Homework answers / question archive / For Questions 1 - 3: For two equally balanced, 6-sided die that are tossed, the following table represents the entire probability space where the first number represents the number on the first die, the second number represents the number on the second die and the third number represents the sum of both numbers (total dots on both die)

For Questions 1 - 3: For two equally balanced, 6-sided die that are tossed, the following table represents the entire probability space where the first number represents the number on the first die, the second number represents the number on the second die and the third number represents the sum of both numbers (total dots on both die)

Math

Share With

For Questions 1 - 3: For two equally balanced, 6-sided die that are tossed, the following table represents the entire probability space where the first number represents the number on the first die, the second number represents the number on the second die and the third number represents the sum of both numbers (total dots on both die). That is, First Number + Second Number = Total Number. Assume an equally likely (uniform) distribution of simple events. 1+1=2 2 +1=3 3+1=4 4+1=5 5+1=6 6+1=7 1+2=3 2+2=4 3+2=5 4+2=6 5+2=7 6+2=8 1+3=4 2+3=5 3+3=6 4+3=7 5+3=8 6+3=9 1+4=5 2+4=6 3+4=7 4+4=8 5+4=9 6+4=10 1+5=6 2+5=7 3 +5=8 4+5=9 5+5=10 6+5=11 1+6=7 2+6=8 3+6=9 4+6=10 5+6=11 6+6=12 / Page 1 of 16 Let A be the event that at least one of the die was a 3 Let B be the event that doubles (both die are the same) was rolled Let C be the event that a sum of 5 was rolled 1. Find: a. (90) N, the total number of possible outcomes from any single event b. (90) P(A) c. (90) P(B) d. (90) P(C) e. (90) P(A AND B) f. Are events A and B mutually exclusive? g. (90) P(A OR B) h. (90) P(NOT(A AND B)) i. (90) P(NOT(A OR B)) j. (90) P(A | B) k. (90) P(B | A) / Page 2 of 16 l. Are events A and B independent? Please show formulas used. 2. Find: a. (90) P(A AND C) b. Are events A and C mutually exclusive? c. (90) P(A OR C) d. (90) P(NOT(A AND C)) e. (90) P(NOT(A OR C)) f. (90) P(A | C) g. (90) P(C | A) h. Are events A and C independent? Please show formulas used. 3. Find: a. (90) P(B AND C) b. Are events B and C mutually exclusive? / Page 3 of 16 c. (90) P(B OR C) d. (90) P(NOT(B AND C)) e. (90) P(NOT(B OR C)) f. (90) P(B | C) g. (90) P(C | B) h. Are events B and C independent? Please show formulas used. i. On any particular roll of 2 die, is the number on the second die independent of the number on the first die? Explain your answer. 4. According to tables provided by the Statistics Quantify Universally Accepted Theory (SQUAT) organization, there is roughly an 45% (p = 0.45) chance that a person aged 20 will get a tattoo before the age of 70. Suppose that three (N = 3) people aged 20 are independently selected at random. Use the Binomial Probability to complete this problem. a. (90) Assumptions: Are N trials to be performed? Are two outcomes, success or failure, possible for each trial? Are the trials independent? / Page 4 of 16 Does the success probability, p, remain the same from trial to trial? Are all four assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps using the binomial probability distribution. b. Step 1 Identify a success c. (90) Step 2 Determine p, the success probability d. (90) Step 3 Determine N, the number of trials Step 4 The binomial probability formula for the number of successes, X is . Plug in p and n and determine the binomial probabilities for this problem e. (90) Find P(X = 2), the probability that exactly 2 people will get a tattoo before the age of 70. f. (90) Find P(X 1), the probability that, at most, one person will get a tattoo before the age of 70 g. (90) Find P(X ≥ 2), the probability that, at least two people will get a tattoo before the age of 70 h. (90) Find / Page 5 of 16 i. Specific to this problem, what does the meaning of the population mean? j. (90) Find 2 k. (90) Find l. Specific to this problem, what is the meaning of the population standard deviation? m. Draw the probability distribution for this problem 0.55 0.50 0.45 0.40 0.20 y t i 0.15 l i b 0.10 a b 0.05 o r 0.00 0123 P 0.35 0.30 0.25 n. Is this distribution left skewed, symmetric or right skewed? / Page 6 of 16 5. Based on a recent add from Symantec, 35% of computers have the Norton antivirus program installed. In a simple random sample of 200 scanned computers, it was found that 56 of them actually had Norton antivirus software programs installed. Use the following steps of the One Proportion z-Hypothesis Test with a 5% level of significance, α, to test the hypothesis that 35% of computers have Norton antivirus software installed. (Note: x = 56 and n = 200) a. (90) Determine whether both assumptions are met: Assumption 1: Simple random sample? Assumption 2: Both np and nq are 10 or greater? Are both assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps of the One Proportion z-Hypothesis Test Procedure. b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA : Tail: c. (90) Step 2 Decide on the significance level, α. d. (90) Step 3 Find the value of the test statistic. / Page 7 of 16 e. (90) Step 4 Find the P-value f. Step 5 Reject H0 or do not reject H0? Why? Explain your reasoning. g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, do 35% of all computers have Norton antivirus software programs installed? Explain your reasoning. 