Business Statistics Unit 7

In this unit we will use sample data to estimate the value of an unknown population mean, percentage, variance, or standard deviation. This unit introduces the concept of confidence intervals and how they are used to estimate population parameters.

In previous units we were able to determine exact values for the mean, percentage, variance, and standard deviation when we were working with population data and sampling distributions. But what happens when you only have one sample from a population? How can you determine the population mean, percentage, variance, or standard deviation? The truth is that the data obtained from one sample is not likely to represent the exact population parameter you are looking for. For example, in unit 6 we found that each time we obtained a sample from a population, the sample means were different.

So how do you determine a population parameter from a single sample? We estimate the population parameter based upon our sample. Estimation is the process of using an estimator obtained from a sample to produce an estimate of a parameter. There are two types of estimates: a point estimate which is a single number or value used to estimate a population parameter, and an interval estimate which is a spread of values used to estimate a population parameter.

The best way to estimate a population parameter is to use an interval estimate. When an interval estimate is used, the possibility of being wrong is identified and known. This is called the error of the estimate and it uses the symbol α called alpha. We also choose the level of accuracy we wish to use to determine our interval estimate. The confidence coefficient refers to the probability of correctly including the population parameter within the interval being produced. This probability is expressed as the level of confidence and common levels of confidence include 90%, 95%, and 99%. For each level of confidence then, there is the possibility of error. To find the amount of error for each confidence level, you simply subtract the level of confidence from 100%. For example, if our confidence level is 95%, the amount of error (α) equals 100% - 95% = 5% error.

Interval estimates are called two-tailed tests. Why? Because the possibility of error exists on both sides of our interval estimate. For any given interval estimate of a population parameter, the actual population parameter may be higher or lower than the estimated interval.

Lets take a look at figure 7.1 which shows the normal distribution and the area represented by a 95% level of confidence and a 5% amount of error.

Figure 7.1. The Normal Distribution with 95% Confidence, 5% Error, Two-Tail Test.

Between our two estimated values of x, we believe that 95% of the time our estimated population parameter will be found. The remaining 5% of the time, our parameter will be higher (2.5%) or lower (2.5%) than our estimated interval values.

The use of an interval estimate and a given level of confidence to estimate a population mean is called a confidence interval. The basic formula for a confidence interval is given in formula box 7.1 below. This formula assumes that you have already calculated the standard error.

In order to use the formula to find the population mean, you need the sample mean, the standard error of the mean, and the z score for the level of confidence you are using. To find the z score for the level of confidence, you must remember that for a two-tail test, which interval estimates are, the possibility of error exists on both sides of the interval estimate.

Looking a figure 7.1 again, since our error totals 5% for a 95% confidence level, the error is divided between both sides of the normal distribution since we are looking at an interval estimate. As a result there is a 2.5% chance of error on each side. To find the z score, we follow part of the process used in previous units where we find the probability first, then look up the value of z in the normal table. In this case our probability will equal .4750 if we are at 95% confidence and using a two tail test. Refer to figure 7.2 for the normal distribution and probabilities associated with 95% confidence and a two tail test.

Once you have the probability, you simply use the normal table and find the z score for the probability closest to the one you are looking for. For example, if our probability equals .4750. then our z score from the normal table equals 1.96. The z score of 1.96 will always be the same, regardless of the problem, as long as the confidence level is 95%, and we are conducting an interval estimate. The common confidence levels and associated z scores are given in Table 7.1 below.

Confidence levels can also be called levels of significance. Each confidence level has a significance level associated with it.

Significance levels are often used in hypothesis testing and also in surveys instead of confidence levels. It is simply using a different term to describe the level of accuracy desired.

σ known and the population is normally distributed, or σ unknown and the sample size > 30

The following steps are used to find the interval estimate of a population mean when the population standard deviation is known and the population is normally distributed, or the population standard deviation is unknown and the sample size is greater than 30.

Practice Exercise 7.1. Finding Interval Estimates for the Population Mean, σ known and the population is normally distributed.

If our sample mean equals 25, the population standard deviation equals 2, the sample size is 100, and the level of confidence equals 99%, find the interval estimate for the population mean.

What if our sample mean equals 25, the population standard deviation equals 2, the sample size is 100, but the level of confidence equals 95%, what will the interval estimate for the population mean be?

