An importer of Herbs and Spices claims that average weight of packets of Saffron is 20 grams. However packets are actually filled to an average weight, 19.5 grams and standard deviation, σ=1.8 gram. A random sample of 36 packets is selected, calculate A- The probability that the average weight is 20 grams or more; B- The two limits within which 95% of all packets weight; C- The two limits within which 95% of all weights fall (n=36); D- If the size of the random sample was 16 instead of 36 how would this affect the results in (a), (b) and (c)? (State any assumptions made)

Samples The means of these samples 1 and 3 2 1 and 5 3 1 and 7 4 1 and 9 5 3 and 5 4 3 and 7 5 3 and 9 6 5 and 7 6 5 and 9 7 7 and 9 8

Example Random samples of size n =2 are drawn from a finite population that consists of the numbers 2, 4, 6 and 8 without replacement. a-) Calculate the mean and the standard deviation of this population b-) List six possible random samples of size n=2 that can be drawn from this population and calculate their means. c-) Use the results in b-) to construct the sampling distribution of the mean. d-) Calculate the standard deviation of the sampling distribution.

Example 2: Tuition Cost The mean tuition cost at state universities throughout the USA is 4,260 USD per year (2002 year figures). Use this value as the population mean and assume that the population standard deviation is 900 USD. Suppose that a random sample of 50 state universities will be selected. A-) Show the sampling distribution of x̄ (where x̄ is the sample mean tuition cost for the 50 state universities) B-) What is the probability that the random sample will provide a sample mean within 250 USD of the population mean? C-) What is the probability that the simple random sample will provide a sample mean within 100 USD of the population mean?

Example 1: A random variable of size 15 is taken from normal distribution with mean 60 and standard deviation 4. Find the probability that the mean of the sample is less than 58.

Example 4: Suppose we have selected a random sample of n=36 observations from a population with mean equal to 80 and standard deviation equal to 6. Q: Find the probability that x̄ will be larger than 82.

Example 5: Ping-Pong Balls The diameter of a brand of Ping-Pong balls is approximately normally distributed, with a mean of 1.30 inches and a standard deviation of 0.04 inch. If you select a random sample of 16 Ping-Pong balls, A-) What is the sampling distribution of the sample mean? B-) What is the probability that sample mean is less than 1.28 inches? C-) What is the probability that sample mean is between 1.31 and 1.33 inches? D-) The probability is 60% that sample mean will be between what two values, symmetrically distributed around the population mean?

Example 6: E-Mails Time spent using e-mail per session is normally distributed, with a mean of 8 minutes and a standard deviation of 2 minutes. If you select a random sample of 25 sessions, A-) What is the probability that sample mean is between 7.8 and 8.2 minutes? B-) What is the probability that sample mean is between 7.5 and 8.0 minutes? C-) If you select a random sample of 100 sessions, what is the probability that sample mean is between 7.8 and 8.2 minutes? D-) Explain the difference in the results of (A) and (C).

Types of Survey Errors Coverage error Non response error Sampling error Measurement error Excluded from frame Follow up on nonresponses Random differences from sample to sample Bad or leading question

Z X Population Distribution Sampling Distribution Standard Normal Distribution ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? X Z

Sampling Distribution Properties As n increases, decreases Larger sample size Smaller sample size

Sampling Distribution Properties Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (has the same mean) Variation:

How Large is Large Enough? For most distributions, n ≥ 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n ≥ 15 For normal population distributions, the sampling distribution of the mean is always normally distributed

Example : Paint Supply The manager of a paint supply store wants to estimate the actual amount of paint contained in 1-gallon cans purchased from a nationally known manufacturer. The manufacturer’s specifications state that the standard deviation of the amount of paint is equal to 0.02 gallon. A random sample of 50 cans is selected, and the sample mean amount of paint per 1-gallon can is 0.995 gallon. A-) Construct a 99% Confıdence Interval estimate for the population mean amount of paint included in a 1-gallon can B-) On the basis of these results, do you think that the manager has a right to complain to the manufacturer? Why? C-) Must you assume that the population amount of paint per can is normally distributed here? Explain. D-) Construct a 95% confidence Interval estimate. How does this change your answer to (B)?

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?

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 – 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 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 50 40 95 120 90 85 110 65 45 ∑x =2700

You recently received a job with a company that manufactures an automobile antitheft device. To conduct an advertising campaign for the product, you need to make a claim about the number of automobile thefts per year. Since the population of various cities in the United States varies, you decide to use rates per 10,000 people. (The rates are based on the number of people living in the cities.) Your boss said that last year the theft rate per 10,000 people was 44 vehicles. You want to see if it has changed. The following are rates per 10,000 people for 36 randomly selected locations in the United States. Assume σ = 30.3 55 42 125 62 134 73 39 69 23 94 73 24 51 55 26 66 41 67 15 53 56 91 20 78 70 25 62 115 17 36 58 56 33 75 20 16

Using this information, answer these questions. What hypothesis would you use? Is the sample considered small or large? What assumption must be met before the hypothesis test can be conducted? Which probability distribution would you use? Would you select a one –or two –tailed test? Why? What critical value(s) would you use? Conduct a hypothesis test. What is your decision? What is your conclusion? Write a brief statement summarizing your conclusion. If you lived in a city whose population was about 50,000, how many automobile thefts per year would you expect to occur?

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 – 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.

Exercise - 1 One Tailed Test TEST at the 5% level whether the single sample value of 172 comes from a normal population with mean µ= 150 and variance σ2=100.

Exercise – 3 The 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