Exam 2 Statistics
A die is tossed 330 times with the results below. x: 1, 2, 3, 4, 5, 6 f: 57, 57, 66, 47, 62, 41 Is this a balanced die? Use a 0.05 level of significance. (a) Identify the null and alternative hypothesis (b) Identify the critical region (c) Find the test statistic (d) What is an appropriate conclusion?
(a) H0: The die is balanced (all faces occur with the same frequency) H1: The die is not balanced (the faces do not all occur with the same frequency (b) χ^2 > 11.07 (c) 7.96 (d) Do not reject H0 and conclude that there is not significant evidence that the die is not balanced. Thus, there is not sufficient evidence to reject the assumption that the die is balanced 10.14.81, HW8, 5
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. No more than 17% of the faculty at the local university contributed to the annual giving fund.
H0: μ = .17, H1: μ > .17, The critical region is in the right tail 10.R.97, HW5, 6
In an experiment to study the dependence of hypertension on smoking habits, the data below were taken on 172 individuals. Test the hypothesis that the presence or absence of hypertension is independent of smoking habits. Use a 0.01 level of significance. Non-smokers, Moderate Smokers, Heavy Smokers Hypertension: 22, 32, 28 No hypertension: 41, 27, 22 (a) State the null and alternative hypothesis for this test (b) Identify the critical region (c) Find the test statistic (d) What is the appropriate conclusion for this test?
(a) H0: The presence or absence of hypertension is independent of smoking habits. H1: The presence or absence of hypertension is not independent of smoking habits. (b) χ^2 > 9.21 (c) 6.45 (d) Do not reject H0 and conclude that there is not significant evidence that the presence or absence of hypertension is not independent of smoking habits 10.14.86, HW8, 6
At a certain college, it is estimated that at most 28% of the students ride bicycles to class. Does this seem to be a valid estimate if, in a random sample of 95 college students, 34 are found to ride bicycles to class? Use a 0.01 level of significance. (a) Let a success be a student that rides a bicycle to class. Identify the null and alternative hypotheses. (b) Identify the critical region. (c) What is the appropriate conclusion for this test?
(a) H0: p = 0.28 H1: p > 0.28 (b) z > 2.33 (c) Do not reject H0 and conclude that there is not sufficient evidence that more than 28% of the students ride bicycles to class. Thus, there is not sufficient evidence to reject the estimate that at most 20% of the students ride bicycles to class 10.9.60, HW5, 5
In a study to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant, it is found that 75 of 120 urban residents favor the construction while only 78 of 150 suburban residents are in favor. Is there a significant difference between the proportions of urban and suburban residents who favor construction of the nuclear plant? Make use of a P-value. (a) Let urban residents correspond to population 1, let suburban residents correspond to population 2, and let a success be a resident who favors the construction of a nuclear power plant. Identify the null and alternative hypotheses. (b) Find the test statistic. (c) Find the P-value. (d) What is the appropriate conclusion for this test?
(a) H0: p1 = p2 H1: p1 ≠ p2 (b) z = 1.73 (c) P = 0.084 (d) Reject H0 and conclude that there is a significant difference between the proportions of urban and suburban residents who favor construction of the nuclear plant. In general, H0 would be rejected for significance levels greater than the P-value (aka reject cause p^ > P-value, 0.567 > 0.084) 10.9.63, HW6, 2
Test the hypothesis that the average content of containers of a particular lubricant is 10.1 liters if the contents of a random sample of 10 containers are 10.1, 9.7, 10.5, 10.4, 9.8, 9.7, 9.4, 9.9, 10.7, and 9.7 liters. Use a 0.10 level of significance and assume that the distribution of contents is normal. (a) Identify the null and alternative hypotheses. (b) Identify the critical region. (c) Find the test statistic. (d) What is the appropriate conclusion for this test?
(a) H0: μ = 10.1 H1: μ ≠ 10.1 (b) t < -1.83 or t > 1.83 (c) -0.83 (d) Do not reject H0 and conclude that the average content of containers of a particular lubricant is not significantly different from 10 ounces 10.7.23, HW6, 8
Engineers at a large automobile manufacturing company are trying to decide whether to purchase brand A or brand B tires for the company's new models. To help them arrive at a decision, an experiment is conducted using 12 of each brand. The tires are run until they wear out, with the accompanying results. Test the hypothesis that there is no difference in the average wear of the two brands of tires. Assume the populations to be approximately normally distributed with equal variances. Use a P-value. Brand A: x1=38,400 kilometers, s1=5300 kilometers Brand B: x1=50,700 kilometers, s1=5700 kilometers (a) Let sample 1 be the mice that received treatment and let sample 2 be the mice that did not receive treatment. State the null and alternative hypotheses. (b) Determine the test statistic
(a) H0: μ1 - μ2 = 0 H1: μ1 - μ2 > 0 (b) t=-.93 10.7.36, HW7, 6
To find out whether a new serum will arrest leukemia, 9 mice, all with an advanced stage of the disease, are selected. Five mice receive the treatment and 4 do not. Survival times, in years, from the time the experiment commenced are provided. At the 0.05 level of significance, can the serum be said to be effective? Assume the populations to be normally distributed with equal variances. Treatment 2.4 5.8 1.1 4.9 1.3 No Treatment 1.6 0.9 3.1 3.4 (a) Let sample 1 be the mice that received treatment and let sample 2 be the mice that did not receive treatment. State the null and alternative hypotheses. (b) Identify the critical region (c) Determine the test statistic (d) State the proper conclusion
(a) H0: μ1 - μ2 = 0 H1: μ1 - μ2 > 0 (b) t>1.895 (c) t=0.70 (d) Do not reject H0. There is insufficient evidence to conclude that the serum is effective 10.7.35, HW7, 5
Five samples of a ferrous-type substance were used to determine if there is a difference between a laboratory chemical analysis and an X-ray fluorescence analysis of the iron content. Each sample was split into two subsamples and the two types of analysis were applied, with the accompanying results. Assuming that the populations are normal, test at the 0.05 level of significance whether the two methods of analysis give, on the average, the same result. Data on back (a) Let sample 1 be the mice that received treatment and let sample 2 be the mice that did not receive treatment. State the null and alternative hypotheses. (b) Identify the critical region (c) Determine the test statistic (d) State the proper conclusion
(a) H0: μD = 0 H1: μD ≠ 0 (b) t<-2.776 or t>2.776 (c) t=-.83 (d) Do not reject H0. There is insufficient evidence to conclude that the two types of analysis give different results 10.7.42, HW7, 7
A study was conducted to determine if the performance of a certain type of surgery on young horses had any effect on certain kinds of blood cell types in the animal. Fluid samples were taken from each of six foals before and after surgery. The samples were analyzed for the number of postoperative white blood cell (WBC) leukocytes. A preoperative measure of WBC leukocytes was also measured, with the accompanying results. Use a paired sample t-test to determine if there is a significant change in WBC leukocytes with the surgery. Data on back (a) Let sample 1 be the presurgery data and let sample 2 be postsurgery data. State the null and alternative hypotheses. (b) Determine the test statistic
(a) H0: μD = 0 H1: μD ≠ 0 (b) t=-2.10 10.R.105, HW7, 8
Past experience indicates that the time required for high school seniors to complete a standardized test is a normal random variable with a standard deviation of 9 minutes. Test the hypothesis that σ < 9 against the alternative that σ < 9 if a random sample of the test times of 26 high school seniors has a standard deviation s equals 6.03. Use a 0.05 level of significance. (a) Identify the null and alternative hypothesis. (b) Identify the critical region (c) Find the test statistic (d) What is an appropriate decision for this test?
(a) H0: σ^2 = 81 H1: σ^2 < 81 (b) χ^2 < 14.61 (c) 11.22 (d) The test statistic is significant at the 0.05 level. Based on the test statistic, reject H0 and conclude that there is sufficient evidence that σ < 9 10.10.68, HW8, 3
A large manufacturing firm is being charged with discrimination in its hiring practices. (a) What hypothesis is being tested if a jury commits a type I error by finding the firm guilty? (b) What hypothesis is being tested if a jury commits a type II error by finding the firm guilty?
(a) The firm is not guilty (b) The firm is guilty 10.3.3, HW5, 3
The proportion of adults living in a small town who are college graduates is estimated to be p = 0.4. To test this hypothesis, a random sample of 250 adults is selected. If the number of college graduates in the sample is anywhere in the fail-to-reject region defined to be 84 ≤ x ≤ 116, where x is the number of college graduates in our sample, we shall not reject the null hypothesis that p = 0.4; otherwise, we shall conclude that p ≠ 0.4. Complete parts (a) through (c) below. Use the normal approximation. (a) Evaluate α assuming that p = 0.4. (b) Evaluate β for the alternatives p = 0.3 and p = 0.5. (c) Is this a good test procedure? Consider a value of alpha to be relatively small if it is less than 0.1000 and relatively large if it is greater than 0.1000.
(a) The probability of committing a type I error, α, is approximately 0.0332. (b) The probability of committing a type II error, beta, for the alternative p = 0.3 is approximately 0.1210 and for the alternative p = 0.5 is 0.1401. (c) This is not a good test procedure, because while α is relatively small, both values of β is relatively large 10.3.7, HW5, 4
An experiment was conducted to compare the alcohol content of soy sauce on two different production lines. Production was monitored eight times a day. The data are shown below. Assume both populations are normal. It is suspected that production line 1 is not producing as consistently as production line 2 in terms of alcohol content. Test the hypothesis that σ1 =σ2 against the alternative that σ1 ≠ σ2. Use a P-value. Production line 1: 0.51, 0.41, 0.43, 0.54, 0.41, 0.50, 0.54, 0.53 Production line 2: 0.37, 0.36, 0.37, 0.39, 0.36, 0.37, 0.38, 0.38 (a) Calculate the test statistic (b) Determine the P-value for this f-test (c) Reach a decision
(a) f = 30.92 (b) P-Value = 0.000 (c) Reject H0. There is sufficient evidence that σ1 ≠ σ2. There is sufficient evidence to conclude that production line 1 is not producing as consistently as production line 2 in terms of alcohol content 10.10.77-T, HW8, 4
A soft-drink machine at a steak house is regulated so that the amount of drink dispensed is approximately normally distributed with a mean of 230 milliliters and a standard deviation of 18 milliliters. The machine is checked periodically by taking a sample of 36 drinks and computing the average content. If x overbar falls in the interval 221 < x < than 239, the machine is thought to be operating satisfactorily; otherwise, the owner concludes that μ ≠ 230 milliliters. (a) Find the probability of committing a type I error when μ = 230 milliliters. (b) Find the probability of committing a type II error when μ = 248 milliliters.
(a) β = 0.0026 (b) β = 0.0013 10.3.16, HW6, 7
A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 8 of each brand, assigned at random to the left and right rear wheels of 8 taxis. The tires are run until they wear out and the distances, in kilometers, are recorded in the accompanying data set. Find a 99% confidence interval for μ1- μ2. Assume that the differences of the distances are approximately normally distributed. Data set on other side of notecard
-2786< μA - μB < 986 9.9.44, HW7, 4
A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 11 of each brand. The tires are run until they wear out. The results are given in the table below. Compute a 95% confidence interval for μA - μB assuming the populations to be approximately normally distributed. You may not assume that the variances are equal. Brand A: x1=35,400 kilometers, s1=4800 kilometers Brand B: x1=38,100 kilometers, s1=6300 kilometers
-7549< μA - μB < 1949 9.9.43, HW7, 3
A manufacturer of MP3 players conducts a set of comprehensive tests on the electrical functions of its product. All MP3 players must pass all tests prior to being sold. Of a random sample of 1100 MP3 players, 11 failed one or more tests. Find a 90% confidence interval for the proportion of MP3 players from the population that pass all tests.
0.985 < p < 0.995 9.11.54, HW5, 1
The measurements given below were recorded for the drying time, in hours, of a certain brand of latex paint. Assuming that the measurements represent a random sample from a normal population, find a 95% prediction interval for the drying time for the next trial of the paint. 6.2, 5.6, 1.9, 4.2, 4.8, 3.3, 4.8, 5.3, 4.5, 2.6, 4.6, 5.9, 5.1, 5.2, 4.4
1.93 < x0 < 7.19 9.7.12, HW6, 5
A random sample of 15 chocolate energy bars of a certain brand has, on average, 310 calories per bar, with a standard deviation of 20 calories. Construct a 99% confidence interval for the true mean calorie content of this brand of energy bar. Assume that the distribution of the calorie content is approximately normal.
