Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Bootstrapping (statistics) wikipedia , lookup
History of statistics wikipedia , lookup
Taylor's law wikipedia , lookup
Foundations of statistics wikipedia , lookup
Psychometrics wikipedia , lookup
Omnibus test wikipedia , lookup
Misuse of statistics wikipedia , lookup
MATH 150-768 Elementary Statistics – Final Review (Spring 2012) Chapter 2. Summarizing and Graphing Data 1. For the data given below, 70 61 74 80 58 66 53 70 65 71 63 60 75 62 68 a. Complete constructing a Frequency distribution by filling in the frequencies Grades 51 – 55 56 – 60 61 – 65 66 – 70 71 – 75 76 – 80 Frequency Ans. 1 Ans. 2 Ans. 4 Ans. 4 Ans. 3 Ans. 1 b. Plot the histogram of the data, in the space below, using the above frequency distribution. Ans. See graph Number of TV Sets 53 58 63 68 73 78 c. Does the frequency distribution in the above problem appear to have a normal distribution? Why? Ans. Yes. Appear to have a normal distribution. d. Identify the class width, class midpoint, class boundaries, lower and upper class limits for the given frequency distribution. Class width: Ans. 5 Class midpoints: Ans. 53, 58, 63, 68, 73, 78 Class boundaries: Ans. 50.5, 55.5, 60.5, 65.5, 70.5, 75.5, 80.5 e. Construct the Stem-and-leaf plot of the original data. Ans. Stem Leaves 5 38 6 0123568 7 00145 8 0 1 Chapter 3. 2. A student obtains the following grades in 8 quizzes. 65 75 80 85 70 90 Find the following: 1) Mean Ans. 76.3 2) Median Ans. 75 3) Mode Ans. 70, 75 (bi-modal) 4) Standard Deviation Ans. 8.3 5) Variance Ans. 69.6 70 75 3. Determine which score corresponds to the higher relative position: a score of 51.5 on a test for which = 47 and s = 9, a score of 5.9 on a test for which = 4.2 and s = 1.2 or a score of 460.8 on a test for which = 444 and s = 42. Ans. a score of 5.9 on a test for which = 4.2 and s = 1.2 (z-scores are, 0.5, 1.4, 0.4) 4. Find the percentile for the data value 6: 4, 12, 9, 6, 4, 4, 12, 6, 4, 12, 2, 12, 15, 5, 9, 4, 12, 9, 6, 12 Ans. the percentile of the data value 6 is 35. (i.e. 35% of the data is less than 6) 5. The following is a table for the grades in an Elementary Algebra test. Construct the boxplot. 70 82 74 80 58 66 63 70 85 75 80 70 75 62 80 Ans. (min = 58, Q1 = 62, Q2 = 74, Q3 = 80, max = 85) 74 58 62 55 60 85 80 65 70 75 80 85 90 Chapter 4. Probability 6. A batch consists of 12 defective coils and 88 ones. Find the probability of getting two good coils when two coils are randomly selected if the first selection is replaced before the second is made. Ans. (0.88)(0.88) = 0.7744 (independent) 7. A sample of 3 different calculators is randomly selected from a group containing 41 that are defective and 22 that have no defects. What is the probability that all four of the calculators selected are defective? Round to four decimal places. Ans. (414039)/(636261) = 0.2684 (dependent) Provide the written description of the complement of the given event. 8. Of ten adults, at least one of them has high blood pressure. Ans. out of ten adults, none of them had high blood pressure. Chapter 5 Discrete Probability Distributions 2 9. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to the three decimal places. n = 4, x = 3, p = 1/6 Ans. 0.015 Find the indicated probability 10. The brand name of a certain chain of coffee shops has 53% recognition rate in the town of Coffleton. An executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in the town. He selects a random sample of 7 Coffleton residents. Find the probability that exactly 4 of 7 Coffleton residents recognize the brand name. Ans. 0.287 Use the given values of n and p to find the minimum usual value – 2 and the maximum usual value + 2. Round your answer to the nearest hundredth unless otherwise noted. 11. n = 104; p = 0.21 Ans. min usual = 13.53, max usual = 30.15 Chapter 6 (6.2 – 6.5) 12. Find the area of the shaded region. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. Ans. 0.7938 85 100 125 Solve the problem. 13. A final exam in Math160 has a mean of 73 and a standard deviation of 7.8. If 24 students are randomly selected, find the probability that the mean of their test score is less than 76. Ans. 0.97 Chapter 7 14. Find the margin of error and construct the CI corresponds to: 90% confidence level, n = 130, x = 65. Ans. 0.428 < p < 0.572 15. Use the given data to find the minimum sample size required to estimate the population proportion: E = 0.03, confidence level = 94%, p̂ = 0.13. Ans. 469 Construct a CI for the population mean . Assume that the population has a normal distribution. 16. A saving and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $664.14 and a standard deviation of $297.29. Find a 98% confidence interval for the true mean checking account balance for local customers. Ans. $453.59 < < $874.69 Chapter 8 Hypothesis Testing Provide an appropriate response. 