In Inferential Statistic, ESTIMATION (i) (ii) is called the True Population Mean and is called the True Population Proportion. You must also remember that are not the only population parameters. There are many other population parameters such as the median, mode, variance etc. but we will study estimation by estimating only the above two population parameters. The following are few examples of estimation: 1. An auto company may want to estimate the mean fuel consumption for a particular model of a car. 2. A manager may want to estimate the average time taken by new employees to learn a job. 3. A statistician may want to estimate the proportion of homes who were victims of flood in the Nuku alofa area in the first 3 months of 2012. Example 1 and 2 above are illustrations of estimating the True Population Mean and Example 3 is an illustration of estimating the True Population Proportion. NOTE WELL : If we can conduct a Census (a survey that include the entire population) each time we want to find a population parameter, then the estimation procedures we are trying to learn would be of no use. But since every Census is almost impossible or very expensive to carry out then Estimation would be the best available method for finding out the characteristics of a population. Margin of Error Whenever we use a point estimate, we calculate a quantity called the Margin of Error associated with that Point Estimate. For the estimation of the population mean, the margin of error is calculated as follows: 1.96 1.96 Here, is called a point estimator of which we will discuss later. Interval Estimate When we make an interval estimate, we usually construct an interval around the point estimate and make a probabilistic claim that the interval constructed contains the corresponding population parameter. Definition: Interval Estimate In interval estimation, an interval is constructed around the point estimate, which we state that this interval is likely to contain the corresponding population parameter. 1
Example Suppose the mean fuel consumption of a specific car model is 72 litres per week. Instead of saying that 72 litres per week is the mean fuel consumption, we can construct an interval by subtracting and adding the same number of litres to the mean, say 15 litres. Then the interval will be: 72 15 72 15.. 57 87 Now we can state that the interval constructed is likely to contain the population mean. We can state now that the mean fuel consumption per week for all cars of this specific model is between 57 litres and 87 litres. The value 57 litres is called the lower limit and 87 litres is the upper limit of the interval. The above procedure is called Interval Estimation. Question usually asked by students What number should we subtract and add to a Point Estimate to obtain an Interval Estimate? The answer to this question depends on two considerations. 1. The standard deviation of the sample mean 2. The level of confidence to be attached to the interval. NOTES The larger the standard deviation of the greater is the number to be added to and subtracted from the Point Estimate Therefore it is obvious that if the range over which can assume values is larger, then the interval constructed around must be wider to include. If we want to have a higher confidence in our interval, then the quantity to be subtracted and added must be larger. We always attach a probabilistic statement to the interval estimation. This probabilistic statement is given by what Statistician called a Confidence Level. An interval constructed based on this confidence level is called a Confidence Interval. Definition: Confidence Level and Confidence Interval Each interval is constructed with regard to a given Confidence level and is called a Confidence Interval. The confidence level associated with a confidence interval states how much confidence we have that this confidence interval contain the True Population Parameter. The confidence level is denoted by % NOTES 2
The Confidence Level is 1 100% and when expressed as a probability then we call it the Confidence Coefficients denoted by 1. The quantity denoted by is called the Significance Level and will be dealt with in detail when discussing Hypothesis Testing. Although any value of the Confidence Level can be chosen for the construction of Confidence Interval, the more common values are 90%, 95% and 99%. The corresponding Confidence Coefficients are: 0.90, 0.95 and 0.99 How to construct a Confidence Interval for the population mean for Large Sample. In the case of : We considered the sample size to be large when 30. The Central Limit Theorem states that for a large sample, the Sampling Distribution of the sample mean is approximately normal irrespective of the shape of the population from which the sample is drawn. Therefore, we will use the Normal Distribution to construct a Confidence Interval for when the sample size 30. We also know that the standard deviation of is If the population standard deviation is not known then we will use the sample standard deviation,, for. Consequently, we use: Confidence Interval for for Large Sample 1 100% The value of z used here is obtained from the Standard Normal Distribution Table for the given Confidence Level. When we considered our Confidence Interval for for large samples, we had the formulas: 3
