Kenwood Academy
AP Statistics Chapter 8
Estimating with Confidence
|Confidence Intervals: The Basics |Objectives: |
| |-Interpret confidence levels and confidence intervals |
| |-Construct a confidence interval |
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| | |
| |Our goal in many statistical settings is to use a sample statistic to estimate a population parameter. In Chapter |
| |4, we learned if we randomly select the sample, we should be able to generalize our results to the population of |
| |interest. |
| |In Chapter 7, we learned that different samples yield different results for our estimate. Statistical inference |
| |uses the language of probability to express the strength of our conclusions by taking chance variation due to random|
| |selection or random assignment into account. |
| |In this chapter, we’ll learn one method of statistical inference – confidence intervals – so we may estimate the |
| |value of a parameter from a sample statistic. As we do so, we’ll learn not only how to construct a confidence |
| |interval, but also how to report probabilities that would describe what would happen if we used the inference method|
| |many times. |
|Point Estimator and | |
|Point Estimate |A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic |
| |from a sample is called the point estimate. Ideally, a point estimate is our “best guess” at the value of an |
| |unknown parameter |
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| |#1,3 |
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|Confidence Interval, Margin of Error, | |
|Confidence Level |A confidence interval for a parameter has two parts: |
| |An interval calculated from the data, which has the form |
| | |
| |estimate [pic] margin of error |
| |or |
| |statistic [pic] (critical value)* (standard deviation of statistic) |
| | |
| |The margin of error tells how close the estimate tends to be to the unknown parameter in repeated random sampling. |
| |“Margin of error plus or minus 3 percentage points” is the shorthand for this statement: |
| |If we took many samples using the same method we used to get this one sample, 95% of the samples would give a result|
| |within plus or minus 3 percentage points of the true population proportion (mean). |
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| |#5 |
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| |A confidence level C: The confidence level tells us how likely it is that the method we are using will produce an |
| |interval that captures the population parameter if we use it many times. |
| |That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. |
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| |#7,9 |
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|Interpreting Confidence Levels and |Confidence level: To interpret a C% confidence level for an unknown parameter (mean/proportion), say |
|Confidence Intervals. |“Intervals produced with this method will capture the true |
| |____mean/proportion in about ___% of all possible samples of this same size from this same population.” |
| | |
| |Confidence interval: To interpret a C% confidence interval for an unknown parameter, say |
| |“We are _____% confident that the interval from _____ to _____ captures the true ___[population parameter in |
| |context]____. |
| | |
| |The price we pay for greater confidence is a wider interval. If we’re satisfied with 80% confidence, then our |
| |interval of plausible values for the parameter will be much narrower than if we insist on 90%, 95%, or 99% |
| |confidence. |
| | |
| |#11,13 |
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|AP Exam Tip |On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure |
| |you understand the difference: the confidence level describes the long-run capture rate of the method, and the |
| |confidence interval gives a set of plausible values for the parameter. |
| | |
| |The confidence interval for estimating a population parameter has the form |
|Calculating a Confidence Interval | |
| |statistic [pic] (critical value)* (standard deviation of statistic) |
| | |
| |where the statistic we use is the point estimator for the parameter. |
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|Facts about CI | |
| |The confidence level decreases. There is a trade-off between the confidence level and the margin of error. To |
| |obtain a smaller margin of error from the same data, you must be willing to accept lower confidence. |
| |The sample size n increases. We like high confidence and small margin of error. A small margin of error says that |
| |we have pinned down the parameter almost precisely. (Remember that [pic] and [pic] . So as the sample size n |
| |increases, the standard deviation of the statistic decreases.) |
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| | |
| |Random: The data come from a well-designed random sample or randomized experiment |
| |Normal: The sampling distribution of the statistic is approximately Normal |
|Conditions for Constructing a CI |For means: The sampling distribution is exactly Normal if the population distribution is Normal. When the population|
| |distribution is not Normal, then the central limit theorem tells us the sampling distribution will be approximately |
| |Normal if n is sufficiently large (n ≥ 30). |
| |For proportions: We can use the Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ |
| |10. |
| | |
| |Independent: Individual observations are independent. |
| |When sampling without replacement, the sample size n should be no more than 10% |
| |of the population size N (the 10% condition) to use our formula for the standard |
| |deviation of the statistic. |
| | |
| |#17 |
| |Objective: |
| |-Carry out the steps in constructing a confidence interval for a population proportion: define the parameter; check |
|8.2 Confidence Intervals: Estimating a |conditions; perform calculations; interpret results in context. |
|Population Proportion |-Determine the sample size required to obtain a level C confidence interval for a population proportion with a |
| |specified margin of error. |
| |-Understand how the margin of error of a confidence interval changes with the sample size and the level of |
| |confidence C. |
| |- choose a sample size |
| |Assignment 8.2 page 496 to #27-47 odd, 49-54 all |
|Standard Error | |
| |When the standard deviation of a statistic is estimated from data, the result is called the standard error of the |
| |statistic. |
| | |
| |The sample proportion [pic]is the statistic we use to estimate p. When the Independent condition is met, the |
| |standard deviation of the sampling distribution of [pic] is |
| |[pic] |
| |If we don’t know the value of p, we replace it with the sample proportion [pic] |
| |[pic] |
| |This quantity is called the standard error (SE) of the sample proportion[pic]. It describes how close the sample |
