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1 Review for Test 5 STA 2023 spr 2014 Name Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of ) r = 0.523, n = 25 Answer: Critical values: r = ±0.396, significant linear correlation Construct a scatterplot for the given data. 2) x y Answer: Determine which scatterplot shows the strongest linear correlation. 3) Which shows the strongest linear correlation? A) B) C) Answer: A 1

2 Find the value of the linear correlation coefficient r. 4) x y ) Answer: -0.2 x y Answer: ) The paired data below consist of the temperatures on randomly chosen days and the amount a certain kind of plant grew (in millimeters): Temp Growth Answer: ) Two separate tests are designed to measure a student's ability to solve problems. Several students are randomly selected to take both tests and the results are shown below. Test A Test B Answer: Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of r given the number of pairs of data n and the significance level. 8) n = 40, = 0.01 Answer: r = ± ) n = 14, = 0.05 Answer: r = ±0.532 Use the given data to find the best predicted value of the response variable. 10) Four pairs of data yield r = and the regression equation y = 3x. Also, y = What is the best predicted value of y for x = 4.3? 11) 12) 13) Answer: Eight pairs of data yield r = and the regression equation y = x. Also, y = What is the best predicted value of y for x = 6.8? Answer: The regression equation relating dexterity scores (x) and productivity scores (y) for the employees of a company is y = x. Ten pairs of data were used to obtain the equation. The same data yield r = and y = What is the best predicted productivity score for a person whose dexterity score is 32? Answer: Six pairs of data yield r = and the regression equation y = 4x - 2. Also, y = What is the best predicted value of y for x = 5? Answer:

3 14) Ten pairs of data yield r = and the regression equation y = 2 + 3x. Also, y = 5.0. What is the best predicted value of y for x = 2? Answer: 5.0 Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. 15) x y Answer: y = 3.0x 16) x y Answer: y = x 17) x y Answer: y = x 18) Managers rate employees according to job performance and attitude. The results for several randomly selected employees are given below. Answer: Performance Attitude y = x Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. 19) A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and production is not really out of control. At the 0.01 level of significance, test the manager's claim. Answer: H0: p = H1: p > Test statistic: z = P-value: p = Critical value: z = Fail to reject null hypothesis. There is not sufficient evidence to warrant rejection of the manager's claim that production is not really out of control. 20) A supplier of digital memory cards claims that no more than 1% of the cards are defective. In a random sample of 600 memory cards, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, test the supplier's claim that no more than 1% are defective. Answer: H0: p = H1: p > Test statistic: z = P-value: p = Critical value: z = Reject null hypothesis. There is sufficient evidence to warrant rejection of the claim that no more than 1% are defective. Note: Since the term "no more than" is used, the translation is p Therefore, the competing hypothesis is p > Find the P-value for the indicated hypothesis test. 21) A medical school claims that more than 28% of its students plan to go into general practice. It is found that among a random sample of 130 of the school's students, 32% of them plan to go into general practice. Find the P-value for a test of the school's claim. Answer:

4 22) In a sample of 88 children selected randomly from one town, it is found that 8 of them suffer from asthma. Find the P-value for a test of the claim that the proportion of all children in the town who suffer from asthma is equal to 11%. Answer: Solve the problem. 23) A manufacturer finds that in a random sample of 100 of its CD players, 96% have no defects. The manufacturer wishes to make a claim about the percentage of nondefective CD players and is prepared to exaggerate. What is the highest rate of nondefective CD players that the manufacturer could claim under the following condition? His claim would not be rejected at the 0.05 significance level if this sample data were used. Assume that a left-tailed hypothesis test would be performed. Answer: 98.2% Determine whether the given conditions justify testing a claim about a population mean µ. 24) The sample size is n = 21, = 6.07, and the original population is normally distributed. Answer: Yes 25) The sample size is n = 19, is not known, and the original population is normally distributed. Answer: Yes Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. 26) Various temperature measurements are recorded at different times for a particular city. The mean of 20 C is obtained for 60 temperatures on 60 different days. Assuming that = 1.5 C, test the claim that the population mean is 22 C. Use a 0.05 significance level. Answer: H0: µ = 22; H1: µ 22. Test statistic: z = P-value: Because the P-value is less than the significance level of = 0.05, we reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the population mean temperature is 22 C. Test the given claim. Use the P-value method or the traditional method as indicated. Identify the null hypothesis, alternative hypothesis, test statistic, critical value(s) or P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. 27) A simple random sample of 15-year old boys from one city is obtained and their weights (in pounds) are listed below. Use a 0.01 significance level to test the claim that these sample weights come from a population with a mean equal to 149 lb. Assume that the standard deviation of the weights of all 15-year old boys in the city is known to be 16.2 lb. Use the traditional method of testing hypotheses Answer: H0: µ = 149 lb H1: µ 149 lb Test statistic: z = 0.91 Critical-values: z = ± Do not reject H0; At the 1% significance level, there is not sufficient evidence to warrant rejection of the claim that these sample weights come from a population with a mean equal to 149 lb. Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither. 28) Claim: µ = 981. Sample data: n = 24, x = 972, s = 26. The sample data appear to come from a normally distributed population with = 28. Answer: Normal 4