The quantities is called the Maximum Error of Estimate and is denoted by. Definition: Maximum Error of Estimate The Maximum Error of Estimation for,, is the quantity that is subtracted and added to the value of to obtain a confidence interval for. NOTES The value of in the confidence interval formula is obtained from the Standard Normal Distribution Table for the given confidence level. The construction of a 95% confidence interval for the population mean we follow the procedures as set out below: 1. Divide 0.95 by 2 giving the value 0.4750 2. Locate o.4750 in the body of the normal distribution table and record the corresponding value of z. This value of z is 1.96 3. Then 1.96 1.96 For a 1 100% confidence level, the area between 1 and the area of the two tails is and each one has area In our 95% confidence interval above, we can work out the value of as follows: 1 0.95 1 0.95 0.05 0.05 0.025 2 2 The company Asian Paints at Small Industries is mixing a new type of paint. Before the company decides the price at which to sell their products, it wants to know the average price of all such paints in the market. The research section of the company took 36 comparable cans of paints and the related information on the price. This information produced a mean price of $70.50 per can of paint for this sample. It is known that the standard deviation of the prices of all paints of this type is $4.50. (i) What is the point estimate of the mean price for all such cans of paints? What is the margin of error for this estimate (ii) Construct a 90% confidence interval for the mean price of all such cans of paints Solution The information given above are as follows: 36 70.50 4.5 The standard deviation of is 4
4.50 0.75 36 (i) The point estimate of the mean price of all such paints is $70.50 i.e. 70.50 The Margin of Error associated with this point estimate of is 1.96 1.960.75 $1.47 The Margin of Error means that the mean price of all such paints is $70.50 give or take $1.47. Also note that the Margin of Error is simply the maximum error of estimate for a 95% confidence interval. (ii) The confidence level is 90% or 0.90. First we find the z value for a 90% confidence interval. First, divide 0.90 by 2:... 0.4500 Next, Locate 0.4500 on the body of the standard normal table. Since 0.4500 is not on the normal table, we can use the number closest to 0.4500 which is either 0.4495 or 0.4505 as an approximation. Any one of these two numbers can be used. If we use 0.4505 as an approximation for 0.4500, then the value of z for this number is 1.65 Finally, we substitute all the values in the confidence interval formula for. The 90% confidence interval for is: 70.50 1.650.75 70.50 1.24 The 90% confidence interval for is: 70.50 1.24 70.50 1.24.. $69.26 $71.74 NOTES The above example shows that we are 90% confident that the mean price of all such cans of paints is between $69.26 and $71.74 Because is a constant, we cannot say that the probability is 0.90 that the interval contains because either it contains or it does not. Consequently the probability is either 0 or 1. The interpretations of the 90% confidence interval is as follows: 5
a. If we take all possible samples of size 36 and construct a 90% confidence interval for around each sample mean we can expect that 90% of all these confidence intervals will include and 10% will not. b. Since we expect many samples of size 36 from the same population, we would have many sample means,,. for each sample. This will in turn tells us to construct a confidence interval around each mean. We conclude that 90% of these intervals will contain and 10% will not. Also, the population standard deviation was known. Sometimes this quantity is not known and therefore we will use the sample standard deviation s to estimate the population standard deviation and estimate the standard deviation by. Construction of a Confidence interval for when is not known. Suppose Small Businesses in Tonga with Negative Assets carried an average of $15,528 debt last year (2011). Assume that this mean was based on a random sample of 400 businesses and the standard deviation for this sample was $4,200. Construct a 99% confidence interval for last year mean debt for all such businesses in Tonga. Solution From the given information, 400 $15,528 $4,200 The confidence level = 99% or 0.99 First we find the standard deviation of. Because is not known, we will use as an estimator of. The value of is 4,200 $210 400 Because the sample is large 30 we will use the normal distribution to determine the confidence interval for. Dividing 0.99 by 2 we get 0.4950. From the normal distribution table, the z-value for 0.4950 is approximately 2.58. The confidence interval is 15,528 2.58210 15,528 541.80 $14,986.20 $16,069.80 Interval Estimation of a Population Mean for Small Samples: Recall that for large samples 30 whether or not is known, the normal distribution is used to estimate the population mean. Sometimes we may be required to select a small sample due to the nature of the experiment or the cost involved in sampling. For example, to carry out a test on a new drug on patients. Research might have to depend on the number of patients available (or willing to participate on the experiment) or the cost involved is too high. In such cases we are forced to take only small samples. 6
If the sample size is small, we can still use the normal distribution in constructing the confidence interval for provided: 1. The population from which the sample was selected is normally distributed. 2. The value of is known. Sometimes we do not the value of and we have to use the sample std. dev. as an estimator of In that case we cannot use the normal distribution for the construction of confidence intervals about. Instead we will use another distribution called the. Conditions under which the is used to make a Confidence Interval about The t-distribution is used to make a confidence interval about if: 1. The population from which the sample is selected is (approximately) normally distributed. 2. The sample size is small 30 3. The population standard deviation is known. Notes on the t-distribution The t-distribution was developed by W.S. Gosset in 1908 and published under the pseudonym Student. As a result it is also called the Student s t-distribution. The t-distribution is symmetric about the mean and never meets the horizontal axis It is bell shaped like the normal curve and the total area under the curve is 1.0 or 100% The t-distribution is flatter than the normal curve i.e. it has a lower height and a wider spread (larger standard deviation) than the normal distribution. As the sample size increases, the t-distribution approaches the Standard Normal Distribution. The units of a t-distribution are denoted by t. The shape of a particular t-distribution depends on the number of Degrees of Freedom denoted by df where 1. The number of Degrees of Freedom is the only parameter of the t-distribution The Mean of the t-distribution is 0 and its Std. Dev. is which always greater than 1. The number of degrees of freedom is defined as the number of observations that can be chosen freely Example Let,, be any four numbers. Suppose we know that their Mean is 20. Then we can have: 20 4 80 7
This equation says that we are free to choose any 3 values for the 4 variables available. Once we choose values for any 3 variables, we automatically know the value for the fourth variable. We say that the number of degrees of freedom is 1 4 1 3. Examples on how to use the t-distribution Table for finding the value of the parameter t. Use the table to find the value of t for: 1. 16 0.05 2. 25 0.005 3. 35.10 Construction of Confidence interval for for small samples To repeat, the following 3 conditions must hold true before we use the t-distribution to construct confidence interval for the population. 1. The population from which the sample is selected is (approximately) normally distributed. 2. The sample size is small 30 3. The population standard deviation is known. The confidence interval for for small samples. The 1 100% The value of t is obtained from the t-distribution table for n 1 degrees of freedom and the given confidence level. Examples of finding Confidence intervals for for small samples will be done in class. Interval Estimation of a Population Proportion: Large Samples First note that any proportion can be expressed as a percentage by multiplying by 100. For this section we will learn how to estimate the population proportion p using the sample proportion. Also, p is a population parameter and is therefore a constant but is a sample statistic and therefore must possess a sampling distribution. We also know that for Large Samples: 1. The Sampling Distribution of the Sample Proportion is approximately normal. 2. The mean of the sampling distribution of is equal to the population proportion p. 8
3. The standard deviation of the sampling distribution of the sample proportion is 1 4. Remember also that in the case of a proportion, When we consider our Confidence Interval for for large samples, we had the formulas: The quantities is called the Maximum Error of Estimate and is denoted by. Definition: Maximum Error of Estimate The Maximum Error of Estimation for,, is the quantity that is subtracted and added to the value of to obtain a confidence interval for. NOTES The value of in the confidence interval formula is obtained from the Standard Normal Distribution Table for the given confidence level. The construction of a 95% confidence interval for the population mean we follow the procedures as set out below: 4. Divide 0.95 by 2 giving the value 0.4750 5. Locate o.4750 in the body of the normal distribution table and record the corresponding value of z. This value of z is 1.96 6. Then 1.96 1.96 For a 1 100% confidence level, the area between 1 and the area of the two tails is and each one has area In our 95% confidence interval above, we can work out the value of as follows: 1 0.95 1 0.95 0.05 0.05 0.025 2 2 The company Asian Paints at Small Industries is mixing a new type of paint. Before the company decides the price at which to sell their products, it wants to know the average price of all such paints in the market. The research section of the company took 36 comparable cans of paints and the related information on the price. This information produced a mean price of $70.50 per can of paint for this sample. It is known that the standard deviation of the prices of all paints of this type is $4.50. 9
(iii) (iv) What is the point estimate of the mean price for all such cans of paints? What is the margin of error for this estimate Construct a 90% confidence interval for the mean price of all such cans of paints Solution The information given above are as follows: 36 70.50 4.5 The standard deviation of is 4.50 0.75 36 (iii) The point estimate of the mean price of all such paints is $70.50 i.e. 70.50 The Margin of Error associated with this point estimate of is 1.96 1.960.75 $1.47 The Margin of Error means that the mean price of all such paints is $70.50 give or take $1.47. Also note that the Margin of Error is simply the maximum error of estimate for a 95% confidence interval. (iv) The confidence level is 90% or 0.90. First we find the z value for a 90% confidence interval. First, divide 0.90 by 2:... 0.4500 Next, Locate 0.4500 on the body of the standard normal table. Since 0.4500 is not on the normal table, we can use the number closest to 0.4500 which is either 0.4495 or 0.4505 as an approximation. Any one of these two numbers can be used. If we use 0.4505 as an approximation for 0.4500, then the value of z for this number is 1.65 Finally, we substitute all the values in the confidence interval formula for. The 90% confidence interval for is: 70.50 1.650.75 70.50 1.24 The 90% confidence interval for is: 70.50 1.24 70.50 1.24 10
.. $69.26 $71.74 NOTES The above example shows that we are 90% confident that the mean price of all such cans of paints is between $69.26 and $71.74 Because is a constant, we cannot say that the probability is 0.90 that the interval contains because either it contains or it does not. Consequently the probability is either 0 or 1. The interpretations of the 90% confidence interval is as follows: c. If we take all possible samples of size 36 and construct a 90% confidence interval for around each sample mean we can expect that 90% of all these confidence intervals will include and 10% will not. d. Since we expect many samples of size 36 from the same population, we would have many sample means,,. for each sample. This will in turn tells us to construct a confidence interval around each mean. We conclude that 90% of these intervals will contain and 10% will not. Also, the population standard deviation was known. Sometimes this quantity is not known and therefore we will use the sample standard deviation s to estimate the population standard deviation and estimate the standard deviation by. Construction of a Confidence interval for when is not known. Suppose Small Businesses in Tonga with Negative Assets carried an average of $15,528 debt last year (2011). Assume that this mean was based on a random sample of 400 businesses and the standard deviation for this sample was $4,200. Construct a 99% confidence interval for last year mean debt for all such businesses in Tonga. Solution From the given information, 400 $15,528 $4,200 The confidence level = 99% or 0.99 First we find the standard deviation of. Because is not known, we will use as an estimator of. The value of is 4,200 $210 400 Because the sample is large 30 we will use the normal distribution to determine the confidence interval for. Dividing 0.99 by 2 we get 0.4950. From the normal distribution table, the z-value for 0.4950 is approximately 2.58. The confidence interval is 15,528 2.58210 15,528 541.80 $14,986.20 $16,069.80 11
Determining the sample size for the estimation of the Mean We have mentioned in earlier lectures several reasons why a Sample Survey is carried out and not a Census. Another important reason is that we do not have enough resources at our disposal. In light of this, we would be more than happy if a small sample can serve our purpose for doing estimate than a larger sample. Searching for larger samples often waste a lot of time and resources. For example, suppose we want to estimate the mean number of all accidents that happens in Tonga for one year. If a sample of 5 accidents can give us the confidence interval that we want, then looking for a sample of 200 accidents would be wasting time and resources. In such cases, if we know the confidence level and the width of the confidence interval that we want, then we can find the approximate size of sample that would produce the required result. Recall that the Maximum Error of Estimate E for is given by:,. How to determine the sample size for the estimation of. (population mean) Given the confidence level and the value of (population standard deviation), the sample size n that will produce a predetermined maximum error E of the confidence interval estimate of is: NOTES If is not known, then a preliminary sample of any size can be taken so that we can use its standard deviation s to replace. However, using s to replace may give a sample size that eventually may produce an error much larger (or smaller) than the predetermined maximum error of estimate. The value of the sample size n must always be rounded up to the next integer. Example How large a sample must be selected so that the maximum error for a 99% confidence interval for is 2.50 if 12.5 Solution The z value for a 99% confidence level is 2.58, E = 2.50, 12.5 then 12