6. Using the same sample data from Problem #5 (56 out of 200 scanned computers actually had Norton antivirus software programs installed), perform a One Proportion z-Interval Procedure to determine the 95% confidence interval for the population proportion, p, of computers that actually have Norton antivirus software programs installed. (Note: x = 56 and n = 200) a. (90) Determine whether both assumptions are met: Assumption 1: Simple random sample? Assumption 2: Both np and nq are 10 or greater? Are both assumptions met? / Page 8 of 16 Even if you do not think the assumptions are met, please proceed with the following steps for the One Proportion z-Interval Procedure for a population proportion. b. (90) Step 1 For a confidence level of 1 – α, find zα/2 c. (90) Step 2 Find the confidence interval for the population proportion. d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this informal test support the same conclusion as the formal results of the hypothesis test of Problem #5? If so, why and if not, why not? 7. Based on a recent simple random sample of CoC students, 80 of 100 CoC students stated that they would rather attend in person classes than have computer based, synchronized instruction. Use the following steps of the One Proportion z Hypothesis Test with a 5% level of significance, α, to test the hypothesis that over 75% of CoC students want to return to having in-person classes. (Note: x = 80 and n = 100) a. (90) Determine whether both assumptions are met: Assumption 1: Simple random sample? Assumption 2: Both np and nq are 10 or greater? Are both assumptions met? / Page 9 of 16 Even if you do not think the assumptions are met, please proceed with the following steps of the One Proportion z-Hypothesis Test Procedure. b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA : Tail: c. (90) Step 2 Decide on the significance level, α. d. (90) Step 3 Find the value of the test statistic. e. (90) Step 4 Find the P-value f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. / Page 10 of16 g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, do over 75% of CoC college students want to return to in-person classes? Explain your reasoning. 8. Using the same sample data from Problem #7 (80 out of 100 CoC students would rather attend in-person classes), perform a One-Proportion z-Interval Procedure to determine the 90% confidence interval for the population proportion, p, of all CoC students that would rather attend in-person classes. (Note: x = 80 and n = 100) a. (90) Determine whether both assumptions are met: Assumption 1: Simple random sample? Assumption 2: Both np and nq are 10 or greater? Are both assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps for the One Proportion z-Interval Procedure for a population proportion. b. (90) Step 1 For a confidence level of 1 – α, find zα/2 / Page 11 of 16 c. (90) Step 2 Find the confidence interval for the population proportion. d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this informal test support the same conclusion as the formal results of the hypothesis test of Problem #7? If so, why and if not, why not? 9. Mean Height of Professional Basketball Players. A simple random sample of 40 from a normal population of professional basketball players’ heights were measured with a sample mean of 79” and a sample standard deviation was 4”. Use the following steps of the One Mean t-Hypothesis Test Procedure with a 5% level of significance, α, to test the hypothesis that the population mean height of all professional basketball players is greater than 78”. (Note: = 79” hours, s = 4” hours and n = 40.) a. (90) Determine whether all 3 assumptions are met: Assumption 1: Simple random sample? Assumption 2: Normal population or large sample size? Assumption 3: σ unknown? Are all 3 assumptions met? / Page 12 of 16 Even if you do not think the assumptions are met, please proceed with the following steps of the One Mean t-Hypothesis Test Procedure. b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA : Tail: c. Step 2 Decide on the significance level, α. d. (90) Step 3 Find the value of the test statistic. e. (90) Step 4 Find the P-value. f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. / Page 13 of 16 g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, is the population mean height of all professional basketball payers greater than 78”? Explain your reasoning. 