Did you notice anything different between our first two problems? The confidence level was lower for the second problem. As a result, our interval estimate was not as wide indicating that we are willing to accept the possibility of more error (from 1% to 5%). What happens if the confidence level stays the same but the sample size increases?

Let's say our sample mean equals 25, the population standard deviation equals 2, the sample size is now 200, and the confidence level equals 95%, what will the interval estimate for the population mean be?

Now what's different? Since our sample size has increased, the standard error decreased, and as a result, the interval estimate width is smaller. Increasing the sample size helped reduce our interval width.

If you are now comfortable with the previous discussion and the steps used to estimate the population mean, go on to Practice Exercise 7.2 for a more application orientated example. If you are still somewhat confused, go back and review the previous material again before moving on.

Practice Exercise 7.2. Interval Estimate for the Population Mean, σ known and the population is normally distributed.

The Rusty Steel Company recently received a shipment of small steel beams. The quality control manager inspects the beams to determine if they meet the established standards. The manager takes a random sample of 25 beams and measures the thickness of the beams. The average thickness is 3.0 inches. From past experience the manager knows that the beam thickness has been normally distributed with a population standard deviation of .02 inches. Construct an interval estimate of the population mean using 95% confidence.

To answer the question, as along as the quality control specifications are below our interval estimate values, the beams are acceptable for use.

It is now time to look at a different situation where we are trying to estimate the population mean, but the population standard deviation is unknown and the sample size is greater than 30. What changes? Technically we have to use the sample standard deviation instead of the population standard deviation. The standard error calculation is done although in this situation it is called the estimated standard error.

To find the estimated standard error, we use the sample standard deviation and the square root of the sample size. Our interval formula also changes to reflect the use of an estimated standard error. However it still works the same way as formula 7.1.

Formula Box 7.3. The Confidence Interval for a Population Mean, σ unknown and n > 30.

Practice Exercise 7.3. Interval Estimate for the Population Mean, σ unknown and n > 30.

The High Dollar Bank and Trust is interested in finding out the average time a customer transaction takes in its drive-up window. During a typical day of transactions, a random sample of 50 customers were selected and their times recorded. The mean transaction time was 7 minutes with a sample standard deviation of 3 minutes. Estimate the population mean using a 95% confidence level.

Did you notice that the process for finding the interval estimate of the population mean in Practice Exercise 7.3 was identical to the process used in Practice Exercise 7.2? We simply substitute the sample standard deviation for the population standard deviation and use the estimated standard error.

In this situation, once again we do not know the population standard deviation. Therefore we will use the sample standard deviation and calculate the estimated standard error. But, since our sample size is less than 30, it is no longer valid for us to use the normal distribution. Instead we must use a new distribution called the t distribution and find a value for t instead of z.

The t distribution has its own set of tables which can be found in your textbook. One of the key differences is that in order to use the t distribution, you need to first calculate the degrees of freedom. The degrees of freedom for a t distribution is equal to n, the sample size, minus 1, (n - 1). The next step is to look at the t table in your textbook. Notice that it has a column for the degrees of freedom (df). You should also notice that it contains other columns with t values listed below. In order to determine which column to use, you need to know your level of confidence.

For 95% confidence, we have 5% of error. Since we are working with confidence intervals we have a two tail test. This means that the possibility of error exists above and below the mean. The amount of error on each side of our distribution equals 2.5% or .025. Since your book provides two separate headings for each column called one tail or two tail, you would go to the column designated for a two tail value of 0.05.

For example, if we have a sample size of 15 and are using 95% confidence, what is our t value? First find your degrees of freedom. 15 - 1 = 14 degrees of freedom. Next we need to find the right column. Since our confidence level is 95% and we have a two tail test, the column we need is the two tails 0.05 column with 14 degrees for freedom. You should find a t value of 2.145.

Let's try another one! What if we have a sample size of 18 and a confidence level of 99%? The degrees of freedom equal 18 - 1 = 17. The column we need to use for 99% confidence, 1% error, two tail test, is the column for two tails, 0.01, with 17 degrees of freedom. Your number is 2.898.

Now that we are able to find the t value, our next step is to examine the interval formula. Since we are using the t distribution, our interval formula has a few minor changes.

