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Published byHarold Mosley Modified over 9 years ago
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Estimation Goal: Use sample data to make predictions regarding unknown population parameters Point Estimate - Single value that is best guess of true parameter based on sample Interval Estimate - Range of values that we can be confident contains the true parameter
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Point Estimate Point Estimator - Statistic computed from a sample that predicts the value of the unknown parameter Unbiased Estimator - A statistic that has a sampling distribution with mean equal to the true parameter Efficient Estimator - A statistic that has a sampling distribution with smaller standard error than other competing statistics
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Point Estimators Sample mean is the most common unbiased estimator for the population mean Sample standard deviation is the most common estimator for (s 2 is unbiased for 2) Sample proportion of individuals with a (nominal) characteristic is estimator for population proportion
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Confidence Interval for the Mean Confidence Interval - Range of values computed from sample information that we can be confident contains the true parameter Confidence Coefficient - The probability that an interval computed from a sample contains the true unknown parameter (.90,.95,.99 are typical values) Central Limit Theorem - Sampling distributions of sample mean is approximately normal in large samples
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Confidence Interval for the Mean In large samples, the sample mean is approximately normal with mean and standard error Thus, we have the following probability statement: That is, we can be very confident that the sample mean lies within 1.96 standard errors of the (unknown) population mean
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Confidence Interval for the Mean Problem: The standard error is unknown ( is also a parameter). It is estimated by replacing with its estimate from the sample data: 95% Confidence Interval for :
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Confidence Interval for the Mean Most reported confidence intervals are 95% By increasing confidence coefficient, width of interval must increase Rule for (1- )100% confidence interval:
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Properties of the CI for a Mean Confidence level refers to the fraction of time that CI’s would contain the true parameter if many random samples were taken from the same population The width of a CI increases as the confidence level increases The width of a CI decreases as the sample size increases CI provides us a credible set of possible values of with a small risk of error
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Confidence Interval for a Proportion Population Proportion - Fraction of a population that has a particular characteristic (falling in a category) Sample Proportion - Fraction of a sample that has a particular characteristic (falling in a category) Sampling distribution of sample proportion (large samples) is approximately normal
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Confidence Interval for a Proportion Parameter: (a value between 0 and 1, not 3.14...) Sample - n items sampled, X is the number that possess the characteristic (fall in the category) Sample Proportion: –Mean of sampling distribution: –Standard error (actual and estimated):
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Confidence Interval for a Proportion Criteria for large samples –0.30 30 –Otherwise, X > 10, n-X > 10 Large Sample (1- )100% CI for :
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Choosing the Sample Size Bound on error (aka Margin of error) - For a given confidence level (1- ), we can be this confident that the difference between the sample estimate and the population parameter is less than z /2 standard errors in absolute value Researchers choose sample sizes such that the bound on error is small enough to provide worthwhile inferences
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Choosing the Sample Size Step 1 - Determine Parameter of interest (Mean or Proportion) Step 2 - Select an upper bound for the margin of error (B) and a confidence level (1- ) Proportions (can be safe and set =0.5): Means (need an estimate of ):
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Small-sample Inference for t Distribution: –Population distribution for a variable is normal –Mean , Standard Deviation –The t statistic has a sampling distribution that is called the t distribution with (n-1) degrees of freedom: Symmetric, bell-shaped around 0 (like standard normal, z distribution) Indexed by “degrees of freedom”, as they increase the distribution approaches z Have heavier tails (more probability beyond same values) as z Table B gives t A where P(t > t A ) = A for degrees of freedom 1-29 and various A
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Probability DegreesofFreedomDegreesofFreedom CriticalValuesCriticalValues Critical Values
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Small-Sample 95% CI for Random sample from a normal population distribution: t.025,n-1 is the critical value leaving an upper tail area of.025 in the t distribution with n-1 degrees of freedom For n 30, use z.025 = 1.96 as an approximation for t.025,n-1
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Confidence Interval for Median Population Median - 50 th -percentile (Half the population falls above and below median). Not equal to mean if underlying distribution is not symmetric Procedure –Sample n items –Order them from smallest to largest –Compute the following interval: –Choose the data values with the ranks corresponding to the lower and upper bounds
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