Chapter7 Sampling distributions
Business Statistics I –QMS102
Chapter8
Chapter8 Sampling distributions Outcomes 1. 2. 3. 4. 5.
Discuss the importance of sampling and the main reasons for sampling Explain the definition of sampling distributions Explain the central limit theorem Calculate the standard error of the mean Use the central limit theorem to determine probability of selecting possible sample means from a specified population
Sampling 1. Select a portion of the population that is most representative of the population 2. Provide sufficient information so that conclusions (inferences) can be drawn about the characteristics of the population Reasons for Sampling 1. To contact the whole population would often be time consuming 2. The cost of studying all the items in a population may be prohibitive 3. Some tests could be destructive in nature If the wine tasters in Niagara-on-the –Lake drank all the wine to evaluate the vintage, they would consume the entire crop, and none would be available for sale 4. The sample results are adequate Example: The Federal Government uses a sample of grocery stores scattered throughout Canada to determine the monthly index of food prices. The prices of bread, milk and other major food items are included in the index. It is unlikely that the inclusion of all grocery stores in Canada would significantly affect the index, since the prices of milk, bread, and other major food usually do not vary by more than a few cents from one chain store to another. Sampling distributions 1. For any population data set, there is only one value of the population mean and population standard deviation. 2. Different samples drawn from the same population may result in different sample mean and sample standard deviation. 3. Sampling distribution is the distribution of all sample statistics.
Fall2012
Page#1
Business Statistics I –QMS102
Chapter8
Sampling Distribution of the Mean 1. The distribution of all possible sample means. N
Population mean
µ=
∑X i =1
i
N
The average of all sample means, µ X
Standard Error of the Mean The value of the standard deviation of all possible sample means
σX =
σ n
The Central Limit Theorem For a large sample size, the sampling distribution of x is approximately normal, irrespective of the shape of the population distribution. The mean and standard deviation of the sampling distribution of x are σ µ X = µ and σ X = n The sample size is usually considered to be large if n ≥ 30 Finding the Z for the Sampling Distribution of the Mean Z=
X − µX
σX
=
X −µ σ n
Note: The sampling distribution of the sample mean will follow a normal probability distribution under two conditions: 1. When the samples are taken from population known to follow a normal distribution. In this case, the size of the sample is not a factor. 2. When the shape of the population distribution is not known or the shape is known to be non-normal but the sample contains at least 30 observations.
Fall2012
Page#2
Business Statistics I –QMS102
Chapter8
Example1 The mean rent for a one-bedroom apartment in downtown Toronto is $1200 per month, with a standard deviation of $250. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 50 one-bedroom apartments and finding the mean to be at least $950 per month?
Fall2012
Page#3
Business Statistics I –QMS102
Chapter8
Example2 Time spent using e-mail per session is normally distributed with µ = 7.8 minutes and σ = 1.9 minutes. If you select a random sample of 16 sessions, Determine the probability that the sample mean is a. between 7.5 and 8.2 minutes b. between 7 and 7.8 minutes c. If you select a random of 100 sessions, what is the probability that the sample mean is between 7.5 and 8.2 minutes? d. Explain the difference in the results of (a) and (c).
Fall2012
Page#4
Business Statistics I –QMS102
Chapter8
Example3 The balances of all savings accounts in Ryerson Bank at Yonge and Dundas have an unknown distribution with its mean equal to $12450 and standard deviation equal to $4200. Determine the probability that the mean balance of a sample of 49 saving accounts selected from this bank will be a. more than $11,500 b. between $12,000 and $13,800 c. within $1500 of the population mean d. more than the population mean by at least $1000
Fall2012
Page#5
Business Statistics I –QMS102
Chapter8
Example4 The occurrence of the first breakdown of an automatic washing machine is normally distributed with a mean of 5.65 years and a standard deviation of 1.42 years. How long (full years) should these washing machines be guaranteed so that no more than a.10% b. 15% would require repairs during the warrantee period?
Fall2012
Page#6
Business Statistics I –QMS102
Chapter8
Example5 Ryerson Inc. is currently evaluating 20 cost-reducing proposals submitted by employees. Past experience has shown that the company implements 35% of such proposals. a. What is the probability that i. more than 8 ii. at least 6 proposals will be implemented? b. What is the probability that at least half of the proposals will be implemented? c. What is the expected number of proposals implemented?
Example6 The weight of mini-boxes of raisin has a mean of 15.4 grams and a standard deviation of 1.9 grams. What is the probability that a case of 100 boxes will weigh more than 1.5kilograms?
Fall2012
Page#7
Business Statistics I –QMS102
Chapter8
Example 7. Records indicate the amount spent by tourists at Eaton Centre follows a normal distribution. Assume that the population mean amount spent by each tourist is $281.60 with a standard deviation of $87.90. a. What is the probability that a randomly selected tourist who will spend less than $200 or more than $300 at Eaton Centre in this Saturday?
b. A sample of 16 tourists is selected, what is the probability that the sample mean amount spent by these tourists is more than the population mean by at least $35.00?