Formula Box 7.4. The Confidence Interval for a Population Mean, σ unknown and n < 30. This formula assumes you will find the standard error first before using the formula.

Notice that the only changes in the formula from Formula 7.3 is that the z symbol has been replaced by the t symbol since we are using a t distribution.

Practice Exercise 7.4. Interval Estimate for the Population Mean, σ unknown and n < 30, the t distribution

The Bone Medical Company makes specialized surgical implants that are used to reconstruct knees and elbows. Each implant is required to be made within highly precise sizes and weights. A random sample of 20 elbow implants were selected by the quality control department. Each implant was weighed and the sample mean was 18 ounces. The sample standard deviation was calculated at .5 ounces. Estimate the population mean based upon the data given using 99% confidence.

The only change in our process for estimating the population mean relates to determining the t value instead of the z value. All of the other steps remain the same.

Practice Exercise 7.5 Interval Estimate for the Population Mean, σ unknown and n < 30, the t distribution

The average starting salary for nursing college graduates in the northwest part of the U.S. was found to be $41,500 with a sample standard deviation of $2,800 in a recent survey conducted of 28 graduates. Using a confidence level of 95%, find the estimate of the true mean starting salary.

We have learned so far how to estimate the population mean based upon a single sample. Now we are going to apply the same concept to estimating the population percentage or proportion. The sampling distribution of p approximates the normal distribution when n, which is the sample size, multiplied by p, the sample percentage/proportion, is greater than or equal to 500, and if n(100 - p) is also greater than or equal to 500.

On this basis we can develop interval estimates of the population percentage π using a sample percentage p. We will first need to calculate the estimated standard error of the percentage, designated as:

Please note that in some statistics text books, the population percentage is represented as p with the sample percentage represented as p "hat". In addition the part of the formula shown as (100 - p) can also be referred to as q. If you work with proportions instead of percentages, to find q, you subtract p from 1 instead of 100. A proportion is simply a decimal instead of a percentage. For example, a percentage value might be 35%. The proportion value is .35. The formula below can be used for either a percentage or a proportion.

Once we have calculated the standard error, we will need to find the z score. We can use the z distribution and z values for estimating the population percentage as long as the distribution is normal. Our z value is once again based upon the level of confidence.

Our third step is to use the interval formula to determine the confidence interval. The interval formula is modified to reflect that we are estimating a population percentage.

Again it needs to be pointed out that the p in this formula represents the sample percentage, Π represents the population percentage, and the standard error of the percentage is already calculated before this formula is used.

Practice Exercise 7.6. Interval Estimate for the Population Percentage/Proportion.

An Internet computer software and hardware company is interested in learning why many of its customers do not make an additional purchase within 90 days of their first purchase. The company sends an email survey to its customers picked randomly from the customer database file. Of the 400 customers who responded to the survey, 180 customers indicated that they found a better price from a competitor. Using a 95% confidence level, find the confidence interval for the percentage of all customers who found a better price from a competitor.

In order to solve this problem we need to find p the sample percentage. Since 180 out of the 400 customers who responded to the survey indicated that they found a better price from a competitor, our p (sample percentage) is equal to 180 / 400 = 45%.

So when we are determining the population percentage interval, we must first make sure that it is valid to use the normal distribution. Then we need to find the value of p, our sample percentage. Then we use our 3 step process to find the interval estimate.

Practice Exercise 7.7. Interval Estimate for the Population Percentage/Proportion.

In a new condo development, some of the residents are interested in changing the landscaping rules. In a random sample of 34 residents, 23 say they will vote in favor of the change. Construct a 95% confidence interval for the population percentage favoring the change in policy.

In order to solve this problem we need to find p the sample percentage. Since 23 out of 34 residents who responded to the survey indicated that they are in favor of the change, our p (sample percentage) is equal to 23 / 34 = 67.65%.

Estimating the population variance or standard deviation is an important statistical technique for business and the public sector. It is used to determine if products being manufactured meet quality control requirements for size, weight, or ingredients. The process used to estimate the confidence interval for a variance or standard deviation requires the use of a new distribution called the Chi-Square distribution.

