Condence Intervals about a Single Parameter:

Transcription

1 Chapter 9 Condence Intervals about a Single Parameter: 9.1 About a Population Mean, known Denition A point estimate of a parameter is the value of a statistic that estimates the value of the parameter. The sample mean X is a point estimate of the population mean : The sample standard deviation s is a point estimate of the population standard deviation : There are several reasons why we use X to estimate : 1. X is an unbiased estimator of, that is, the expected value of X is ; the parameter we are trying to estimate. Note an unbiased estimator of a parameter is an estimator that does not systematically overestimate or underestimate the value of the parameter it estimates. 2. The sample mean X is a consistent estimator of ; that is the larger the sample used, the closer the value of the sample mean gets to the value of the population mean. 3. The sample mean X is an ecient estimator of ; that is in repeated samples, the majority of the sample means will be \close" to the value of the population mean. Note there is a margin of error in using the sample mean as a point estimate of the value of the population mean. Is this a problem? Denition A condence interval estimate of a parameter conists of an interval of numbers, along with a probability that the interval contains the unknown parameter. The level of condence in a condence interval is a probability that represents the percentage of intervals that will contain if a large number of repeated samples are obtained. The level 109

2 of condence is denoted (1 ) 100%: A condence interval has the form Point estimate Margin of error. From the previous chapter we know that if the data within our sample is normally distributed, then the sample mean based on the sample will be normally distributed as well. From the previous chapter we also know that if a sample is suciently large (n 30) then the Central Limit Theorem states that X is approximately normally distributed. We can use these two facts to create a condence interval for : x z =2 pn ; x + z =2 pn where P (Z z =2 ) = =2; n is the sample size, and is the population standard deviation. The quantity E = z =2 pn is referred to as the (1 ) 100% error margin. Note: z 0:025 = 1:96: We call z =2 a critical z-value. In order for us to compute a (1 ) 100% condence interval for ; the following must be true: we have to know AND the population was normally distributed or the sample size n was greater than or equal to 30 AND the sample was a simple random sample. Example The mean height of 100 women in the United States was found to be 63.5" with a standard deviation of 2.5". Find a 95% condence interval for the population mean. A (1 ) 100% condence interval means that if we obtained many simple random samples of size n from the population whose mean is unknown, then approximately (1 ) 100% of the intervals will contain : Example For the previous example, compute a 90% error margin. 110

3 Example Assume that the standard deviation of the amount of copper precipitate from a chemical experiment is 4.7 grams. How many times should the experiment be repeated if one wants to be 99% sure that the true mean amount of precipitate is estimated within 1.8 grams? Formula Determine a formula for the sample size n using the (1 margin formula. ) 100% error 111

4 9.2 About a Population Mean, unknown Suppose that we do not have at least 30 measurements. As well, suppose that the population standard deviation is not available to us and we would like to use the sample standard deviation s in place of. Suppose X 1 through X n are drawn from a normal population with mean and standard deviation. Then the random variable T = X s= p n has a Student's t-distribution with (n 1) degrees of freedom. We use t = x s= p n to transform a particular sample mean value x with sample standard deviation s and (n 1) degrees of freedom. Remark We can only use the Student's t-distribution for SAMPLE MEANS!!!!!! Remark We can only use the Student's t-distribution if the distribution of the underlying random variable is NORMAL!!! Property (Student's t-distribution) 1. The t-distribution is dierent for dierent values of n (the sample size). 2. The t-distribution is centered at 0 and is symmetric about The area under the curve is 1. The area under the curve to the right of 0 equal the area under the curve to the left of 0 equals 1/2. 4. As t increases without bound, the graph approaches, but never equals, zero. As t decreases without bound, the graph approaches, but never equals, zero: 5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of ; thereby introducing further variabilitity into the t-statistic. 6. As the sample size n increases, the density curve of t gets closer to the standard normal curve. This result occurs because, as the sample size n increases, the values of s get closer to the values of ; by the Law of Large Numbers. Example Find the t-value such that the area under the t-distribution to the right of the t-value is 0.10, assuming 15 degrees of freedom. 112

5 Theorem (t-interval) Suppose a simple random sample of size n is taken from a population with unknown mean and unknown standard deviation : A (1 ) 100% condence interval for is given by x t =2 (df) p s ; x + t =2 (df) s p n n where t =2 (df) is computed with df degrees of freedom. Example Suppose x = and s = 4522 was computed from a data set comprising of 15 data points (assuming normality). Compute the corresponding 90% condence interval for the population mean : How important is normality when using the t-distribution to determine a condence interval for the population mean? Example The management of Disney World wanted to estimate the mean waiting time at the Dumbo ride. They randomly selected 15 riders and measured the amount of time the riders spent waiting in line. The data was to used to compute an average waiting time of minutes with an estimated standard deviation of minutes. Determine the width of the 95% condence interval. 113