2.58 12.5 2.50 6.6564156.25 6.25 1,040.0625 6.25 166.41 167 167 How to determine the sample size for the estimation of. (population proportion) Recall that the Maximum Error of Estimate E for is given by: How to determine the sample size for the estimation of. (population proportion) Given the confidence level and the value of, the sample size n that will produce a predetermined maximum error E of the confidence interval estimate of is: NOTES The formula above requires that we must know p and q to determine the value of n. However the values of p and q are not known to us and we have to choose one of the following two alternatives. 1. Most Conservative Estimate of the sample size n: Use p = 0.5 and q = 0.5. For a given E these values of p and q will give us the largest sample size compared to any other pair of p and q since the product 0.50.5 is the greatest compared to any other pair of p and q. 13
2. We take a preliminary sample and calculate for this sample and use them instead of p and q to find the sample size n. Example A company had just installed a new machine for making spare parts for clocks. The spare parts produced by this machine sometimes good and sometimes defective. The Manager wants to estimate the proportion of parts that are defective. He also wants this estimate to be within 2% of the population proportion for a 95% confidence level. What is the most conservative estimate of the sample size that will limit the Maximum Error to within 2% of the population proportion? Solution The company Manager wants the 95% confidence interval to be: 0.02 0.02 For a 95% confidence level the value of z is 1.96 and for the most conservative estimate of n we use p = 0.5 and q = 0.5 Therefore the required sample size is: 1.96 0.50.5 0.02 2,401 Thus, if the company takes a sample of 2,401parts, there is a 95% chance that the estimate of p will be within 2% of the population proportion. Before we move on to our next topic which is Hypothesis Testing we shall summarize our work on ESTIMATION by revisiting terminologies and key formulas that were used in Estimation. Terms and Definitions: 1 Confidence Interval: An interval constructed around the value of a sample statistic to estimate the value of a population parameter 2 Confidence Level: The Confidence level is denoted by 1 100% and it states how much confidence we have that the confidence interval contain the true population parameter. 3 Degrees of Freedom(df): The number of observations that can be chosen freely. For the estimation of using the t-distribution, the degrees of freedom are n-1. 4 Estimate: The value of a sample statistic that is used to find the corresponding population parameter. 14
5 Estimation: The procedure by which a numerical value or values are assigned to a population parameter based on the information collected from a sample. 6 Estimator: The sample statistic that is used to estimate a population parameter. 7 Interval Estimate: An interval constructed around the point estimate that is likely to contain the corresponding population parameter. Each interval estimate has a confidence level. 8 Maximum Error of Estimate: The quantity that is subtracted from and added to the value of a sample statistic to obtain a confidence interval for the corresponding population parameter. 9 Point Estimate: The value of a sample statistic assigned to the corresponding population parameter. 10 t Distribution: A continuous distribution with a specific type of bell shaped curve Formulas with its mean equal to 0 and standard deviation equal to 2 1 Margin of Error associated with the point estimate of 1.96 1.96 2 The 1 100% confidence interval for for a large sample 30 3 The 1 100% confidence interval for for a small sample 30 when the population is (approximately) normally distributed and is not known is: 4 Margin of error associated with the point estimate of p 1.96 5 5 The 1 100% confidence interval for for a large sample 30 is 6 Maximum Error, E, of the estimate for is 15
7 Required sample size for a predetermined maximum error for estimating is 8 Maximum Error, E, of the estimate for the population proportion is 9 Required sample size for a predetermined maximum error for estimating is Use 0.5 & 0.5 for the most conservative estimate. Use the values of & if the estimate is to be based on preliminary sample. 16