Chapter 9 Estimation Using a Single Sample 1

Chapter 9 Estimation Using a Single Sample 1 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Point Estimation A point estimate of a population characteristic is a single number that is based on sample data and represents a plausible value of the characteristic. 2 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example A sample of 200 students at a large university is selected to estimate the proportion of students that wear contact lens. In this sample 47 wore contact lens. Let p = the true proportion of all students at this university who wear contact lens. Consider “success” being a student who wears contact lens. 3 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example A sample of weights of 34 male freshman students was obtained. 185 202 197 188 166 148 161 139 214 170 231 180 174 177 283 207 176 194 175 170 184 180 184 176 202 151 189 167 179 178 176 168 177 155 If one wanted to estimate the true mean of all male freshman students, you might use the sample mean as a point estimate for the true mean. 4 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example After looking at a histogram and boxplot of the data (below) you might notice that the data seems reasonably symmetric with a outlier, so you might use either the sample median or a sample trimmed mean as a point estimate. Calculated using Minitab 5 140 180 220 260 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Bias A statistic with mean value equal to the value of the population characteristic being estimated is said to be an unbiased statistic. A statistic that is not unbiased is said to be biased. Sampling distribution of a unbiased statistic Original distribution 6 Sampling distribution of a biased statistic © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Criteria Given a choice between several unbiased statistics that could be used for estimating a population characteristic, the best statistic to use is the one with the smallest standard deviation. Unbiased sampling distribution with the smallest standard deviation, the Best choice. 7 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Large-sample Confidence Interval for a Population Proportion A confidence interval for a population characteristic is an interval of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the value of the characteristic will be captured inside the interval. 8 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Confidence Level The confidence level associated with a confidence interval estimate is the success rate of the method used to construct the interval. 9 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Recall For the sampling distribution of p, p = p, and for large* n The sampling distribution of p is approximately normal. Specifically when n is large*, the statistic p has a sampling distribution that is approximately normal with mean p and standard deviation. 10 * np 10 and np(1 -p) 10 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Some considerations 11 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Some considerations This interval can be used as long as np 10 and np(1 -p) 10 12 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

The 95% Confidence Interval 13 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example For a project, a student randomly sampled 182 other students at a large university to determine if the majority of students were in favor of a proposal to build a field house. He found that 75 were in favor of the proposal. Let p = the true proportion of students that favor the proposal. 14 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example - continued So np = 182(0. 4121) = 75 >10 and n(1 -p)=182(0. 5879) = 107 >10 we can use the formulas given on the previous slide to find a 95% confidence interval for p. 15 The 95% confidence interval for p is (0. 341, 0. 484). © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

The General Confidence Interval The general formula for a confidence interval for a population proportion p when 1. p is the sample proportion from a random sample , and 2. The sample size n is large (np 10 and np(1 -p) 10) is given by 16 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Finding a z Critical Value Finding a z critical value for a 98% confidence interval. 2. 33 Looking up the cumulative area or 0. 9900 in the body of the table we find z = 2. 33 17 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Some Common Critical Values Confidence z critical level value 80% 95% 98% 99. 9% 18 1. 28 1. 645 1. 96 2. 33 2. 58 3. 09 3. 29 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Terminology The standard error of a statistic is the estimated standard deviation of the statistic. 19 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Terminology The bound on error of estimation, B, associated with a 95% confidence interval is (1. 96)·(standard error of the statistic). The bound on error of estimation, B, associated with a confidence interval is (z critical value)·(standard error of the statistic). 20 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Sample Size The sample size required to estimate a population proportion p to within an amount B with 95% confidence is The value of p may be estimated by prior information. If no prior information is available, use p = 0. 5 in the formula to obtain a conservatively large value for n. Generally one rounds the result up to the nearest integer. 21 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Sample Size Calculation Example If a TV executive would like to find a 95% confidence interval estimate within 0. 03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if a prior estimate for p was 0. 15. We have B = 0. 03 and the prior estimate of p = 0. 15 A sample of 545 or more would be needed. 22 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Sample Size Calculation Example revisited Suppose a TV executive would like to find a 95% confidence interval estimate within 0. 03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if we have no reasonable prior estimate for p. We have B = 0. 03 and should use p = 0. 5 in the formula. 23 The required sample size is now 1068. Notice, a reasonable ball park estimate for p can lower the needed sample size. © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Another Example A college professor wants to estimate the proportion of students at a large university who favor building a field house with a 99% confidence interval accurate to 0. 02. If one of his students performed a preliminary study and estimated p to be 0. 412, how large a sample should he take. We have B = 0. 02, a prior estimate p = 0. 412 and we should use the z critical value 2. 58 (for a 99% confidence interval) The required sample size is 4032. 24 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

One-Sample z Confidence Interval for m 2. The sample size n is large (generally n 30), and 3. s , the population standard deviation, is known the general formula for a confidence interval for a population mean is given by 25 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

One-Sample z Confidence Interval for m If n is small (generally n < 30) but it is reasonable to believe that the distribution of values in the population is normal, a confidence interval for (when is known) is Notice that this formula works when is known and either 1. n is large (generally n 30) or 2. The population distribution is normal (any sample size. 26 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example A certain filling machine has a true population standard deviation = 0. 228 ounces when used to fill catsup bottles. A random sample of 36 “ 6 ounce” bottles of catsup was selected from the output from this machine and the sample mean was 6. 018 ounces. Find a 90% confidence interval estimate for the true mean fills of catsup from this machine. 27 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Example I (continued) The z critical value is 1. 645 90% Confidence Interval (5. 955, 6. 081) 28 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Unknown - Small Size Samples [All Size Samples] An Irish mathematician/statistician, W. S. Gosset developed the techniques and derived the Student’s t distributions that describe the behavior of 29 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

t Distributions If X is a normally distributed random variable, the statistic follows a t distribution with df = n-1 (degrees of freedom). 30 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

t Distributions This statistic is fairly robust and the results are reasonable for moderate sample sizes (15 and up) if x is just reasonable centrally weighted. It is also quite reasonable for large sample sizes for distributional patterns (of x) that are not extremely skewed. 31 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

t Distributions 32 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

t Distributions Notice: As df increase, t distributions approach the standard normal distribution. Since each t distribution would require a table similar to the standard normal table, we usually only create a table of critical values for the t distributions. 33 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

34 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

One-Sample t Procedures Suppose that a SRS of size n is drawn from a population having unknown mean . The general confidence limits are and the general confidence interval for is 35 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Confidence Interval Example Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are 82 66 90 84 75 88 80 94 110 91 Find a 90% confidence interval estimate for the true mean number of hours of television watched per week by shut-ins. 36 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Confidence Interval Example Calculating the sample mean and standard deviation we have n = 10, = 86, s = 11. 842 We find the critical t value of 1. 833 by looking on the t table in the row corresponding to df = 9, in the column with bottom label 90%. Computing the confidence interval for is 37 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Confidence Interval Example To calculate the confidence interval, we had to make the assumption that the distribution of weekly viewing times was normally distributed. Consider the normal plot of the 10 data points produced with Minitab that is given on the next slide. 38 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Confidence Interval Example Notice that the normal plot looks reasonably linear so it is reasonable to assume that the number of hours of television watched per week by shut-ins is normally distributed. Typically if the p-value is more than 0. 05 we assume that the distribution is normal 39 Anderson-Darling Normality Test A-Squared: 0. 226 P-Value: 0. 753 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.