In order to use the Chi-Square distribution, you need to understand that this distribution and the table of values for the Chi-square distribution, are used a little differently than the z or t tables. Refer to the Chi-Square Distribution tables in your book. Notice that you have a column for the degrees of freedom. The values in this column are found by taking the sample size (n), and subtracting 1 from the value of n. So if our sample size is 15, the degrees of freedom are equal to 14. Now look across the top at the probabilities listed. You will notice that each column of values has a different probability listed. The values start at 0.995 and go all of the way to 0.005.

df	Area or probability (α) in the right tail
df	0.995	0.99	0.975	0.95	0.90	0.75	0.50	0.25	0.10	0.05	0.025	0.01	0.005
1	.0000	.0000	.0000	.0003	.0158	.102	.455	1.323	2.71	3.84	5.02	6.63	7.88
2	.0100	.0201	.0506	.103	.211	.575	1.386	2.77	4.61	5.99	7.38	9.21	10.60
3	.0717	.115	.216	.352	.584	1.213	2.37	4.11	6.25	7.81	9.35	11.34	12.84
4	.207	.297	.484	.711	1.064	1.923	3.36	5.39	7.78	9.49	11.14	13.28	14.86

The mean of a Chi-Square distribution is equal to the degrees of freedom. In addition, as we change sample sizes a different Chi-Square distribution is used. In order to find the proper columns to use, we must once again know our level of confidence. For example, if our level of confidence is equal to 95%, the amount of error (α) is equal to 5%. For confidence intervals, our Chi-Square distribution has two tails indicating that our error of 5% must be divided into two tails, or 2.5% for each tail. When using the Chi-Square distribution tables, our first probability can be found by using our error of 2.5%, converting it into a probability (.025) and finding the .025 column in the Chi-Square table.

However there are actually two different values we need. One value is called the lower limit, and it is found by finding the error amount, dividing it by 2, and using the appropriate column. In our example using 95% confidence, 5% error, the column we should use to find our lower limit value is the .025 column. The upper limit value can be found by taking our lower limit probability and subtracting it from 1. In our example, we had a lower limit probability of .025. Therefore the upper limit probability is equal to 1 - .025, which equals .975. So we use the .975 column to find our upper limit probability value.

Let's use an example to practice finding the Chi-Square probability values. Suppose we have a confidence level of 99% and a sample size of 5. Find the Chi-Square probability values. To find our first value, we need to know the amount of error. For 99% confidence, we have 1% error. For a two tail test, our 1% error is divided into two tails. Therefore we have .5% of error in each tail or .005. So to find our lower limit value, we use the .005 probability column. Our degrees of freedom equal 5 - 1 = 4. Looking at the .005 column and 4 degrees of freedom, our probability for the lower limit equals 14.86.

To find the upper limit value, we take 1 - .005 = .995. Looking at the .995 column using 4 degrees of freedom our upper limit value is .207. Now that we have these values we can use the formula in Formula Box 7.7 to find our confidence interval for a variance or standard deviation.

Formula Box 7.7. The Confidence Interval for a Variance or Standard Deviation.

Formula: confidence interval for a variance or standard deviation

The lower limit value has the symbol . The upper limit value has the symbol . Now that we have the formula for a confidence interval for a variance or a standard deviation, we are ready to solve a problem.

Practice Exercise 7.8. Finding the Confidence Interval for a Variance.

The manager of the local post office measures the waiting time for a random sample of 25 customers and finds that the variance is 12 minutes. Assuming the waiting times are normally distributed, form a 95% confidence interval for the population variance.

Our first step is to find our upper and lower limit values. Since we are using 95% confidence, our total error equals 5% divided by two equals 2.5% in each tail, or .025. Our degrees of freedom equal 25 - 1 = 24. Using the Chi-Square table, our lower and upper limit values are:

Lower limit value: 39.40

Upper limit value: 12.40

Our next step is to use the formula to solve for the population variance.

(25 - 1)12 / 39.40 < σ2 < (25 - 1)12 / 12.40

Our interval estimate for the population variance is: 7.3096 < σ2 < 23.2258

As a third step, its a good idea to refer back to the problem to make sure you have found what the problem asked for. Since we were looking for the population variance, we have solved the problem.

