effective in reducing weight. The following represents the weights of 8 people before and after the exercise program. Is there sufficient evidence to support the claim that there is a difference in weights before and after the program? Use a 0.05 significance level. Before After Assumptions: Null and alternative hypotheses: Calculations: * d d t and degrees of freedom = n 1 s n Graphs and critical points: Conclusion:

53 Problem 36: Two population proportions inferences: According to the PEW Research Center for the People & the Press in 2000 approximately 50% of the people surveyed said that the immigrants strengthen the U.S. with their hard work and talents whereas in 2006 approximately 41% responded similarly. Let us assume that each year the survey was conducted on 2,000 randomly selected adults in the U.S. Based on this information would you conclude that there is a progressively negative attitude towards immigrants in the U.S.? (Use significance level of 0.05, i.e. 95% confidence level) Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: z * pˆ pˆ p p pq pq n n 1 2 x x x x where pˆ and pˆ and p and q 1 p n1 n2 n1 n2

54 Graphs and critical points: Conclusion: Create a confidence interval: pˆ pˆ E p p pˆ pˆ E where margin of error: E z /2 pˆ qˆ pˆ qˆ n n

55 Problem 37: A 50-year long study by the British researchers shows that on the average smokers live 10 years less than the nonsmokers do. You being the curious cat collected some data from a reliable source and found that 621 smokers had average life span of 77 while 831 nonsmokers had an average life of 71. You also have the information that the sample standard deviation for life expectancy for both the smokers and nonsmokers is 10. Based on this information would you reject the null hypothesis that there is no difference in the average life expectancy between the smokers and nonsmokers? Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: t * x x s n s n Graphs and critical points: Conclusion:

56 Problem 38: F-distribution: A new drug was tested on treatment population and placebo population. Test the claim that variances for two populations differ. Significance level is Treatment group: sample size = 17, mean = 23.84, standard deviation = 2.31 Placebo group: sample size = 36, mean = 21.97, standard deviation = 2.01 Assumptions: Null and alternative hypotheses: Calculations: Test Statistic: F s 2 * 1 2 s2 Graph and critical values: Conclusion:

57 Problem 39: Following represents the amount of tips paid for different bills at a restaurant. Bill (in $) Tips (in $) (a) Use hypothesis testing to determine whether there is linear correlation between the bill and the tip paid? * r t 2 1 r Degree of freedom (n 2) n 2 (b) Find the equation of the regression line for the above set of data. (c) Predict tip for $95. (d) Predict tip for $72

58 Problem 40:ANOVA: Use F-distribution Given below are electricity consumptions for four different cities for five different months. Use a 0.05 significance level to test the null hypothesis that different cities have the same mean for electricity consumption. Jan. May July Oct. Nov City A: City B: City C: City D: Assumptions: Null and Alternative hypotheses: Calculations: Graphs: Conclusions: