1. In a random sample of n = 500 families owning television sets in the city of Hamilton, Canada, it is found that x = 340 subscribed to HBO.
(a) Find a 95% confidence interval for the actual proportion of families in this city who subscribe to HBO.
(b) How large sample is required if we want 95% confident that our estimate of p is within 0.02?
(c) How large sample is required if we want to be at least 95% confident that our estimate of p is within 0.02?
2. A manufacturer of compact disk players uses a set of comprehensive tests to access the electrical function of its product. All compact disk players must pass all tests prior to being sold. A random sample of 500 disk players resulted in 15 failing one or more tests. Find a 90% confidence interval for the proportion of compact disk players from the population that pass all tests.
3. A study is to be made to estimate to percentage of citizens in a town who favor having their water fluoridated. How large a sample is needed if one wishes to be at least 95% confident that our estimate is within 1% of the true percentage?
4. A sample of 81 college students finds that 27 attend 3 or more fun games each summer. Find a 95% confidence interval for the true population proportion of KSU students that attend 3 or more Braves games each summer.
5. A random sample of 140 college students finds that 113 of those students polled avoid classes that start before 9:30 AM. Construct a 99% confidence interval for the true population proportion of students who avoid classes that start before 9:30 AM.
6. In a random sample of 1,000 homes in a certain city, it is found that 228 are heated by oil. Find the 99% confidence interval for the proportion of homes in this city that are heated by oil.
7. Compute a 98% confidence interval for the proportion of defective items in a process when it is found that a sample of 100 yields 8 defectives.
8. A national electronics chain wishes to estimate the percentage of its customers who would pay a yearly membership fee in order to receive a 15% discount on all purchases of books, CD’s, DVD’s, games and software. Find the sample size needed to ensure that the sample estimate differs from the true population percentage by no more than 2.5%. Test at 95% confidence.
9. A study is to be made to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents in this city and its suburbs that favor the construction of the nuclear power plant?
15 Hypothesis Testing
Read and analyze each problem carefully. Perform a complete hypothesis testing for each item.
15.1 Test on the Mean of a Single Population
1. A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230.
2. A random sample of 100 recorded deaths in the United States during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean life span today is greater than 70 years? Use 0.05 level of significance.
3. The mean weight of the baggage carried into an airplane by individual passengers at Roxas City Airport is 19.8 kilograms. A statistician takes a random sample of 110 passengers and obtains a sample mean weight of 18.5 kilograms with standard deviation of 8.5 kilograms. Test the claim at α = 0.01 level of significance.
4. A national magazine claims that the average college student watches less television than the general public. The national average is 29.4 hours per week, with a standard deviation of 2 hours.
A sample of 20 college students has a mean of 27 hours. Is there enough evidence to support the claim at α = 0.01?
5. A job placement director claims that the average starting salary for nurses is $24,000. A sample of 10 nurses has a mean of $23,450 and a standard deviation of $400. Is there enough evidence to reject the director’s claim at α = 0.05?
6. According to the Department of Education, high school teachers work an average of 40 hours per week during the school year. A district supervisor of a certain schools surveyed 28 randomly selected teachers and found that they work an average of 42.6 hours a week and the standard deviation was 3.75 hours. Test if the mean number of hours worked by teachers in the supervisor’s school district differs from the national average. Use α = 0.01.
15.2 Test on the Difference of Means of Two Population
1. An agronomist randomly selected 20 matured calamansi trees of one variety and have a mean height of 10.8 feet with standard deviation of 1.25 feet, while 12 randomly selected calamansi trees of another variety have a mean height of 9.6 feet with standard deviation of 1.45 feet. Test whether the difference between the two sample means is significant. Use α = 0.05.
2. To compare freshmen’s knowledge of mathematics in two departments of the College of Business Administration, a certain professor in Statistics got a sample of economics and accountancy stu-dents and gave them special examination. A sample of 25 economics major stustu-dents had a mean score of 85.85 with standard deviation of 7.5. A sample of 28 accounting major students had a mean score of 90.5 with a standard deviation of 10.3. Is there a significant difference between the two sample means? Use 0.05 level of significance.
3. The daily sales of two newspaper vendors were recorded on a random basis. The result of two samples are as follows:
• Vendor I: Php 108, Php 125, Php 130, Php 116, Php 120, Php 119
• Vendor II: Php 113, Php 120, Php 120, Php 110, Php 125, Php 120
Is there a significant difference between the mean sales of the two newspaper vendors? Test at 0.01 significance level.
16 Correlation and Regression
Read, analyze and solve each problem carefully.
1. The data on yearly consumption of cigarettes in the Philippines and the percentage of the country’s population admitted to mental institutions as psychiatric cases were collected for 8 years. The correlation coefficient r = 0.61. What can we conclude about the data?
2. The data below consists of age and the income in thousands of dollars. Find the value of r and interpret the result.
Age 60 63 51 25 47 56 19 24 25 20 66 19 48 52 27
Income 43.4 18.8 14.4 29.4 19.4 83 10.4 12.6 36.4 29.6 17.2 17.2 67 33 37.4 3. A teacher is interested in knowing whether or not two IQ tests produce linearly related scores. A
sample of 10 students was taken randomly. Five students took Test 1 and 5 students took Test 2 in the morning. In the afternoon, those who took Test 1 took Test 2 and vice versa. The results are shown in the table below:
Student A B C D E F G H I J
Test 1 125 145 110 120 124 110 121 142 100 126 Test 2 114 127 126 116 108 100 129 131 96 113 (a) Plot a scatter diagram for these data.
(b) Solve for the value of r.
4. The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows
x 77 50 71 72 81 94 96 99 67 y 82 66 78 34 47 85 99 99 68 (a) Find the equation of the regression line.
(b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report but was ill at the time of the final examination.
(c) Estimate the midterm mark of the a student who received a grade of 65 on the final exami-nation.
5. A student counted the number of words in an essays she had written, recording the total every 10 lines.
No. of lines x 10 20 30 40 50 60 70 80 No. of words y 75 136 210 291 368 441 519 588 (a) Draw the scatter diagram to show the data.
(b) Calculate the equation of the regression lie.
(c) How many words (approximately) has she written if she writes:
i. 65 lines ii. 100 lines iii. 1,000 lines
References:
• Altares, P., et. al. (2013). Elementary Statistics with Computer Applications (2nd Edition). Rex Printing Company, Inc., Quezon City.
• Mendenhall, W., et. al. (1999). Introduction to Probability and Statistics. Brooks/Cole Publishing Company, USA.
• Reyes, C. and Saren, L. (2003). Elementary Statistics. National Bookstore, Mandaluyong City.
• Walpole, R., et. al. (2005). Probability and Statistics for Engineers and Scientists (7th Edition).
Pearson Education, Inc., New Jersey.