STATISTICAL INTERVALS 2.pptx staticssmsokso

Editor's Notes

• #1: Thus the interval estimate indicates, by its length, the accuracy of the point estimate.
• #2: In an interval estimate, the parameter is specified as being between two values. For example, an interval estimate for the average age of all students might be 21.9 m 22.7, or 22.3 0.4 years. Either the interval contains the parameter or it does not. A degree of confidence (usually a percent) must be assigned before an interval estimate is made. For instance, you may wish to be 95% confident that the interval contains the true population mean. Another question then arises. Why 95%? Why not 99 or 99.5%? If you desire to be more confident, such as 99 or 99.5% confident, then you must make the interval larger. For example, a 99% confidence interval for the mean age of college students might be 21.7 m 22.9, or 22.3 0.6. Hence, a tradeoff occurs. To be more confident that the interval contains the true population mean, you must make the interval wider.
• #4: Intervals constructed in this way are called confidence intervals. Three common confidence intervals are used: the 90%, the 95%, and the 99% confidence intervals. The wider the confidence interval is, the more confident we can be that the given interval contains the unknown parameter. Of course, it is better to be 95% confident that the average life of a certain television transistor is between 6 and 7 years than to be 99% confident that it is between 3 and 10 years. Ideally, we prefer a short interval with a high degree of confidence.
• #6: The term za2(s ) is called the margin of error (also called the maximum error of the estimate). For a specific value, say, a 0.05, 95% of the sample means will fall within this error value on either side of the population mean, as previously explained. See Figure 7–1. When n 30, s can be substituted for s, but a different distribution is used.
• #10: Hence, one can say with 95% confidence that the interval between 52.3 and 55.7 days does contain the population mean, based on a sample of 50 automobiles.
• #11: Hence, one can be 99% confident (rounded values) that the mean number of customers that the store averages is between 350 and 374 customers per hour.
• #12: Anotherway of looking at a confidence interval is shown in Figure 7–2.According to the central limit theorem, approximately 95% of the sample means fall within 1.96 standard deviations of the population mean if the sample size is 30 or more, or if s is known when n is less than 30 and the population is normally distributed. If it were possible to build a confidence interval about each sample mean, as was done in Examples 7–1 and 7–2 for m, then 95% of these intervals would contain the population mean, as shown in Figure 7–3. Hence, you can be 95% confident that an interval built around a specific sample meanwould contain the population mean. If you desire to be 99% confident, you must enlarge the confidence intervals so that 99 out of every 100 intervals contain the population mean.
• #14: Since other confidence intervals (besides 90, 95, and 99%) are sometimes used in statistics, an explanation of how to find the values for za2 is necessary. As stated previously, the Greek letter a represents the total of the areas in both tails of the normal distribution. The value for a is found by subtracting the decimal equivalent for the desired confidence level from 1. For example, if you wanted to find the 98% confidence interval, you would change 98% to 0.98 and find a 1 0.98, or 0.02. Then a2 is obtained by dividing a by 2. So a2 is 0.022, or 0.01. Finally, z0.01 is the z value that will give an area of 0.01 in the right tail of the standard normal distribution curve. See Figure 7–4. Once a2 is determined, the corresponding za2 value can be found by using the procedure shown in Chapter 6, which is reviewed here. To get the za2 value for a 98% confidence interval, subtract 0.01 from 1.0000 to get 0.9900. Next, locate the area that is closest to 0.9900 (in this case, 0.9901) in Table E, and then find the corresponding z value. In this example, it is 2.33. See Figure 7–5. For confidence intervals, only the positive z value is used in the formula. When the original variable is normally distributed and s is known, the standard normal distribution can be used to find confidence intervals regardless of the size of the sample. When n 30, the distribution of means will be approximately normal even if the original distribution of the variable departs from normality. When s is unknown, s can be used as an estimate of s, but a different distribution is used for the critical values. This method is explained in Section 7–2.
• #16: When s is known and the sample size is 30 or more, or the population is normally distributed if the sample size is less than 30, the confidence interval for the mean can be found by using the z distribution, as shown in Section 7–1. However, most of the time, the value of is not known, so it must be estimated by using s, namely, the standard deviation of the sample. When s is used, especially when the sample size is small, critical values greater than the values for are used in confidence intervals in order to keep the interval at a given level, such as the 95%. These values are taken from the Student t distribution, most often called the t distribution. To use this method, the samples must be simple random samples, and the population from which the samples were taken must be normally or approximately normally distributed, or the sample size must be 30 or more.
• #20: For example, if the mean of 5 values is 10, then 4 of the 5 values are free to vary. But once 4 values are selected, the fifth value must be a specific number to get a sum of 50, since 50 5 10. Hence, the degrees of freedom are 5 1 4, and this value tells the researcher which t curve to use. The symbol d.f. will be used for degrees of freedom. The degrees of freedom for a confidence interval for the mean are found by subtracting 1 from the sample size. That is, d.f. n 1. Note: For some statistical tests used later in this book, the degrees of freedom are not equal to n - 1.
• #21: When d.f. is greater than 30, it may fall between two table values. For example, if d.f. it falls between 65 and 70. Many texts say to use the closest value, for example, 68 is closer to 70 than 65; however, in this text a conservative approach is used. In this case, always round down to the nearest table value. In this case, 68 rounds down to 65. Note: At the bottom of Table F where d.f. is large or , the za2 values can be found for specific confidence intervals. The reason is that as the degrees of freedom increase, the t distribution approaches the standard normal distribution.