10. Using the same sample data from Problem #9 (40 professional basketball players’ heights were measured with a sample mean of 79” and a sample standard deviation was 4”), follow the steps of the One Mean t-Interval Procedure to determine the 90% confidence interval for the population mean height of all professional basketball players. (Note: = 79” hours, s = 4” hours and n = 40.) a. (90) Determine whether all 3 assumptions are met: Assumption 1: Simple random sample? Assumption 2: Normal population or large sample size? Assumption 3: σ unknown? Are all 3 assumptions met? / Page 14 of 16 Even if you do not think the assumptions are met, please proceed with the following steps for the One-Sample t-Interval Procedure for a population mean. b. (90) Step 1 For a confidence level of 1 – α, find tα/2 c. (90) Step 2 Find the confidence interval for the population mean. d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this informal test support the same conclusion as the formal results of the hypothesis test of Problem #9. If so, why and if not, why not? / Page 15 of 16 11. Extra Credit. Under what two conditions can the population distribution of sample means (for any given sample size) be considered normal? a. b. Last (not for any credit). Statistics alone cannot prove ______________. Page 16 of 16 1. Are women community college students more likely to transfer to a 4-year college than men? Independent data was obtained from simple random samples from California community colleges throughout the state. For women, there were 450 out of 800 that transferred and for men, there were 375 out of 700 that transferred to a 4-year college. Use the following steps of the Two Proportion z-Hypothesis Test with a 0.05 level of significance, α, to test the claim (hypothesis) that women are more likely to transfer to a 4-year college than men. (Note: x1 = 450, n1 = 800, x2 = 375 and n2 = 700) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? Assumption 3: Are the number of successes and the number of failures 10 or greater for both samples? Are all three assumptions met? / Page 1 of 17 Even if you do not think the assumptions are met, please proceed with the following steps of the Two Proportion z-Hypothesis Test Procedure. b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA: Tail: c. Step 2 Decide on the significance level, α. (90) d. Step 3 Find the value of the test statistic. (90) e. Step 4 Find the P-value. (90) f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. / Page 2 of 17 g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, are women more likely than men to transfer to a 4year college from a California community college? Explain your reasoning. 2. Using the same sample data from Problem #1 (450 out of 800 women and 375 out of 700 men) to perform a Two-Proportion z-Interval Procedure to determine the 90% confidence interval for the difference between the two population proportions of women and men that transferred to a 4-year college from California community colleges. Note: x1 = 450, n1 = 800, x2 = 375 and n2 = 700) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? Assumption 3: Are the number of successes and the number of failures 10 or greater for both samples? Are all three assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps for the Two Proportion z-Interval Procedure for a population proportion. b. Step 1 For a confidence level of 1 – α, find zα/2 (90) / Page 3 of 17 c. Step 2 Find the confidence interval for the difference of two population proportions. (90) d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this confidence interval support the same conclusion as the formal results of the hypothesis test of Problem #1? If so, why and if not, why not? 3. Some people believe that the population mean average cholesterol for men and women is the same while others think that men’s cholesterol is higher. Independent data was obtained from simple random samples of men and women. The mean cholesterol level of 40 men was 395 with a standard deviation of 292 and the mean cholesterol level of 50 women was 240 with a standard deviation of 185. Use the following steps of the Two Mean t-Hypothesis Test (independent samples or nonpooled) with a 0.05 level of significance, α, to test the claim (hypothesis) that the mean average cholesterol levels for men and women are the same (equal). (Note: 1 = 395, s1 = 292, n1 = 40, 2 = 240, s2 = 185 and n2 = 50) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? Assumption 3: Are the samples are from normal populations or a sample size of at least 30? / Page 4 of 17 Assumption 4: Are the population standard deviations unknown? Are all four assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps of the Two Mean t-Hypothesis Test Procedure (independent samples or nonpooled). b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA: Tail: c. Step 2 Decide on the significance level, α. (90) d. Step 3 Find the value of the test statistic. (90) / Page 5 of 17 e. Step 4 Find the P-value. (90) f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, is the mean average cholesterol level for men and women the same (equal)? Explain your reasoning. 