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• In estimation procedures, statistics calculated from random samples are used to estimate the value of population parameters.
• If we know that 42% of a random sample drawn from a city vote Liberal, we can estimate the percentage of all city residents who vote Liberal.
• The logic behind estimation is that information from samples is used to estimate information about the population.
• Statistics are used to estimate parameters.
• The Sampling Distribution is the link between sample and population.
• The value of the parameters is unknown but characteristics of the Sampling Distribution are defined by theorems.
• A point estimate is a sample statistic used to estimate a population value.
• Confidence intervals (for means or proportions) consist of a range of values.
• An estimator of a mean (or a proportion) is unbiased if the mean of its sampling distribution is equal to the population mean.
• The smaller the standard error (S.D. of the sampling distribution,) the more the samples are clustered about the mean of the sampling distribution.
• As sample size N increases, the standard error ( ) will decrease.
• The 95% confidence level means that we are willing to accept a probability of being wrong 5% of the time (or alpha ( α ) = .05)
• This probability (the area under the curve) will be divided evenly between the upper and lower tail of the distribution (.025 on either side of the curve.)
• The .95 in the middle section of the confidence level is our confidence level.
• The cut-off between our confidence level and +/ - .025 is represented by a Z-value of +/ - 1.96.
• The procedure for confidence intervals for proportions involves setting alpha = .05, finding the associated Z score, and substituting the sample information into formula: c.i = ± σ ± 1.96(n/√n).
• We can estimate that households in this community average 6.0 ± .44 hours of TV watching each day.
• The confidence interval for proportions can also be stated as 38% ≤ P u ≤ 46%.
• For a random sample of 178 households, average television viewing time was 6 hours/day with s = 3.
• If alpha is equal to 0.05, half of the probability is placed in the lower tail and half in the upper tail of the distribution.
• Changing back to %, we estimate that 42% ± 4% of the city residents voted Liberal.
• The Z-score for 95% confidence level (α +.05) is +/ - 1.96.
• The formula for calculating sample sizes for means or proportions involves n = minimum required sample size, Z = determined by your alpha level, σ or P u = population standard deviation (use s if unknown) or population proportion, and ME = margin of error (in +/ - actual units of your desired estimate).
• If 42% of a random sample of 764 people from a city voted Liberal, what % of the entire city voted Liberal?
• In other words, even if the statistic is as much as ± 1.96 standard deviations from the mean of the sampling distribution the confidence interval will still include the value of μ.
• Substituting all information into formula and solving, the confidence interval for means is 6.0 ± 1.96(3/√177) = 6.0 ± 1.96(3/13.30) = 6.0 ± 1.96(.23) = 6.0 ± .44.
• The confidence interval for means can also be stated as 5.56 ≤ μ ≤ 6.44.
• Substitute values into formula and solve: c.i = ± σ ± 1.96(n/√n).
• Nanos* Monarchy Poll (from CTV article) had a sample size of 1000, a level of confidence of 19/20 or 95%, an alpha of .05, and results of Support 69%, Oppose 29%.