• #26: Students sometimes have difficulty deciding whether to use za2 or ta2 values when finding confidence intervals for the mean. As stated previously, when s is known, za2 values can be used no matter what the sample size is, as long as the variable is normally distributed or n30. When s is unknown and n30, then s can be used in the formula and ta2 values can be used. Finally, when s is unknown and n 30, s is used in the formula and ta2 values are used, as long as the variable is approximately normally distributed. These rules are summarized in Figure 7–8.
• #27: One of the most common types of confidence intervals is one that uses proportions. Many statistical studies involve finding a proportion of the population that has a certain characteristic. In this section, you will learn how to find the confidence interval for a population. A USA TODAY Snapshots feature stated that 12% of the pleasure boats in the United States were named Serenity. The parameter 12% is called a proportion. It means that of all the pleasure boats in the United States, 12 out of every 100 are named Serenity. A proportion represents a part of a whole. It can be expressed as a fraction, decimal, or percentage. In this case, 12% 0.12 or . Proportions can also represent probabilities. In this case, if a pleasure boat is selected at random, the probability that it is called Serenity is 0.12. Proportions can be obtained from samples or populations. The following symbols will be used. For example, in a study, 200 people were asked if they were satisfied with their jobs or professions; 162 said that they were. In this case, n 200, X 162, and ˆp Xn 162200 0.81. It can be said that for this sample, 0.81, or 81%, of those surveyed were satisfied with their jobs or professions. The sample proportion is 0.81. The proportion of people who did not respond favorably when asked if they were satisfied with their jobs or professions constituted ˆ q, where ˆq (n X)n. For this survey, ˆq (200 162)200 38200, or 0.19, or 19%. When ˆp and ˆq are given in decimals or fractions, ˆp ˆq 1. When ˆp and ˆq are given in percentages, ˆp ˆq 100%. It follows, then, that ˆq 1 ˆ p, or ˆp 1 ˆ q, when ˆp and ˆq are in decimal or fraction form. For the sample survey on job satisfaction, ˆq can also be found by using ˆq 1 ˆ p, or 1 0.81 0.19.
• #28: Rounding Rule for a Confidence Interval for a Proportion Round off to three decimal places.
• #31: In Sections 7–1 through 7–3 confidence intervals were calculated for means and proportions. This section will explain how to find confidence intervals for variances and standard deviations. In statistics, the variance and standard deviation of a variable are as important as the mean. For example, when products that fit together (such as pipes) are manufactured, it is important to keep the variations of the diameters of the products as small as possible; otherwise, they will not fit together properly and will have to be scrapped. In the manufacture of medicines, the variance and standard deviation of the medication in the pills play an important role in making sure patients receive the proper dosage. For these reasons, confidence intervals for variances and standard deviations are necessary.
• #35: Recall that s2 is the symbol for the sample variance and s is the symbol for the sample standard deviation. If the problem gives the sample standard deviation s, be sure to square it when you are using the formula. But if the problem gives the sample variance s2, do not square it when you are using the formula, since the variance is already in square units. Rounding Rule for a Confidence Interval for a Variance or Standard Deviation When you are computing a confidence interval for a population variance or standard deviation by using raw data, round off to one more decimal place than the number of decimal places in the original data. When you are computing a confidence interval for a population variance or standard deviation by using a sample variance or standard deviation, round off to the same number of decimal places as given for the sample variance or standard deviation.
• #38: Many practical problems are phrased in terms of individual measurements rather than parameters of distributions. We take two such examples. The first one will require, what is to be called a prediction interval, instead of a confidence interval. A consumer is considering buying a car. Then this person should be far more interested in knowing whether a full tank on a particu- lar automobile will suffice to carry her/him the 500 kms to her/his destination than in learning that there is a 95% confidence inter- val for the mean mileage of the model, which is possible to use to project the average or total gasoline consumption for the ma- nufactured fleet of such cars over their first 5000 kilometers of use. A different situation appears in the following. This will require, what is to be called a tolerance interval, instead of a confidence interval. A design engineer is charged with the problem of determining how large a tank the car model really needs to guarantee that 99% of the cars produced will have a cruising range of 500 kilometers. 1 What the engineer really needs is a tolerance interval for a fraction of 100 × = 99% mileages of such automobiles. In many applications, the objective is to predict a single value of a variable to be observed at some future time, rather than to estimate the mean value of that variable. The interval containing the next single response. What if you want to make a claim about the resistance of a future cable, or the average resistance of a group of cables that you are going to manufacture in the future? The confidence intervals for the mean and standard deviation (Figure 2) that you calculated refer to the population of cables manufactured during the month in which the 40 cables were sampled, not to an individual observation or group of observations in the future. A prediction interval for a single future observation resembles a confidence interval for the mean, but it is wider because it takes into account the prediction noise by adding a 1 to the expression inside the square root:
• #45: Tolerance Interval The interval which contains at least a given proportion of the population. A tolerance interval is an enclosure interval for a specified proportion of the sampled population, not its mean or standard deviation. For a specified confidence level, you may want to determine lower and upper bounds such that 99 percent of the population is contained within them. Tolerance bounds allow you to set up specification limits by finding the lower and upper values, which correspond to stated yield or process capability goals.
• #46: a (1-α) proportion of the measurements can be estimated with 100( 1-Ƴ )% confidence