4. Using the same sample data from Problem #3 (the mean cholesterol level of 40 men was 395 with a standard deviation of 292 and the mean cholesterol level of 50 women was 240 with a standard deviation of 185), follow the steps of the Two Mean t-Interval Procedure (independent samples or non-pooled) to determine the 95% confidence interval for the difference between the two population mean cholesterol level of men and women. (Note: 1 = 395, s1 = 292, n1 = 40, 2 = 240, s2 = 185 and n2 = 50) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? / Page 6 of 17 Assumption 3: Are the samples are from normal populations or a sample size of at least 30? Assumption 4: Are the population standard deviations unknown? Are all four assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps for the Two-Proportion t-Interval Procedure (independent samples or nonpooled) for a population mean. b. Step 1 For a confidence level of 1 – α, find tα/2 (90) c. Step 2 Find the confidence interval for the difference of two population means. (90) d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this confidence interval support the same conclusion as the formal results of the hypothesis test of Problem #3? If so, why and if not, why not? / Page 7 of 17 5. A simple random sample of the ratings of male and female teachers from Rate My Professor.com for this school was performed. Averaged data for 12 female professors and 15 male professors’ ratings were independently obtained from normal populations and the average rating for female professors was 3.8 with a sample standard deviation of 0.5 (for a sample of 12) and the average rating for male professors was 3.9 with a standard deviation of 0.4 (for a sample of 15). Use the following steps of the Two Mean t-Hypothesis Test (independent samples or non-pooled) with a 0.05 level of significance, α, to test the claim (hypothesis) that female professors are rated higher than male professors. (Note: 1 = 3.8, s1 = 0.5, n1 = 12, 2 = 3.9, s2 = 0.4 and n2 = 15) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? Assumption 3: Are the samples are from normal populations or a sample size of at least 30? Assumption 4: Are the population standard deviations unknown? Are all four assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps of the Two Mean t-Hypothesis Test Procedure (independent samples or nonpooled). / Page 8 of 17 b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA: Tail: c. Step 2 Decide on the significance level, α. (90) d. Step 3 Find the value of the test statistic. (90) e. Step 4 Find the P-value. (90) f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. / Page 9 of 17 g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, are the ratings of female professors higher than male professors at this school? Explain your reasoning. 6. Using the same sample data from Problem #5 (the average rating for female professors was 3.8 with a standard deviation of 0.5 and a sample size of 12 and the average rating for male professors was 3.9 with a standard deviation of 0.4 with a sample size of 15), follow the steps of the Two Mean t-Interval Procedure (independent samples or non-pooled) to determine the 90% confidence interval for the difference between the two population means of the rating of female and male professors. (Note: 1 = 3.8, s1 = 0.5, n1 = 12, 2 = 3.9, s2 = 0.4 and n2 = 15) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample? Assumption 2: Independent samples? Assumption 3: Are the samples are from normal populations or a sample size of at least 30? Assumption 4: Are the population standard deviations unknown? Are all four assumptions met? / Page 10 of 17 Even if you do not think the assumptions are met, please proceed with the following steps for the Two-Proportion t-Interval Procedure (independent samples or nonpooled) for a population mean. b. Step 1 For a confidence level of 1 – α, find tα/2 (90) c. Step 2 Find the confidence interval for the difference of two population means. (90) d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this confidence interval support the