By now you may wondering how you can use the formula to find the standard deviation instead of the variance. Actually the process is quite simple. Once you have found the variance, you take the square root of each interval value and you will then have the standard deviation. Let's do a practice problem finding the interval estimate for the population standard deviation.

Practice Exercise 7.9. Finding the Confidence Interval for the Standard Deviation.

The Acme Tool Company makes specialty tools for a variety of customers. One of its customers has complained that the specialty tools being made by Acme are not the proper size frequently. The quality control manager at Acme takes a random sample of 20 tools and finds that the sample mean for the tool width is 8.1 inches and the sample standard deviation equals .60 inches. Estimate the population standard deviation at 99% confidence. If the customer requires each tool to be within .25 inches of a mean of 8 inches, are the standards being met?

Our first step is to find the upper and lower limit values. Since we are using 99% confidence, our total amount of error equals 1% which is divided into two tails each with .5% of error, or .005. The two columns we will be using are the .005 column and the .995 column. The degrees of freedom equal 20 - 1 = 19. Using the Chi-Square table, our lower and upper limit values are:

Lower limit value: 38.60

Upper limit value: 6.84

Our next step is to use the formula to solve for the population variance.

(20 - 1)(.60)2 / 38.60 < σ2 < (20 - 1)(.60)2 / 6.84 Did you notice that we needed to square .60? That is because we were given the sample standard deviation and the formula requires the sample variance.

Our interval estimate for the population variance is 0.1772 < σ2 < 1.00.

Now we need to go back to the question. We were looking for the interval estimate for the standard deviation not the variance, so we have one step to go. We need to take the square root of our interval estimate.

√0.1772 < σ < √1.00

Our answer is .4210 < σ < 1.00. Our estimated population standard deviation is between .4210 inches and 1 inch. Are the customer's standards being met? Even though the mean meets the standards, our standard deviation clearly indicates that some tools are larger or smaller than the standards. The quality control manager needs to work with production to solve the problem!

NOTE: As a general rule the lower limit value is always larger than the upper limit value. Therefore if you get confused about where to place a number, the larger number goes into the left denominator side of the formula, and the smaller number goes into the right denominator side of the formula.

We have worked with samples to find the mean, percentage, standard deviation, and variance. But how do you determine the proper sample size? Remember that as your sample size increases, the standard error becomes smaller. So if we want to have as little error as possible, we must increase our sample size. Our tolerance for error may be based upon some existing quality control specification. For example, if your company produces radios for GM Corvettes, the radio must fit into the slot in the dash. A radio that is too wide, tall, or deep will not fit into the slot. GM may specify for example, that the radio must be between 8.4 and 8.6 inches wide with an average width of 8.5. This indicates that we can be .1 inches above and below our mean value. The error that we can tolerate is 0.1 inches.

Another issue we need to consider is the characteristics of the population being sampled. Do you know anything about the population standard deviation? Often we do not have any information about it. We may have to rely upon a guess or perhaps we can take a preliminary sample from the population and calculate a sample standard deviation which we can use to help us determine the proper sample size.

Finally we have to decide upon the level of confidence we wish to use. The most commonly used level of confidence is 95% however we might need to be more certain so it is not uncommon to use 99% confidence.

Once you have decided upon the amount of error that is acceptable, selected the level of confidence, and have determined the population or sample standard deviation, you can then use a formula to calculate the sample size.

Formula Box 7.8. Determining the Sample Size to Estimate the Population Mean (µ).

Practice Exercise 7.10. Determining the Sample Size to Estimate the Population Mean.

If you are working with a percentage or proportion value instead of a mean value, you need to use a different formula to determine the sample size.

Formula Box 7.9. Determining the Sample Size to Estimate the Population Percentage/Proportion.

The homework problems for unit 7 can be found by going to the Course Documents link in Blackboard, and clicking on the link for Unit 7. Look for the Unit 7 - Assignment link. Once you have completed your homework assignment, you will need to post your answers on Blackboard®. Once you have signed into Blackboard, simply go to the Unit 7 - Assignment 7 - Post Answers Here link and post your answers. Immediate feedback is provided once you have completed the posting of all of your answers and clicked on submit. Make sure you print the entire submitted homework assignment to assist you with quizzes and tests.

Confidence Level	z Score or Value
90%	1.645
95%	1.96
99%	2.575