Students require different amounts of sleep. A sample of 59

Students require different amounts of sleep. A sample of 59 students at a large midwest university reported the following hours of sleep the previous night. 4.5 4.7 5.0 5.0 5.3 5.5 5.5 5.7 5.7 5.7 6.0 6.0 6.0 6.0 6.3 6.3 6.3 6.5 6.5 6.5 6.7 6.7 6.7 6.7 7.0 7.0 7.0 7.0 7.3 7.3 7.3 7.3 7.5 7.5 7.5 7.5 7.7 7.7 7.7 7.7 8.0 8.0 8.0 8.0 8.3 8.3 8.3 8.5 8.5 8.5 8.5 8.7 8.7 9.0 9.0 9.0 9.3 9.3 10.0

One company, actively pursuing the making of green gasoline, starts with biomass in the form of sucrose and converts it into gasoline using catalytic reactions. At one step in a pilot plant process, the output includes carbon chains of length 3. Fifteen runs with same catalyst produced the product volumes (liter) 2.79 2.88 2.09 2.32 3.51 3.31 3.17 2.79 3.94 2.34 3.62 3.22 2.80 2.70 3.62 While mean product volume is the prime parameter, it is also important to control variation. Conduct a test with the intent of showing that the population standard deviation σ is less than .8 liter. Use α 0.05

A brochure inviting subscriptions for a new diet program states that the participants are expected to lose over 22 pounds in five weeks. Suppose that, from the data of the fiveweek weight losses of 56 participants, the sample mean and standard deviation are found to be 23.5 and 10.2 pounds, respectively. Could the statement in the brochure be substantiated on the basis of these findings? Test with α 0.05. Also calculate the P-value and interpret the result.

In a large-scale, cost-of-living survey undertaken last January, weekly grocery expenses for families with one or two children were found to have a mean of 148 and a standard deviation of 25. To investigate the current situation, a random sample of families with one or two children is to be chosen and their last week's grocery expenses are to be recorded. (a. How large a sample should be taken if one wants to be 95% sure that the error of aestimation of the population mean grocery expenses per week for families with one or two )children does not exceed 2? (Use the previous s as an estimate of the current σ .) (b. A random sample of 100 families is actually chosen, and from the data of their last week's bgrocery bills, the mean and the standard deviation are found to be 155 and 22, )respectively. Construct a 98% confidence interval for the current mean grocery expense per week for the population of families with one or two children.

Students are asked about the number of songs they downloaded from a pay-for-songs Web site the last month. From a random sample of 39 students, the sample mean was 4.7 with a standard deviation of 3.2. a.Obtain a 95% confidence interval for, µ , the mean number of songs downloaded by the population of all students. b. Does µ lie in your interval obtained in part (a)?

Biological oxygen demand (BOD) is an index of pollution that is monitored in the treated effluent of paper mills on a regular basis. From 43 determinations of BOD (in pounds per day) at a particular paper mill during the spring and summer months, the mean and standard deviation were found to be 3246 and 757, respectively. The company had set the target that the mean BOD should be 3000 pounds per day. Do the sample data indicate that the actual amount of BOD is significantly off the target? (Use α 0.05)

Refer to previous example. Along with the determinations of BOD, the discharge of suspended solids (SS) was also monitored at the same site. The mean and standard deviation of the 43 determinations of SS were found to be 5710 and 1720 pounds per day, respectively. Do these results strongly support the company's claim that the true mean SS is lower than 6000 pounds per day? (Answer by calculating and interpreting the p -value.)

Eighteen samples of seaweed, each weighing 50-kilograms, are collected to study the feasibility of extracting protein for use in animal feed. The 18 determinations of protein yield have sample mean 3.6 kilograms and standard deviation .8 kilogram. Determine a 95% confidence interval for the mean protein yield per 50 kilograms of seaweed. What did you assume about the population?

The petal width (mm) of one kind of iris has a normal distribution. Suppose that, from a random sample of widths, the @uni:href based 90% confidence interval for the population mean width is (16.8, 19.6) mm. Answer each question “yes,” “no,” or “can't tell,” and justify your answer. On the basis of the same sample: (a) Would H0 : µ 20 be rejected in favor of H0 : µ 20 at α 0.10 ? (b) Would H0 : µ 18 be rejected in favor of H0 : µ 18 at α 0.10 ? (c) Would H0 : µ 17 be rejected in favor of H0 : µ 17 at α 0.05? (d) Would H0 : µ 22 be rejected in favor of H0 : µ 22 at α 0.01?

One semester, an instructor taught the same computer course at two universities located in different cities. He was able to give the same final at both locations. The student's scores provided the summary statistics. Sample:1 Sample:2

Rural and urban students are to be compared on the basis of their scores on a nationwide musical aptitude test. Two random samples of sizes 90 and 100 are selected from rural and urban seventh-grade students. The summary statistics from the test scores are Urban Sample size Rural 90 Mean 76.4 81.2 8.2 7.6 Standard deviation 100 Construct a 98% confidence interval for the difference in population mean scores between urban and rural students.

Psychologists have made extensive studies on the relationship between child abuse and later criminal behavior. Consider a study that consisted of the follow-ups of 52 boys who were abused in their preschool years and 67 boys who were not abused. The data of the number of criminal offenses of those boys in their teens yielded the following summary statistics. Notabused Abused Mean 2.48 1.57 Standard deviation 1.94 1.31 Construct a 98% confidence interval for the difference in population mean scores between urban and rural students.