same conclusion as the formal results of the hypothesis test of Problem #5? If so, why and if not, why not? 7. The ACT exam is used by many colleges to test the readiness of high school students for college. A local high school offers an ACT prep class and wants to know if it really helps. A simple random sample of 40 students was selected. They took the ACT exam before and after taking the ACT prep class. For each student, the difference between the before and after scores were measured (d = before – after). The mean of the differences was -1.5 ACT points with a standard deviation of 2.3 ACT points. Use a 5% significance level to test the claim that taking the prep class is effective in raising ACT scores. (Note: dbar = -1.5, sd = 2.3 and n = 40) / Page 11 of 17 a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample of paired data? Assumption 2: The difference of population means is normally distributed or the sample size is large. Are both assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps of the Two Mean Paired t-Hypothesis Test Procedure (Dependent samples or paired data). b. Step 1 State the null and alternative hypotheses. Is this hypothesis test left tailed, two tailed or right tailed? H0 : HA: Tail: c. Step 2 Decide on the significance level, α. (90) d. Step 3 Find the value of the test statistic. (90) / Page 12 of 17 e. Step 4 Find the P-value. (90) f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. g. Step 6 Interpret the results of the hypothesis. h. Based on this analysis, is taking this prep class effective in raising ACT scores? Please explain your answer. 8. Using the same sample data from Problem #7 (the average difference was -1.5 ACT points with a sample standard deviations of 2.3 ACT points for 40 students), follow the steps of the Two Mean Paired t-Interval Procedure (dependent samples or paired data) to determine the 90% confidence interval for the difference of ACT scores for before and after taking the prep class. (Note: dbar = -1.5, sd = 2.3 and n = 40) a. Determine whether the assumptions are met: (90) Assumption 1: Simple random sample of paired data? / Page 13 of 17 Assumption 2: The difference of population means is normally distributed or the sample size is large. Are both assumptions met? Even if you do not think the assumptions are met, please proceed with the following steps for the Two Mean Paired t-Interval Procedure (dependent samples or paired data) for a population mean. b. Step 1 For a confidence level of 1 – α, find tα/2 (90) c. Step 2 Find the confidence interval for the difference of two population means. (90) d. Step 3 Interpret the confidence interval e. Based on your confidence interval, does this confidence interval support the conclusion as the hypothesis test from Problem #7? If so, why and if not, why not? / Page 14 of 17 9. Four part-time students in three different majors were asked to estimate the amount of time they spend on homework per week (in hours). The results are shown. English Math 7 3 6 6 6 5 5 8 Political Science 4 7 6 7 At the 0.05 significance level, perform a One-Way ANOVA hypothesis test to determine if the population means are equal. That is, determine if the mean time spent on homework of all part-time students majoring in English, Math and Political Science are (statistically) equal. Assume that these independent simple random samples are from normal populations and that their standard deviations are equal. a. Are the assumptions met? (90) 1. Simple random sample? 2. Independent samples? 3. Normal populations? 4. Equal population standard deviations? Are all four assumptions met? / Page 15 of 17 Even if you do not think that all assumptions are met, please proceed with the One-Way ANOVA hypotheses test to determine if the population means are equal. b. Step 1 State the null and alternative hypotheses. H0 : HA: c. Step 2 Decide on the significance level, α. (90) d. Step 3 Find the value of the test statistic, MSTR / MSE. (90) e. Step 4 Find the P-value. (90) f. Step 5 Reject H0 or do not reject H0? Why? That is, explain your reasoning. g. Step 6 Interpret the results of the hypothesis. / Page 16 of 17 h. Is the mean number of study hours of at least one of the three majors statistically different than the others? If so, without doing any other analysis, which major has a different mean amount of study time? 10. Ethics in Statistics Do you think it is important to be ethical in the process of data collection, data analysis, and the reporting of statistical studies? Explain you answer. 11. Extra Credit. Discuss the interdependence between the confidence level, the confidence interval and the precision of any estimate. (How are they related?) Last (not for any credit). Statistics alone cannot prove ______________.