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sample proportion

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The sample proportion is the best point estimate of the population proportion.

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point estimate

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single value (or point) used to approximate a population parameter.

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confidence interval (or interval estimate)

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range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

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confidence level, degree of confidence, or the confidence coefficient.

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is the probability 1 - α (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times.

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Margin of Error for Proportions

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E = z(α/2) * ( [p̂ * q̂] / n ) ^(1/2)

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Confidence Interval for Estimating a Population Proportion p

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p̂ - E < p̂ < p̂ + E
where
E = z(α/2) * ( [p̂ * q̂] / n ) ^(1/2)

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Confidence Interval for Estimating a Population Proportion p notation

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p̂ - E < p̂ < p̂ + E
p̂ +/- E
(p̂ - E, p̂ + E)

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Procedure for Constructing a Confidence Interval for p

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1.Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np >= 5, and nq >= 5 are both satisfied.)
2. Refer to Table A-2 and find the critical value z(α/2) that corresponds to the desired confidence level.
3. Evaluate the margin of error
4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p - E and p + E. Substitute those values in the general format for the confidence interval:
p̂ - E < p̂ < p̂ + E
5. Round the resulting confidence interval limits to three significant digits.

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Sample Size for Estimating Proportion p

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When an estimate of p̂ is known
n = (z(α/2)^2 * p̂ * q̂) / E^2
When no estimate of p is known:
n = (z(α/2)^2 * 0.25) / E^2

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Round-Off Rule for Determining Sample Size

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If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

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Finding the Point Estimate and E from a Confidence Interval

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Point estimate of
p̂ = (upper confidence limit + lower confidence limit) / 2
Margin of error
E = (upper confidence limit - lower confidence limit) / 2

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Point Estimate of the Population Mean

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The sample mean xbar is the best point estimate of the population mean µ.

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Confidence Interval for Estimating a Population Mean (with σ Known)

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1. The sample is a simple random sample. (All samples of the same size have an equal chance of being selected.)
2. The value of the population standard deviation σ is known.
3. Either or both of these conditions is satisfied: The population is normally distributed or n > 30.

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Confidence Interval for Estimating a Population Mean (with σ Known)

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xbar - E < μ < xbar + E
or
xbar +/- E
or
(xbar - E, xbar + e)
where
E = z(α/2) * ( σ / (n)^(1/2) )

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confidence interval limits

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xbar - E, xbar + E

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Sample Mean

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1. For all populations, the sample mean x is an unbiased estimator of the population mean xbar, meaning that the distribution of sample means tends to center about the value of the population mean μ.
2. For many populations, the distribution of sample means x tends to be more consistent (with less variation) than the distributions of other sample statistics.

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Procedure for Constructing a Confidence Interval for µ (with Known σ)

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1. Verify that the requirements are satisfied.
2. Refer to Table A-2 or use technology to find the critical value z(α/2) that corresponds to the desired confidence level
3. Evaluate the margin of error
E = z(α/2) * ( σ / (n)^(1/2) )
4. Find the values of xbar - E and xbar + E. Substitute those values in the general format of the confidence interval
5. Round using the confidence intervals round-off rules.

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Round-Off Rule for Confidence Intervals Used to Estimate µ

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When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data.
When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample mean.

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Finding a Sample Size for Estimating a Population Mean

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n = ( [z(α/2) * σ] / E)^2

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Round-Off Rule for Sample Size n

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If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

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Sample Mean

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The sample mean is the best point estimate of the population mean.

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Student t Distribution

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If the distribution of a population is essentially normal, then the distribution of
t = (xbar - μ) / [ s / n^(1/2) ]

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degrees of freedom

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The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df.
degrees of freedom = n - 1
in this section

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Procedure for Constructing a Confidence Interval for µ (With σ Unknown)

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1. Verify that the requirements are satisfied.
2. Using n - 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t(α/2) that corresponds to the desired confidence level.
3. Evaluate the margin of error E = t(α/2) • [ s / n^(1/2 ] .
4. Find the values of xbar - E and xbar + E. Substitute those values in the general format for the confidence interval:
xbar - E < μ < xbar + E
5. Round the resulting confidence interval limits

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Important Properties of the Student t Distribution

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1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12).
2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.
3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a σ = 1).
5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.

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Choosing the Appropriate Distribution

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Use the normal (z) distribution
If σ known and normally distributed population or σ known and n > 30
Use t distribution
if σ not known and normally distributed population or σ not known and n > 30
Use a nonparametric method or bootstrapping
If Population is not normally distributed and n ≤ 30

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Finding the Point Estimate and E from a Confidence Interval

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Point estimate of µ:
xbar = (upper confidence limit + lower confidence limit) / 2
Margin of Error:
E = (upper confidence limit - lower confidence limit) / 2

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Chi-Square Distribution

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In a normally distributed population with variance σ^2 assume that we randomly select independent samples of size n and, for each sample, compute the sample variance s2 (which is the square of the sample standard deviation s). The sample statistic x^2 (pronounced chi-square) has a sampling distribution called the chi-square distribution.

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Chi-Square Distribution formula

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X^2 = ( [ n -1 ] * s^2) / σ^2

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Properties of the Distribution of the Chi-Square Statistic

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1. The chi-square distribution is not symmetric, unlike the normal and Student t distributions.
As the number of degrees of freedom increases, the distribution becomes more symmetric.
2. The values of chi-square can be zero or positive, but they cannot be negative.
3. The chi-square distribution is different for each number of degrees of freedom, which is df = n - 1. As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.
In Table A-4, each critical value of X^2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value.

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Confidence Interval for Estimating a Population Standard Deviation or Variance

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( [n - 1] * s^2 ) / X(r)^2 < σ^2 < ( [ n - 1 ] * s^2) / X(L)^2

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Procedure for Constructing a Confidence Interval for σ or σ^2

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1. Verify that the required assumptions are satisfied.
2. Using n - 1 degrees of freedom, refer to Table A-4 or use technology to find the critical values X(r)^2 and X(L)^2 that correspond to the desired confidence level/
3. Evaluate the upper and lower confidence interval limits using this format of the confidence interval:
( [n - 1] * s^2 ) / X(r)^2 < σ^2 < ( [ n - 1 ] * s^2) / X(L)^2
4. If a confidence interval estimate of is desired, take the square root of the upper and lower confidence interval limits and change σ^2 to σ.
5. Round the resulting confidence level limits. If using the original set of data to construct a confidence interval, round the confidence interval limits to one more decimal place than is used for the original set of data. If using the sample standard deviation or variance, round the confidence interval limits to the same number of decimals places.

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Confidence Intervals for Comparing Data Caution

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Confidence intervals can be used informally to compare the variation in different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of variances or standard deviations.

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Critical Value

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A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.