Random samples of 250 persons in the 30- to 40-year age group and 250 persons in the 60- to 70-year age group are asked about the average number of hours they sleep per night, and the following summary data are recorded. Hours of Sleep Age 8 8 Total 30-40 173 77 250 60-70 120 130 250 Total 293 207 500 Do these data demonstrate that the proportion of persons who have 8 hours of sleep per night is significantly higher for the age group 30 to 40 than that for the age group 60 to 70? Answer by calculating the p-value.

Exercise - 1 A package-filling process at a Cement company fills bags of cement to an average weight of µ but µ changes from time to time. The standard deviation is σ 3 pounds. A sample of 25 bags has been taken and their mean was found to be 150 pounds. Assume that the weights of the bags are normally distributed. Find the 90% confidence limits for µ.

STEP BY STEP Critical Value Approach to Hypothesis Testing 1- State Ho and H1 2- Choose level of significance, α Choose the sample size, n 3- Determine the appropriate test statistics and sampling distribution. 4- Determine the critical values that divide the rejection and non-rejection areas. 5- Collect the sample data, organize the results and compute the value of the test statistics. 6- Make the statistical decision and state the managerial conclusion If the test statistics falls into non-rejection region, DO NOT REJECT Ho If the test statistics falls into rejection region, REJECT Ho The managerial conclusion is written in the context of the real world problem.

Exercise –Hourly wage The president of a company states that the average hourly wage of his/her employees is 8.65 TRL. A sample of 50 employees has the distribution shown below. At α 0.05, is the president’s statement believable? Assume σ 0.105 TRL Class 8.35-8.43 8.44-8.52 8.53-8.61 8.62-8.70 8.71-8.79 8.80-8.88 Total: Freq. 2 6 12 18 10 2 50 M 8.39 8.48 8.57 8.66 8.75 8.84 fM 16.78 50.88 102.84 155.88 87.5 17.68 431.56 fM2 140.7842 431.4624 881.3388 1349.9208 765.625 156.2912 3725.4224

Exercise – Athletic Shoes A researcher claims that the average cost of men s athletic shoes is less than 80 USD. He selects a random sample of 35 pairs of shoes from a catalog and finds the following costs. Is there enough evidence to support the researcher s claim at α 0.10. 60 70 75 55 80 55 50 40 80 70 50 95 120 90 75 85 80 60 110 65 80 85 85 45 75 60 90 90 60 95 110 85 45 90 70 x 2630 (x x̄ )² 12824

Exercise –INFECTIONS A medical investigation claim that the average number infections per week at a hospital is 16.3. A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8 Is there evidence to reject the investigator’s claim at α 0.05? Assume the variable is normally distributed .

Exercise –Internet Access Z-test for Proportion Of 2000 adults, 1540 said that they wanted Internet Access so, they could check personal email while on vacation. A survey conducted in the previous year indicated that 75% of adults wanted Internet Access. Is there evidence that the percentage of adults who wanted Internet Access has changed from the previous year

Exercise – Starting Salary A job placement director claims that the average starting salary for nurses is 24,000 USD. A sample of 10 nurses salaries has a mean of 23,450 USD and a standard deviation of 400 USD. Is there enough evidence to reject the director s claim at α 0.05?

Exercise – Attorney Advertisements An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α 0.05, is there enough evidence to support the attorney s claim? Use the p-value method.

Exercise – Sugar Sugar is packed in 5 kg bags. An inspector suspects the bags may not contain 5 kg. A sample of 50 bags produces a mean of 4.6 kg and a standard deviation of 0.7 kg. Is there enough evidence to conclude that the bags do not contain 5 kg as stated at α 0.05? Also find the 95% CI of the true mean.

researcher thinks that if expectant mother use vitamin pills, the birth weight of the babies will increase . The average birth weight of the population is 3.5 kg. A H0:µ 3.5 and H1 : µ 3.5

An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of compact disks by using robots instead of humans for certain tasks. The mean number of defective disks per 1000 is 18. H0:µ 18 and H1 : µ 18

A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores was 73. H0:µ 73 and H1 : µ 73

ACCEPT H0 H0 IS TRUE H0 IS FALSE REJECT H0 CORRECT DECISION TYPE I ERROR (α ERROR) TYPE II ERROR (β ERROR) CORRECT DECISION If the null hypothesis is true and accepted or false and rejected the decision is in either case CORRECT. If the null hypothesis is true and rejected or false and accepted the decision is in either case in ERROR.

Example : Fast-Food Restaurant You are manager of a fast-food restaurant. You want to determine whether the waiting time to place an order has changed in the past month from its previous population mean value of 4.5 minutes. A-) State the null Hypothesis and Alternative Hypothesis From past experience, you can assume that the population is normally distributed with the standard deviation of 1.2 minutes. You select a sample of 30 orders during one-hour period. The sample mean is 5.1 minutes. B- Determine whether there is evidence at the 0.05 level of significance that the population mean waiting time to place an order has changed in the past month from its previous population mean value of 4.5 minutes. C- Find and use p-Value approach to test the Hypothesis.