6 Now suppose that one of the data points used in the above calculations was identied to be an outlier. The outlier value was removed from the sample and the average waiting time and estimated standard deviation was recomputed to be 27 minutes and 7.75 minutes respectively. Determine the width of the 95% condence interval. What should we do if the assumptions required to compute a t-interval are not met? 114

7 9.3 About a Population Proportion Theorem (Point Estimate of a Population Proportion) Suppose a simple random smaple of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The best point estimate of p; denoted bp; (the proportion of the population with the desired characteristic) is given by bp = x n where x is the number of individulas in the sample with the specied characteristic. Example In a poll conducted May 7-10, 2000, by ABC News, a simple random sample of 1068 American adults was asked \Have you ever been shot at?". Of the 1068 American adults surveyed, 96 responded yes. Obtain a point estimate for the population proportion of American adults who have been shot at. Theorem (Sampling distribution of bp) For a simple random sample of size n such that n 0:05N (where N is the population size), the sampling r distribution of bp is p(1 p) approximately normal with mean bp = p and standard deviation bp = ; provided n np(1 p) 10: Theorem Suppose a simple random sample of size n is taken from a population: A (1 ) 100% condence interval for p is given by r r! bp(1 bp) bp(1 bp) bp z =2 ; bp + z =2 n n where nbp(1 bp) 10 must be true. 115

8 Example For the above poll, compute a 95% condence interval for the population proportion p: Theorem (Sample size for estimating the population proportion p) The sample size required to obtain at (1 ) 100% condence interval for p with a margin of error E is given by: z= n = bp(1 bp) rounded to the next integer, if bp is available, or E z= n = 0:25 rounded to the next integer, if bp is NOT available. E Example A sociologist wishes to estimate the percentage of the American population living in poverty. What size of sample should be obtained if she wishes to be within 2 percentage points with 99% condence if: 1. she uses the 1999 estimate of 11.8% obtained from the Current Population Survey. 2. she does not use the prior estimate. 116

9 The above technique implements a hypothesis test but happens if one needs a (1 )100% condence interval for p? When n^p(1 ^p) < 10; as long as n 10; one can use the \Plus Four" condence interval to determine a (1 ) 100% condence interval for p. If the point estimate used in the \Plus Four" condence interval is ~p = x + 2 n + 4 and the standard error is r ~p(1 ~p) S:E: = n + 4 ; then the (1 ) 100% condence interval for p is r r! ~p(1 ~p) ~p(1 ~p) ~p z =2 n + 4 ; ~p + z =2 n + 4 and holds when n 10: (9.1) Example In a medical trial, patients are randomly assigned to the treatment or placebo group. Though the trial is double-blind, some patients can determine whether or not they are getting the active treatment, a fact which is quite often ignored when the results of the study are presented. In a simple random sample of 97 articles, only 7 of the articles discussed the success of \blinding". Determine a 95% condence interval for the true proportion of studies that discuss the success of \blinding". 117

10 9.4 About a Population Standard Deviation Recall that we use the symbol `s' to represent the sample standard deviation and `` to represent the population standard deviation. We can use s 2 (the sample variance) to estimate 2 (the population variance), that is, we can use s 2 to make inferences about 2. Let x 1, x 2,..., x n be a sample from a normal population with mean and standard deviation. Then the sampling distribution of is the 2 distribution with df = n 1: (n 1) s2 2 = Basic properties of 2 curves: P (xi x) 2 1. The total area under the curve is one. 2. The curve starts at ZERO, rises quickly to its maximum, and then extends indefinitely to the right, only approaching zero from above in the limit. 3. The curve is NOT symmetric; it is skewed to the right. 4. As the degrees of freedom increase, the 2 curve looks increasingly like a normal curve Finding Critical Values for the 2 distribution Example Find the critical values that separate the middle 90% of the 2 distribution from the 5% area in each tail, assuming 15 degrees of freedom. 118

11 9.4.3 Condence Intervals for 2 The 100(1 )% condence interval for 2 is given by s 2 (n 1) 2 =2 (df) < 2 < (n 1) 2 1 s 2 =2 (df) assuming that the data used to compute s 2 was normally distributed!!!! For the above to be a valid condence interval, the sample data must have been drawn from a normally distributed population!!!! Example An experimenter times 3 chinchillas going through a maze and obtains the following times (in minutes): 4.1, 5.2, and 4.8. Assuming that such times are normally distributed, calculate a 90% condence interval for the true variance. Example If a sample of four instrument readings on the same object yielded the measurements 353, 351, 351, and 355, calculate a 95% condence interval for the true standard deviation. 119