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AP Statistics Section 10.2 A CI for Population Mean When is Unknown

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In Section 10.1B, we constructed a confidence interval for the population mean when we knew the population standard deviation. It is extremely unlikely that we would actually know the population standard deviation, however.

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In this section, we will discover how to construct a confidence interval for an unknown population mean when we don’t know the standard deviation. We will do this by estimating from the data.

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This need to estimate with s introduces additional error into our calculations. To account for this, we will use a critical value of t * instead of z * when computing our confidence interval.

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Note the following properties of a t distribution:

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The density curves of the t distributions are similar in shape to the standard Normal, or z, distribution (i.e.

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Unlike the standard Normal distribution, there is a different t distribution for each sample size n. We specify a particular t distribution by giving its __________________ ( _____ ). When we perform inference about using a t distribution, the appropriate degrees of freedom is equal to ______. We will write the t distribution with k degrees of freedom as _____.

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The spread of the t distributions is slightly greater than that of the z distribution. The t distributions are less concentrated around the mean and have more probability in the tails. This is what accounts for the increased error in using s instead of.

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As the degrees of freedom increase, the t curve approaches the standard Normal curve ever more closely. This happens because s approximates more accurately as the sample size increases.

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Table B, gives the values of t * for various degrees of freedom and various upper-tail probabilities. When the actual degrees of freedom does not appear in Table B, use the largest degrees of freedom that is less than your desired degrees of freedom.

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Example: Determine the appropriate value of t * for a confidence interval for with the given confidence level and sample size. a) 98% with n = 22

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Example: Determine the appropriate value of t * for a confidence interval for with the given confidence level and sample size. b) 90% with n = 38

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TI 84:

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As before, we need to verify three important conditions before we estimate a population mean.

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SRS: Our data are a SRS of size n from the population of interest or come from a randomized experiment. This condition is very important.

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Normality of : The population has a Normal distribution or : Use t procedures if sample data appears roughly Normal. : The t procedures can be used except in the presence of outliers or strong skewness in the sample data. The t procedures are robust. : The t procedures can be used even for clearly skewed distributions. However, outliers are still a concern. You may still refer to the Central Limit Theorem in this situation.

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Independence: The method for calculating a confidence interval assumes that individual observations are independent. To keep the calculations reasonably accurate when we sample without replacement from a finite population, we should verify that the population size is at least _______________________(________).

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Example: A number of groups are interested in studying the auto exhaust emissions produced by motor vehicles. Here is the amount of nitrogen oxides (NOX) emitted by light-duty engines (grams/mile) from a random sample of size n = 46. Construct and interpret a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this type.

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Parameter: The population of interest is ____________________. We want to estimate, the ____________________________.

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Conditions: Since we do not know, use ______________________ SRS: Normality of : Independence:

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Calculation:

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Interpretation:

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TI 83/84:

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Standard Error When the standard deviation of a statistic, i.e., is estimated from the data, the result is called the standard error of the statistic.

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Some textbooks simply refer to standard error as the standard deviation of the sampling distribution,, whether it is estimated from the data or not.

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