Confidence Intervals. Chapter 10

Transcription

1 Confidence Intervals Chapter 10

2 Confidence Intervals : provides methods of drawing conclusions about a population from sample data. In formal inference we use to express the strength of our conclusions about a population from sample data. When you use you are acting as if the data were a or come from a experiment.

3 Example Suppose you want to estimate the mean SAT-M score for more than 350,000 high school seniors in California. Only 49% of California students take the SAT. These self-selected seniors are planning on attending college and so are not representative of all California seniors. You know better than to make inference about the population based on any sample data. So, you give the test to an SRS of 500 California seniors. The mean for your sample is x-bar = 461. (Let us suppose we know the standard deviation of SAT-M scores in the population of all California seniors is s = 100.) What can you say about the mean score m of the population of all 350,000 seniors?

4 Example What are the basic facts about the sampling distribution of x-bar? 1. The tells us the mean (x-bar) of 500 scores has a distribution close to. 2. The of a normal sampling distribution is the as the ( ) of an entire population. 3. The of x-bar for an SRS of 500 students is

5 Example The rule says that in 95% of all samples, the mean score x-bar for the sample will be within standard deviations of the population mean. So the mean of 500 SAT-M scores will be within points of m in of all samples. Whenever x-bar is within 9 points of the unknown m, m is within 9 points of all observed x-bars. This happens in of all samples. So in 95% of all samples, the unknown m lies within and.

6 Example

7 Example Our sample gave x-bar = 461. We say that we are confident that the unknown mean SAT-M score for all California high school seniors lies between and. Either you do or do not catch the in your interval. If you are 95% confident that the unknown m lies between a and b (where a and b are values) you are saying that we got these numbers by that gives correct results of the time.

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9 A LEVEL C CONFIDENCE INTERVAL for a parameter has two parts: 1. An interval calculated from the data, usually of the form ± The estimate is our for the value of the unknown. The margin of error shows we believe our guess is, based on the of the estimate. 1. A confidence level C, which gives the probability that the interval will the value in samples.

10 CONSTRUCTING A CONFIDENCE INTERVAL The construction of a for a population mean is appropriate when: 1. The data comes from an from the population of interest. 2. The sampling distribution of x-bar is.

11 CRITICAL VALUES The number z* with probability p lying to its under the standard normal curve is called the of the standard normal distribution.

12 Example To find an 80% confidence interval, we must capture the middle 80% of the normal sampling distribution of x-bar. To do so we need to leave out in each tail. So z* is the point with area to its right (or to its left) on the standard normal curve. For 80% z* =, so and on a standard normal curve an 80% confidence interval lies between and

13 Example

14 Using the z table Confidence level Tail Area z* 90% % %

15 Practice Find z* for a 75% confidence interval.

16 CONFIDENCE INTERVAL FOR A POPULATION MEAN An SRS of size n from a population having an and a standard deviation s has a level C confidence interval for m as below: x z This interval is when the population distribution is normal and approximately correct for large values of n. * s n

17 1. Conditions for a Z-interval: 2. 3.

18 Steps to Construct ANY P: Confidence Interval PANIC A: N: I: C:

19 Example 1 Every week, a website displays the average price of a gallon of regular gasoline nationwide. This is based on a random poll of 80 gas stations. Suppose the average price was $2.94 and historically the standard deviation of a gallon of gas was 8.3 cents. What is the 95% confidence interval for the price of a gallon of gasoline nationwide?

20 Example 2 A study of the career paths of hotel general managers sent questionnaires to an SRS of 175 hotels belonging to major U.S. hotel chains. There were 123 responses. The average time these 123 managers spent with their company was years. Give the following confidence intervals for the mean number of years of majorchain hotel managers have spent with their current company. (Take it as known that the standard deviation of time spent with their company for all general managers is 2.9 years). a.) 90% b) 95% c) 99%

21 Behavior of Confidence Intervals The margin of error is A margin of error says that we have gotten really close to the parameter. We want confidence and margin of errors which make small intervals. Margin of error gets smaller when 1. z* gets same as smaller confidence level C (to obtain a margin of error you must be willing to accept confidence) 2. gets smaller a small means smaller variation 3. gets larger

22 SAMPLE SIZE MANUFACTURING To determine the sample size n that will yield a confidence interval for population mean with specified margin of error, ME, set the expression for the margin of error to be or to and solve:

23 Example 3 The standard deviation of the lifetime of a certain brand of tire is 2,825 miles. The tire company wishes to estimate the average lifetime of that tire with 95% confidence with a margin of error of 1,000 miles. How many times should it sample?

24 CAUTIONS!! 1. The data must be an from the population. 2. The formula is for probability sampling designs more complex than an SRS. 3. There is no correct method for inference from data haphazardly collected with of unknown. Fancy formulas cannot rescue badly produced data. 4. Because is strongly influence by a few extreme observations, can have a effect on confidence intervals. You should search for outliers and try to them or justify their.

25 CAUTIONS!! 5. If the is small and the population the true confidence level will be different from the value C used in computing the interval. Examine your data carefully for and other signs of. 6. You must know the of the population. This unrealistic requirement has little use in statistical practice. We will learn in the next chapter what to do when is unknown.

26 You measure the blood pressure of 25 employees of a company. A 95% confidence interval for the mean BP is (122, 138). What is a valid interpretation of the confidence interval? 1. 95% of the sample of employees has a BP between 122 and % of the population of employees has a BP between 122 and The probability that the population BP is between 122 and 138 is If the procedure were repeated many times, 95% of the sample means would be between 122 and If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean BP 6. We are 95% confident that the true population BP is between 122 and If the procedure were repeated many times, 95% of the time the true population would be between 122 and The probability that our procedure will generate a confidence interval containing the true mean is 95%.