294.6 < μ < 325.4 9.7.12, HW6, 4
A random sample of 30 chocolate energy bars of a certain brand has, on average, 300 calories per bar, with a standard deviation of 45 calories. Assume that the distribution of the calorie content is approximately normal. Construct a 99% confidence interval for σ.
33.50 < σ < 66.90 9.13.75, HW8, 1
A random sample of 22 graduates of a certain secretarial school typed an average of 74.6 words per minute with a standard deviation of 8.8 words per minute. Assuming a normal distribution for the number of words typed per minute, compute the 95% prediction interval for the next observed number of words per minute typed by a graduate of the secretarial school.
55.88 < x0 < 93.32 9.7.12, HW6, 6
A random sample of 21 graduates of a certain secretarial school typed an average of 74.7 words per minute with a standard deviation of 8.6 words per minute. Assuming a normal distribution for the number of words typed per minute, find a 95% confidence interval for the average number of words typed by all graduates of this school.
70.78 < μ < 78.62 9.7.10,HW6,3
Two catalysts in a batch chemical process are being compared for their effect on the output of the process reaction. A sample of 16 batches was prepared using catalyst 1 and gave an average yield of 96 with a sample standard deviation of 7. A sample of 18 batches was prepared using catalyst 2 and gave an average yield of 84 and a sample standard deviation of 2. Find a 90% confidence interval for the difference between the population means, assuming that the populations are approximately normally distributed with equal variances.
9.09< μ1 - μ2 < 14.91 9.9.38, HW7, 1
The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections. Find a 99% confidence interval for the difference μ2-μ1 between in the mean recovery times for the two medications, assuming normal populations with equal variances. Medication 1: n1=21, x1=12, s1^2=0.8 Medication 2: n2=10, x2=18, s2^2=0.9
9.09< μ1 - μ2 < 14.91 9.9.41, HW7, 2
The accompanying data represent the running times of files produced by two motion-picture companies. Assume that the running-time differences are approximately normally distributed. Construct a 90% confidence interval for (σ1)^2/(σ2)^2. If a 90% confidence interval for μ1-μ2 were created, should it be assumed that (σ1)^2 = (σ2)^2? Company 1: 108, 99, 106, 85, 93 Company 2: 102, 87, 124, 89, 179, 84, 117
CI: 0.017 < (σ1)^2/(σ2)^2 < 0.464 Since the interval does not include 1, it should not be assumed that (σ1)^2 = (σ2)^2 9.13.79, HW8, 2
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. At least 67% of next year's new cars will be in the compact and subcompact category.
H0: μ = .67, H1: μ < .67, The critical region is in the left tail 10.R.97, HW5, 6
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. The proportion of voters favoring the incumbent in the upcoming election is 0.68.
H0: μ = .68, H1: μ ≠ .68, The critical region is in both tails 10.R.97, HW5, 6
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. The mean snowfall at a certain lake during the month of February is 24.1 centimeters.
H0: μ = 24.1, H1: μ ≠ 24.1, The critical region is in both tails 10.R.97, HW5, 6
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. The average rib-eye steak at a certain steakhouse weighs at least 300 grams.
H0: μ = 300, H1: μ < 300, The critical region is in the left tail 10.R.97, HW5, 6
State the null and alternative hypotheses to be used in testing the given claim and determine generally where the critical region is located. On the average, children attend schools within 6.8 kilometers of their homes in the suburbs of a certain major city.
H0: μ = 6.8, H1: μ > 6.8, The critical region is in the right tail 10.R.97, HW5, 6
Ten engineering schools in a country were surveyed. The sample contained 300 electrical engineers, 60 being women; 225 chemical engineers, 20 being women. Compute a 90% confidence interval for the difference between the proportions of women in these two fields of engineering. Is there a significant difference between the two proportions?
The 90% confidence interval is 0.062 < p1 - p2 < 0.160 There is a reason to believe that there is a significant difference between the two proportions, because the confidence interval does not contain 0 9.11.66, HW6, 1
A study is to be made to estimate the proportion of residents of a certain city and its suburbs who favor the construction of a nuclear power plant near the city. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.10 of the true proportion of residents who favor the construction of the nuclear power plant?
The sample should be 97 people 9.11.64, HW5, 2