Fall2012
Page#8
Chapter8
Chapter8 Sampling distributions Outcomes 1. 2. 3. 4. 5.
Discuss the importance of sampling and the main reasons for sampling Explain the definition of sampling distributions Explain the central limit theorem Calculate the standard error of the mean Use the central limit theorem to determine probability of selecting possible sample means from a specified population
Sampling 1. Select a portion of the population that is most representative of the population 2. Provide sufficient information so that conclusions (inferences) can be drawn about the characteristics of the population Reasons for Sampling 1. To contact the whole population would often be time consuming 2. The cost of studying all the items in a population may be prohibitive 3. Some tests could be destructive in nature If the wine tasters in Niagara-on-the –Lake drank all the wine to evaluate the vintage, they would consume the entire crop, and none would be available for sale 4. The sample results are adequate Example: The Federal Government uses a sample of grocery stores scattered throughout Canada to determine the monthly index of food prices. The prices of bread, milk and other major food items are included in the index. It is unlikely that the inclusion of all grocery stores in Canada would significantly affect the index, since the prices of milk, bread, and other major food usually do not vary by more than a few cents from one chain store to another. Sampling distributions 1. For any population data set, there is only one value of the population mean and population standard deviation. 2. Different samples drawn from the same population may result in different sample mean and sample standard deviation. 3. Sampling distribution is the distribution of all sample statistics.
Fall2012
Page#1
Business Statistics I –QMS102
Chapter8
Sampling Distribution of the Mean 1. The distribution of all possible sample means. N
Population mean
µ=
∑X i =1
i
N
The average of all sample means, µ X
Standard Error of the Mean The value of the standard deviation of all possible sample means
σX =
σ n
The Central Limit Theorem For a large sample size, the sampling distribution of x is approximately normal, irrespective of the shape of the population distribution. The mean and standard deviation of the sampling distribution of x are σ µ X = µ and σ X = n The sample size is usually considered to be large if n ≥ 30 Finding the Z for the Sampling Distribution of the Mean Z=
X − µX
σX
=
X −µ σ n
Note: The sampling distribution of the sample mean will follow a normal probability distribution under two conditions: 1. When the samples are taken from population known to follow a normal distribution. In this case, the size of the sample is not a factor. 2. When the shape of the population distribution is not known or the shape is known to be non-normal but the sample contains at least 30 observations.
Fall2012
Page#2
Business Statistics I –QMS102
Chapter8
Example1 The mean rent for a one-bedroom apartment in downtown Toronto is $1200 per month, with a standard deviation of $250. The distribution of the monthly costs does not follow the normal distribution. In fact, it is positively skewed. What is the probability of selecting a sample of 50 one-bedroom apartments and finding the mean to be at least $950 per month?
Fall2012
Page#3
Business Statistics I –QMS102
Chapter8
Example2 Time spent using e-mail per session is normally distributed with µ = 7.8 minutes and σ = 1.9 minutes. If you select a random sample of 16 sessions, Determine the probability that the sample mean is a. between 7.5 and 8.2 minutes b. between 7 and 7.8 minutes c. If you select a random of 100 sessions, what is the probability that the sample mean is between 7.5 and 8.2 minutes? d. Explain the difference in the results of (a) and (c).
Fall2012
Page#4
Business Statistics I –QMS102
Chapter8
Example3 The balances of all savings accounts in Ryerson Bank at Yonge and Dundas have an unknown distribution with its mean equal to $12450 and standard deviation equal to $4200. Determine the probability that the mean balance of a sample of 49 saving accounts selected from this bank will be a. more than $11,500 b. between $12,000 and $13,800 c. within $1500 of the population mean d. more than the population mean by at least $1000
Fall2012
Page#5
Business Statistics I –QMS102
Chapter8
Example4 The occurrence of the first breakdown of an automatic washing machine is normally distributed with a mean of 5.65 years and a standard deviation of 1.42 years. How long (full years) should these washing machines be guaranteed so that no more than a.10% b. 15% would require repairs during the warrantee period?
Fall2012
Page#6
Business Statistics I –QMS102
Chapter8
Example5 Ryerson Inc. is currently evaluating 20 cost-reducing proposals submitted by employees. Past experience has shown that the company implements 35% of such proposals. a. What is the probability that i. more than 8 ii. at least 6 proposals will be implemented? b. What is the probability that at least half of the proposals will be implemented? c. What is the expected number of proposals implemented?
Example6 The weight of mini-boxes of raisin has a mean of 15.4 grams and a standard deviation of 1.9 grams. What is the probability that a case of 100 boxes will weigh more than 1.5kilograms?
Fall2012
Page#7
Business Statistics I –QMS102
Chapter8
Example 7. Records indicate the amount spent by tourists at Eaton Centre follows a normal distribution. Assume that the population mean amount spent by each tourist is $281.60 with a standard deviation of $87.90. a. What is the probability that a randomly selected tourist who will spend less than $200 or more than $300 at Eaton Centre in this Saturday?
b. A sample of 16 tourists is selected, what is the probability that the sample mean amount spent by these tourists is more than the population mean by at least $35.00?
Fall2012
Page#8
