Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval Statistics

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 2 Where We’ve Been Populations are characterized by numerical measures called parameters Decisions about population parameters are based on sample statistics Inferences involve uncertainty reflected in the sampling distribution of the statistic

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 3 Where We’re Going Estimate a parameter based on a large sample Use the sampling distribution of the statistic to form a confidence interval for the parameter Select the proper sample size when estimating a parameter

7.1: Identifying the Target Parameter The unknown population parameter that we are interested in estimating is called the target parameter. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 4 ParameterKey Word or PhraseType of Data µMean, averageQuantitative pProportion, percentage, fraction, rate Qualitative

7.2: Large-Sample Confidence Interval for a Population Mean A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 5

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 6 Suppose a sample of 225 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours.  What can we conclude about all college students’ television time?

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 7 Assuming a normal distribution for television hours, we can be 95%* sure that *In the standard normal distribution, exactly 95% of the area under the curve is in the interval … +1.96

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 8 An interval estimator or confidence interval is a formula that tell us how to use sample data to calculate an interval that estimates a population parameter.

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 9 The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter. The confidence level is the confidence coefficient expressed as a percentage. (90%, 95% and 99% are very commonly used.) 95% sure

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 10 The area outside the confidence interval is called α 95 % sure So we are left with (1 – 95)% = 5% = α uncertainty about µ

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 11 Large-Sample (1-α)% Confidence Interval for µ If  is unknown and n is large, the confidence interval becomes

7.2: Large-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 12 If n is large, the sampling distribution of the sample mean is normal, and s is a good estimate of  For the confidence interval to be valid … the sample must be random and … the sample size n must be large.

7.3: Small-Sample Confidence Interval for a Population Mean Large Sample Sampling Distribution on  is normal Known  or large n Standard Normal (z) Distribution Small Sample McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 13 Sampling Distribution on  is unknown Unknown  and small n Student’s t Distribution (with n-1 degrees of freedom)

7.3: Small-Sample Confidence Interval for a Population Mean Large SampleSmall Sample McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 14

7.3: Small-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 15 For the confidence interval to be valid … the sample must be random and … the population must have a relative frequency distribution that is approximately normal* * If not, see Chapter 14

7.3: Small-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 16 Suppose a sample of 25 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours.  What can we conclude about all college students’ television time?

7.3: Small-Sample Confidence Interval for a Population Mean McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 17 Assuming a normal distribution for television hours, we can be 95% sure that

7.4: Large-Sample Confidence Interval for a Population Proportion McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 18 Sampling distribution of  The mean of the sampling distribution is p, the population proportion.  The standard deviation of the sampling distribution is where  For large samples the sampling distribution is approximately normal. Large is defined as

7.4: Large-Sample Confidence Interval for a Population Proportion McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 19 Sampling distribution of

7.4: Large-Sample Confidence Interval for a Population Proportion McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 20 where and We can be 100(1-α)% confident that

7.4: Large-Sample Confidence Interval for a Population Proportion A nationwide poll of nearly 1,500 people … conducted by the syndicated cable television show Dateline: USA found that more than 70 percent of those surveyed believe there is intelligent life in the universe, perhaps even in our own Milky Way Galaxy. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 21 What proportion of the entire population agree, at the 95% confidence level?

7.4: Large-Sample Confidence Interval for a Population Proportion McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 22 If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results where

7.4: Large-Sample Confidence Interval for a Population Proportion McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 23 Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is.002. If 100 firms are sampled and one had declared bankruptcy, what is the 95% CI on the proportion of profitable firms that will tank the next year?

7.5: Determining the Sample Size To be within a certain sampling error (SE) of µ with a level of confidence equal to 100(1-α)%, we can solve for n: McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 24

7.5: Determining the Sample Size The value of  will almost always be unknown, so we need an estimate:  s from a previous sample  approximate the range, R, and use R/4 Round the calculated value of n upwards to be sure you don’t have too small a sample. McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 25

7.5: Determining the Sample Size McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 26 Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?

7.5: Determining the Sample Size McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 27 Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?

7.5: Determining the Sample Size McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 28 For a confidence interval on the population proportion, p, we can solve for n: To estimate p, use the sample proportion from a prior study, or use p =.5. Round the value of n upward to ensure the sample size is large enough to produce the required level of confidence.

7.5: Determining the Sample Size McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 29 How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within.01 with 90% confidence, if an initial estimate of.10 is used for p?

7.5: Determining the Sample Size McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals 30 How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within.01 with 90% confidence, if an initial estimate of.10 is used for p?