5 29) Claim: µ = 120. Sample data: n = 11, x = 100, s = The sample data appear to come from a normally distributed population with unknown µ and. Answer: Student t Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion. 30) Test the claim that for the population of female college students, the mean weight is given by µ = 132 lb. Sample data are summarized as n = 20, x = 137 lb, and s = 14.2 lb. Use a significance level of = 0.1. Answer: = 0.1 Test statistic: t = 1.57 P-value: p = Critical values: t = ±1.729 Because the test statistic, t < 1.729, we fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that µ = 132 lb. 31) Test the claim that for the adult population of one town, the mean annual salary is given by µ = $30,000. Sample data are summarized as n = 17, x = $22,298, and s = $14,200. Use a significance level of = Answer: = 0.05 Test statistic: t = P-value: p = Critical values: t = ±2.120 Because the test statistic, t < , we reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that µ = $30,000. Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim. 32) Use a significance level of = 0.05 to test the claim that µ = The sample data consist of 15 scores for which x = 39 and s = 7.8. Use the traditional method of testing hypotheses. Answer: H0: µ = H1: µ Test statistic: t = Critical values: t = ± Reject H0. There is sufficient evidence to warrant rejection of the claim that the mean is ) In tests of a 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. Tests on a random sample of 10 modified components resulted in the following times (in hours) between failures 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 of testing hypotheses. Answer: H0: µ = 520 hrs. H1: µ > 520 hrs. Test statistic: t = < P-value < Reject H0. There is sufficient evidence to support the claim that the mean is greater than 520 hours. 34) Use a significance level of = 0.01 to test the claim that µ > The sample data consist of 9 scores for which x = 3.1 and s = 0.6. Use the traditional method of testing hypotheses. Answer: H0: µ = H1: µ > Test statistic: t = Critical value: t = Fail to reject H0. There is not sufficient evidence to support the claim that the mean is greater than

6 Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. 35) n = 51, x = 27; 95% confidence Answer: < p < ) Of 234 employees selected randomly from one company, 12.82% of them commute by carpooling. Construct a 90% confidence interval for the true percentage of all employees of the company who carpool. Answer: 9.23% < p < 16.4% Solve the problem. 37) In a certain population, body weights are normally distributed with a mean of 152 pounds and a standard deviation of 26 pounds. How many people must be surveyed if we want to estimate the percentage who weigh more than 180 pounds? Assume that we want 96% confidence that the error is no more than 4 percentage points. Answer: 658 Use the confidence level and sample data to find a confidence interval for estimating the population µ. Round your answer to the same number of decimal places as the sample mean. 38) Test scores: n = 92, x = 90.6, = 8.9; 99% confidence Answer: 88.2 < µ < ) A random sample of 130 full-grown lobsters had a mean weight of 21 ounces and a standard deviation of 3.0 ounces. Construct a 98% confidence interval for the population mean µ. Answer: 20 oz < µ < 22 oz Use the given information to find the minimum sample size required to estimate an unknown population mean µ. 40) How many women must be randomly selected to estimate the mean weight of women in one age group. We want 90% confidence that the sample mean is within 3.7 lb of the population mean, and the population standard deviation is known to be 28 lb. Answer: 155 Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume that the population has a normal distribution. 41) A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 225 milligrams with s = 15.7 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs. Answer: mg < µ < mg 42) The football coach randomly selected ten players and timed how long each player took to perform a certain drill. The times (in minutes) were: Determine a 95% confidence interval for the mean time for all players. Answer: 8.15 min < µ < min Find the indicated probability. 43) A die with 6 sides is rolled. What is the probability of rolling a number less than 5? Answer: 2 3 6

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