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Confidence Intervals

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General Mean ( )

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Computation First, edit and summarise the data. Obtain: sample size (n), sample mean (m) and sample standard deviation(sd). Note the indicated confidence level for the desired interval. Obtain the confidence coefficient (Z) via table look-up. Compute the interval as lower = m ─ Z*(sd/ n) And upper = m + Z*(sd/ n). Write the interval as [lower,upper].

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0 Z - Z “Central Probability Mass” ProbCent “Right Tail Probability Mass” ProbRt “Left Tail Probability Mass” ProbLt Confidence Multiplier Success Rate for corresponding Family of Intervals

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Confidence Coefficient Examples Using the table linked here, find coefficients for confidence levels 50%, 70%, 80%, 90%, 99%.here Z ProbRT ProbCent 50%: 0.70 0.24196 0.51607; Z ≈.70 70%: 1.05 0.14686 0.70628; Z ≈ 1.05 80%: 1.30 0.09680 0.80640; Z ≈ 1.30 90%: 1.65 0.049471 0.90106; Z ≈ 1.65 99%: 2.60 0.004661 0.99068 ; Z ≈ 2.60

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Interpretive Base: General Mean Briefly identify the population of interest. Briefly identify the population mean of interest. Briefly describe the family of samples. Briefly describe the family of intervals. Apply confidence level to the family of intervals. Interpret the computed interval.

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Populations, Samples and Families 1 We begin with a population T and a population mean . For any fixed sample size, n, the Family of Samples consists of the collection of all possible random samples of size n from T. Each individual member of this Family of Samples is a single random sample of size n from T. Family of Sample Means: A sample mean, m, can be computed from each member of the Family of Samples. If we compute a sample mean from each member of the Family of Samples, we obtain a Family of Means. Each member of this Family is a single sample mean computed from a member of the Family of Samples. When the Family of Sample Means is based on large (n>30) random samples from a population T, the members of the Family of Sample Means tend to cluster around the population mean for T. That is, a large proportion of the members of the Family of Sample Means are relatively close to the population mean for T.

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Populations, Samples and Families 2 Family of Intervals (for the Mean): For a fixed multiplier Z, an interval, m ± Z(sd/ n), can be computed from each member of the Family of Samples. If we compute an interval from each member of the Family of Samples, we obtain a Family of Intervals. Each member of this Family is a single interval of the form m ± Z(sd/ n) computed from a member of the Family of Samples. When the Family of Intervals is based on large (n>30) samples, a fixed percentage of the members of the Family of Intervals contain the population mean for T. The actual value of this percentage depends on the multiplier Z.

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Proportion (P)

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Computation First, edit and summarise the data. Obtain: sample size (n) and sample event count (n E ). Compute p = (n E /n) and sdp = ( p*(1 ─ p) / n). Note the indicated confidence level for the desired interval. Obtain the confidence coefficient (Z) via table look-up. Compute the interval as lower = p ─ (Z*sdp) And upper = p + (Z*sdp). Write the interval as [lower,upper].

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Interpretive Base: Proportion Briefly identify the population of interest. Briefly identify the population proportion of interest. Briefly describe the family of samples. Briefly describe the family of intervals. Apply confidence level to the family of intervals. Interpret the computed interval.

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Populations, Samples and Families 1 We begin with a population T and a population proportion P. For any fixed sample size, n, the Family of Samples consists of the collection of all possible random samples of size n from T. Each individual member of this Family of Samples is a single random sample of size n from T. Family of Sample Proportions: A sample proportion, p, can be computed from each member of the Family of Samples. If we compute a sample proportion from each member of the Family of Samples, we obtain a Family of Proportions. Each member of this Family is a single sample proportion computed from a member of the Family of Samples. When the Family of Sample Proportions is based on large (n>30) random samples from T, the members of the Family of Sample Proportions tend to cluster around the population proportion P. That is, a large proportion of the members of the Family of Sample Proportions are relatively close to the population proportion P.

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Populations, Samples and Families 2 Family of Intervals (for Proportion): For a fixed multiplier Z, an interval, p ± (Z*sdp), can be computed from each member of the Family of Samples. If we compute an interval from each member of the Family of Samples, we obtain a Family of Intervals. Each member of this Family is a single interval of the form p ± (Z*sdp) computed from a member of the Family of Samples. When the Family of Intervals is based on large (n>30) samples, a fixed percentage of the members of the Family of Intervals contain the population proportion P. The actual value of this percentage depends on the multiplier Z.

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Confidence Intervals: Basic Elements of Interpretation Briefly identify the population of interest: “The population consists of …” Briefly identify the population mean of interest: “We seek to estimate the population mean …” Briefly describe the family of samples: Each member of the Family of Samples is a single random sample of n=? Members of the population. The FoS consists of every possible random sample of this type …”

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Confidence Intervals: Basic Elements of Interpretation Briefly describe the family of intervals (FoS). “Each member of the FoS yields the following statistics: n (sample size), m (sample mean) and sd (sample std deviation. Each member of the FoS yields an interval of the form: [m – Z*(sd/ n), m + Z*(sd/ n)]. These intervals collectively form a Family of Intervals (FoI). Each member of the FoI is an interval derived from a member of the FoS.” Apply confidence level to the family of intervals (FoI). “Approximately ??% of these intervals contain the true population mean …, and the approximately 100-??% fail.” Interpret the computed interval. “If our single interval resides in the ??% supermajority, then the population mean is contained in the interval, and is …”

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