3 17. In a population, 11% of people are left handed. In a simple random sample of 160 people selected from this population, the proportion of left handers is 0.10. What is the number of left handers in the sample and what notation is given to that number? What are the values of p and p̂ ? Ans. 16 left handers in the sample, x = 16, p = 0.11, p̂ = 0.10 18. In a sample of 47 adults selected randomly from one town, it is found that 9 of them have been exposed to a particular strain of the flu. Find the p-value for the test of the claim that the proportion of all adults in the town that have been exposed to this strain of the flu is 8%. Use the significance level of 0.05. Ans.claim: p = 0.08 Alternative: p 0.08 Hypothesis test: H0: p = 0.08 H1: p 0.03 9 0.08 pˆ p Test statistic: z = 2.8174 47 pq 0.08(0.92) n 47 P-value: = 2 Normalcdf(2.8174, ∞, 0,1) 0.0048 < 0.05 Reject H0 Final conclusion: There is sufficient evidence to warrant rejection of the claim that p = 0.08. 19. In tests of computer component, it is found that the mean time between failures is 520 hours. A modification is made which is supposed to increase the time between failures. Tess on a random sample of 10 modified components resulted in the following times (in hours) between failures: (mean is 537.1, sd = 20.7) 518 548 561 523 536 499 538 557 528 563 At the 0.05 significance level, test the claim that for the modified components, the mean time between failures is greater than 520 hours. Use the P-value method. Ans. claim: > 520 Alternative: ≤ 520 Hypothesis test: H0: = 520 H1: > 520 x 537.1 520 Test statistic: t = 2.612 s 20.7 n 10 P-value: 0.014 Reject H0 Final conclusion: there is sufficient evidence to support the claim that > 520. Chapter 9 20. Given sample sizes and number of successes to find the pooled estimate p . Round your answer to the nearest thousandth. n1 = 34, x1 = 15; n2 = 414, x2 = 105. Ans. (x1 + x2)/(n1 + n2) = 120/448 0.268 4 21. Assume that you plan to use a significance level of = 0.05 to test the claim that p1 = p2. Use the given sample sizes and numbers of successes to find the P-value for the hypothesis test. n1 = 100, x1 = 41; n2 = 140, x2 = 35. Ans. p1 = 0.41; p2 = 35/140 = 0.25; p = 76/240 0.317; claim: p1 = p2 Alternative: p1 p2 Hypothesis test: H0: p1 = p2 H1: p1 p2 (Two tail test) Test statistic z 2.626; P-value = 2Normalcdf(2.626, 10000, 0, 1) 0.0086 < 0.05 Reject H0 Final conclusion: The is sufficient evidence to warrant rejection of the claim that p1 = p2. 22. Test the indicated claim about the means of two populations. Assume that all requirements are met. Do not assume that 1 = 2. A researcher was interested in comparing the response times of two different cab companies. Company A and B were each called at 50 randomly selected times. The calls to company A were made independently of calls to company B. The response time for each call were recorded. The summary statistics were as follows: Company A Company B Mean response time 7.6 minutes 6.9 minutes Standard deviation 1.4 minutes 1.7 minutes Use a 0.02 significance level to test the claim that the mean response time for company A is the same as the mean response time for company B. Use the P-value method. Ans. claim: 1 = 2 Alternative: 1 2 Hypothesis test: H0: 1 = 2 H1: 1 2 (two tail test) = 0.02 Critical value = 2.403 x x 7.6 6.9 Test statistic t = 1 2 2.248 s12 s22 1.4 2 1.7 2 n1 n2 50 50 P-value = 0.029 (> 0.02) Fail to reject H0 Final conclusion: there is not sufficient evidence to support the claim that 1 = 2. (The following will be worth 1/4 of the final) 23. The two data sets are dependent. Find d to the nearest tenth. A 70 67 56 63 51 B 22 24 29 25 22 Ans. 37.0 5 24. Assume that all requirements are met. The table below shows the weights of seven subjects before and after following a particular diet for two months. subject A B C D E F G before 180 188 172 193 195 168 158 after 173 179 179 198 181 170 146 Using a 0.01 level of significance, test the claim that the diet is effective in reducing weight. Ans. Claim: d > 0 Alternative: d ≤ 0 Hypothesis test: H0: d = 0 H1: d > 0 (original claim) (right tail) = 0.01 d 0 40 Test statistic: t = 1.327 sd 8.52 n 8 Critical Value = 2.998 Fail to reject H0. There is not sufficient evidence to support the claim that the diet is effective in reducing weight. Chapter 10 25. Given that LCC r = –0.844, and n = 5. Find the critical values of r and determine whether or not the given r represents a significant linear correlation. Using = 0.05. Ans. critical values = 0.878. Since | r | < 0.878, the given r does not represent a significant correlation. 26. Describe the error in the stated conclusion. Given: there is no significant linear correlation between scores of a math test and scores on a verbal test. Conclusion: there is no relationship between scores on the math test and scores on the verbal test. Ans. there can be other relations (e.g. non-linear) 27. Use the given data to find the best predicted value of the response variable. Eight pairs of data yield r = 0.708 and the regression equation ŷ = 55.8 + 2.79x. Also, y = 71.125. What is the best predicted value of y for x = 5.7? Ans. since r > 0.707 (at = 0.05). The data are linearly correlated. So the best predicted value of y is 55.8 + 2.79(5.7) 71.7. 28. Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. hours 0 3 4 5 12 Score 8 2 6 9 12 Ans. ŷ = 4.88 + 0.525x 29. Using the same data as in the above problem. Find the residual at the sample point (3, 2) Ans. Residual 2 – (4.88 + 0.5253) –4.455 (residual = observed – predicted) 6