1. Introduction to Statistical Inferences8
Introduction to Statistical Inferences
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2. 8.2 Estimation of Mean  ( Known)
Copyright © Cengage Learning. All rights reserved.

3. Estimation of Mean  ( Known)
The assumption for estimating mean  using a known
 The sampling distribution of has a normal distribution.
Note The word assumptions is somewhat of a misnomer. It does not mean that we “assume” something to be the situation and continue but rather that we must be sure that the conditions expressed by the assumptions do exist before we apply a particular statistical method.

4. Estimation of Mean  ( Known)

5. Estimation of Mean  ( Known)
Therefore, we can satisfy the required assumption by either (1) knowing that the sampled population is normally distributed or (2) using a random sample that contains a sufficiently large amount of data.
The first possibility is obvious. We either know enough about the population to know that it is normally distributed or we don’t.
The second way to satisfy the assumption is by applying the CLT. Inspection of various graphic displays of the sample data should yield an indication of the type of distribution the population possesses.

6. Estimation of Mean  ( Known)
The CLT can be applied to smaller samples (say, n = 15 or larger) when the data provide a strong indication of a unimodal distribution that is approximately symmetrical.
If there is evidence of some skewness in the data, then the sample size needs to be much larger (perhaps n  50).
If the data provide evidence of an extremely skewed or J-shaped distribution, the CLT will still apply if the sample is large enough. In extreme cases, “large enough” may be unrealistically or impracticably large.

7. Estimation of Mean  ( Known)
Note There is no hard-and-fast rule defining “large enough”; the sample size that is “large enough” varies greatly according to the distribution of the population.
The 1 –  confidence interval for the estimation of mean  is found using formula (8.1).

8. Estimation of Mean  ( Known)
Here are the parts of the confidence interval formula:
is the point estimate and the center point of the confidence interval.
2. z(/2) is the confidence coefficient. It is the number of multiples of the standard error needed to formulate an interval estimate of the correct width to have a level of confidence of 1 – .

9. Estimation of Mean  ( Known)
Figure 8.4 shows the relationship among the level of confidence 1 –  (the middle portion of the distribution), /2 (the “area to the right” used with the critical-value notation), and the confidence coefficient z(/2) (whose value is found using Table 4B of Appendix B).
Alpha, , is the first letter of the Greek alphabet and represents the portion associated with the tails of the distribution.
Confidence Coefficient z(/2)
Figure 8.4

10. Estimation of Mean  ( Known)
is the standard error of the mean, or the standard deviation of the sampling distribution of sample means.
is one-half the width of the confidence interval
(the product of the confidence coefficient and the standard error) and is called the maximum error of estimate, E.

11. Estimation of Mean  ( Known)
is called the lower confidence limit (LCL),
and is called the upper confidence limit
(UCL) for the confidence interval.
The estimation procedure is organized into a five-step process that will take into account all of the preceding information and produce both the point estimate and the confidence interval.

12. Estimation of Mean  ( Known)
The Confidence Interval: A Five-step Procedure
Step 1 The Set-Up:
Describe the population parameter of interest.
Step 2 The Confidence Interval Criteria: a. Check the assumptions.
b. Identify the probability distribution and the formula to be used. c. State the level of confidence, 1 – .

13. Estimation of Mean  ( Known)
Step 3 The Sample Evidence:
Collect the sample information.
Step 4 The Confidence Interval:
a. Determine the confidence coefficient.
b. Find the maximum error of estimate.
c. Find the lower and upper confidence limits.
Step 5 The Results:
State the confidence interval.

14. Example 4 – Demonstrating the Meaning of a Confidence Interval
Single-digit random numbers, like the ones in Table 1 in Appendix B, have a mean value  = 4.5 and a standard deviation  = Draw a sample of 40 single-digit numbers from Table 1 and construct the 90% confidence interval for the mean. Does the resulting interval contain the expected value of , 4.5?
If we were to select another sample of 40 single-digit numbers from Table 1, would we get the same result?
Random Sample of Single-Digit Numbers [TA08-01]
Table 8.1

15. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
What might happen if we selected a total of 15 different samples and constructed the 90% confidence interval for each?
Would the expected value for  —namely, 4.5—be contained in all of them?
Should we expect all 15 confidence intervals to contain 4.5? Think about the definition of “level of confidence”; it says that in the long run, 90% of the samples will result in bounds that contain .
In other words, 10% of the samples will not contain . Let’s see what happens.

16. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
First we need to address the assumptions; if the assumptions are not satisfied, we cannot expect the 90% and the 10% to occur. We know:
(1) The distribution of single-digit random numbers is rectangular (definitely not normal),
(2) the distribution of single-digit random numbers is symmetrical about their mean,
(3) the distribution for very small samples (n = 5) is a distribution that appeared to be approximately normal, and
(4) there should be no skewness involved.

17. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
Therefore, it seems reasonable to assume that n = 40 is large enough for the CLT to apply.
The first random sample was drawn from Table 1 in Appendix B: The sample statistics are n = 40, x = 159, and = 3.98.

18. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
Here is the resulting 90% confidence interval: to
3.23 to 4.73 is the 90% confidence interval for .

19. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
Figure 8.5 shows this confidence interval, its bounds, and the expected mean .
The expected value for the mean, 4.5, does fall within the bounds of the confidence interval for this sample.
Let’s now select 14 more random samples from Table 1 in Appendix B, each of size 40.
The 90% Confidence Interval
Figure 8.5

20. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
Table 8.2 lists the mean from the first sample and the means obtained from the 14 additional random samples of size 40.
Fifteen Samples of Size 40 [TA08-02]
Table 8.2

21. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
The 90% confidence intervals for the estimation of  based on each of the 15 samples are listed in Table 8.2 and shown in Figure 8.6.
We see that 86.7% (13 of the 15) of the intervals contain  and 2 of the 15 samples (sample 7 and sample 12) do not contain .
Confidence Interval from Table 8.2
Figure 8.6

22. Example 4 – Demonstrating the Meaning of a Confidence Interval
cont’d
The results here are “typical”; repeated experimentation might result in any number of intervals that contain 4.5.
However, in the long run, we should expect approximately –  = 0.90 (or 90%) of the samples to result in bounds that contain 4.5 and approximately 10% that do not contain 4.5.

23. Sample Size

24. Sample Size
The confidence interval has two basic characteristics that determine its quality: its level of confidence and its width.
It is preferable for the interval to have a high level of confidence and be precise (narrow) at the same time.
The higher the level of confidence, the more likely the interval is to contain the parameter, and the narrower the interval, the more precise the estimation.

25. Sample Size
The maximum error part of the confidence interval formula specifies the relationship involved.

26. Sample Size This formula has four components:
(1) the maximum error E, half of the width of the confidence interval;
(2) the confidence coefficient, z(/2), which is determined by the level of confidence;
(3) the sample size, n; and
(4) the standard deviation,. The standard deviation  is not a concern in this discussion because it is a constant (the standard deviation of a population does not change in value).

27. Sample Size
That leaves three factors. Inspection of formula (8.2) indicates the following: increasing the level of confidence will make the confidence coefficient larger and thereby require either the maximum error to increase or the sample size to increase; decreasing the maximum error will require the level of confidence to decrease or the sample size to increase; and decreasing the sample size will force the maximum error to become larger or the level of confidence to decrease.

28. Sample Size
We thus have a “three-way tug-of-war,” as pictured in Figure 8.7. An increase or decrease in any one of the three factors has an effect on one or both of the other two factors.
The “Three-Way Tug-of-War” between 1 – , n, and E
Figure 8.7

29. Sample Size
The statistician’s job is to “balance” the level of confidence, the sample size, and the maximum error so that an acceptable interval results.
Note When we solve for the sample size n, it is customary to round up to the next larger integer, no matter what fraction (or decimal) results.

30. Sample Size
Using the maximum error formula (8.2) can be made a little easier by rewriting the formula in a form that expresses n in terms of the other values.
If the maximum error is expressed as a multiple of the standard deviation , then the actual value of  is not needed in order to calculate the sample size.

31. Now you are ready to use the sample size formula (8.3):
Example 7 – Determining the Sample Size Without a Known Value of Sigma ()
Find the sample size needed to estimate the population mean to within of a standard deviation with 99% confidence.
Solution:
Determine the confidence coefficient (using Table 4B): 1 –  = 0.99, z(/2) = The desired maximum error is E =
Now you are ready to use the sample size formula (8.3):

32. Example 7 – Solution
cont’d
= [(2.58)(5)]2
= (12.90)2
= 166.4
= 167