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.

Exercise - 3

An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the upper end point in a 99% confidence interval for the average income?

Exercise - 4

An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the width of the 90% confidence interval?

Exercise - 5

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant for a 95% confidence interval, approximately how large a sample did her assistant use to determine the interval estimate?

Exercise – Sports

A researcher hypothesizes that the average number sports that colleges offer for males is greater than the average number sports that colleges offer for females. A sample of the number of sports offered by colleges is shown.At α = 0.10, is there enough evidence to support the claim

6 11 11 8 15 6 8 11 13 8

6 14 8 12 18 7 5 13 14 6

6 9 5 6 9 6 5 5 7 6

6 9 18 7 6 10 7 6 5 5

15 6 11 5 5 16 10 7 8 5

9 9 5 5 8 7 5 5 6 5

8 9 6 11 6 9 18 13 7 10

9 5 11 5 8 7 8 5 7 6

7 7 5 10 7 11 4 6 8 7

10 7 10 8 11 14 12 5 8 5

Male Female

STEP BY STEPCritical Value Approach to Hypothesis Testing1- 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 wageThe 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 Freq.8.35-8.43 28.44-8.52 68.53-8.61 128.62-8.70 188.71-8.79 108.80-8.88 2

Total: 50

M fM fM2 _______8.39 16.78 140.78428.48 50.88 431.46248.57 102.84 881.33888.66 155.88 1349.92088.75 87.5 765.6258.84 17.68 156.2912

431.56 3725.4224

Exercise – Athletic ShoesA researcher claims that the average cost of men`s athletic shoes is less than 80 USD. He selects a random sample of 36 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. Assume σ=19.2

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 70

∑x =2700

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 AccessZ-test for Proportion

Of 2000 adults, 1540 said that they wanted Internet Access so, they could check personal e-mail 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 – Life Guards

A researcher wishes to test the claim that the average age of lifeguards in a city is greater than 24 years. She selects a sample of 36 guards and finds the mean of the sample to be 24.7 years with a standard deviation of 2 years. Is there evidence to support the claim at α = 0.05?Use p-value method.

Exercise – Assist. Prof.

A researcher reports that the average salary of assistant professors is more than 42,000 TL. A sample of 30 assistant professors has a mean salary of 43,260 TL. At α = 0.05, Test the claim that assistant professors earn more than 42,000 TL a year. The population standard deviation is 5,230 TL.

Exercise – Wind Speed

A researcher claims that the average wind speed in a certain city 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the sample is 0.6 mile per hour. At α = 0.05, is there enough evidence to reject the claim?Use p-value method.

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.

A 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. 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

CORRECT DECISION TYPE I ERROR(α ERROR)

TYPE II ERROR(β ERROR) CORRECT DECISION

H0 IS TRUE

H0 IS FALSE

REJECT H0 ACCEPT H0

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 RestaurantYou 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 25 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.

Exercise –McDonaldOne Tailed Test

In one past study, McDonald’s had a mean service time of 174.22 seconds. Suppose that this company began a quality improvement effort to reduce the service time and selected a sample of 25 stores. The sample mean has been calculated as 162.96 seconds and sample standard deviation is 20.2 seconds. You wish to determine whether the new drive-through process has a mean that is less than 174.22 seconds.

Exercise –Fast FoodOne Tailed Test for Proportion

A fast food chain has developed a new process to ensure that orders at the drive-through are filled correctly. The business problem is defined as determining whether the new process can increase the percentage of orders processed correctly. The previous process filled orders correctly 85% of the time. Data are collected from a sample of 100 orders using the new process. The results indicate that 94 orders were filled correctly. At the 0.01 level of significance, can you conclude that the new process has increased the proportion of orders filled correctly?

Exercise - 2One Tailed Test

TEST at the 1% level whether the single sample value 54 has been drawn from a normal population with mean 65 and variance 30 or whether the mean is less than 65.

Exercise – 3The manager of the women`s dress department of a department store wants to know whether the true average number of women`s dresses sold per day is 24.If in a random sample of 36 days the average number of dresses sold is 23 with a standard deviation of 7 dresses,Is there, at the 0.05 level of significance, sufficient evidence to reject the null hypothesis that µ=24?

Exercise – 4

Exercise – 5

Exercise – 6

Exercise – 7