| |proportion [pic]will be, on average, to the population proportion p in repeated SRSs of size n. |
| | |
| |One-Sample z Interval for a Population Proportion: |
| |Choose an SRS of size n from a large population with unknown p of successes. A level C confidence interval for p is|
| | |
| |Formula: Estimate ( margin of error |
| |Statistic ( (critical value)*(Standard deviation of statistic) |
|Construct a Confidence Interval for p |[pic] |
| |Calculator: STAT ( TESTS (1-PropZ Interval ( Data or Stats |
| | |
| |C.I. Tail area z*(Critical Value) |
| |80% 0.1 1.282 |
| |90% 0.05 1.645 |
| |95% 0.025 1.960 |
| |99% 0.005 2.576 |
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| | |
| |Conditions for constructing a CI for p: |
| |1. Data must come from a random sample or randomized experiment |
| |2. At least 10 successes and failures; that is [pic]and [pic] |
| |3. Observations are independent; 10% condition is met |
| | |
| |State: What parameter do you want to estimate, and at what confidence level? |
| |Plan: Identify the appropriate inference method. Check conditions. |
| |Do: If the conditions are met, perform the calculations. |
| |Conclusion: Interpret your interval in the context of the problem |
| |Choosing a Sample Size for Desired Margin of Error |
|Confidence Intervals: Four-Step Process |To determine the sample size n that will yield a level C confidence interval for a population proportion p with a |
| |maximum margin of error ME, solve the following inequality for n: |
| |[pic] |
|Choosing a Sample Size |where[pic]is a guessed value for the sample proportion. The margin of error will always be less than or equal to ME|
| |if you take the guess[pic]to be 0.5 |
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| |Objective: |
| |- When [pic]is known, construct and interpret a one-sample z interval for a population mean |
| |- When [pic]is unknown, construct and interpret a one-sample t interval for a population mean |
| |- -Determine the sample size required to obtain a level C confidence interval for a population mean with a |
| |specified margin of error. |
| |Assignment 8.3 page 518 #55-73 odd, 75-80 |
|8.3 Confidence Intervals: Estimating a | |
|Population Mean |One-Sample z Interval for a Population Mean: |
| |Choose an SRS of size n from a population with unknown mean [pic]and known standard deviation [pic]of successes. A |
| |level C confidence interval for [pic] is |
| |Formula: [pic] |
|The z distribution |Calculator: STAT ( TESTS (Z Interval ( Data or Stats |
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| |Conditions for constructing a CI for[pic] : |
| |1. Random: Data must come from a random sample or randomized experiment |
| |2. Normal: Population distribution is normal or a large sample ([pic]) |
| |3. Independent: Observations are independent; 10% condition is met |
| |Choosing a Sample Size for Desired Margin of Error When Estimating[pic] |
| |To determine the sample size n that will yield a level C confidence interval for a population mean [pic] with a |
| |maximum margin of error ME, solve the following inequality for n: |
| |[pic] |
| |(Always roundup when finding n) |
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|Choosing a Sample Size | |
| |When we do not know σ, we substitute the standard error (SE) [pic] of x for its s.d. [pic]. The statistic that |
| |results does not have a normal dist. It has a distr. called the t distribution. |
| | |
| |The t distribution has a different shape than the standard Normal curve: still symmetric with a single peak at 0, |
| |but with much more area in the tails. |
| |[pic] |
| | |
| |The statistic t has the same interpretation as any standardized statistic: it says how far [pic]is from its[pic] in |
|The t distribution |standard deviation units. There is a different t distribution for each sample size. We specify a particular t |
| |distribution by giving its degrees of freedom (df). |
| | |
| |The t Distribution; Degrees of Freedom |
| |Choose an SRS of size from a large population with mean [pic]and standard deviation [pic] |
| |[pic] with degrees of freedom df = n-1. |
| | |
| |The density curves of the t distr. are similar in shape to the standard normal curve. They are symmetric about |
| |zero, single peaked, and bell shaped. |
| |The spread of the t distribution is a bit greater than the standard normal distr. The t distribution has more |
| |probability in the tails and less in the center than the normal. |
| |As the degrees of freedom increase, the t density curve approaches the N(0,1) curve. This happens because [pic]for |
| |the fixed parameter [pic] |
| |causes little extra variation when the sample is large. |
| | |
| |One-Sample t Interval for a Population Mean: |
| |Choose an SRS of size n from a population with unknown mean. A level C confidence interval for [pic] is |
| |Formula: [pic] with df = n – 1 |
| |Where t* is the critical value for [pic]distribution. |
| | |
| |Table B gives critical values for the t distribution. By looking down any column, you can check that the t critical|
| |values approach the normal values as the degrees of freedom increase. When the actual df does not appear in the |
| |table, use the greatest df available that is less than the greatest df available. |
| | |
| |Calculator: STAT ( TESTS (T Interval ( Data or Stats |
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| |Conditions: |
| |1. Random: Data must come from a random sample or randomized experiment |
| |2. Normal: Population distribution is normal or a large sample ([pic]) |
|Facts about the t distribution: |3. Independent: Observations are independent; 10% condition is met |
| |An inference procedure is called robust if the probability calculations involved in that procedure remain fairly |
| |accurate when a condition for using the procedure is violated. The t procedures are not robust against outliers, |
| |because [pic]and [pic]are not resistant to outliers. |
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| |Using One sample t Procedures |
| |Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more|
| |important than the assumption that the population distribution is normal. |
| |Sample size less than 15. Use t procedure if the data are close to Normal (roughly symmetric, single peak, no |
| |outliers). If the data are clearly skewed or if outliers are present, do not use t. |
| |Sample size at least 15. The t procedures can be used except in the presence of outliers or strong skewness. |
|t Confidence Interval |Large samples. The t procedures can be used even for clearly skewed distributions when the sample is large, roughly|
| |n ( 30. |
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|Robust procedures | |
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