ࡱ> TVSzg bjbjVV xr<r<& XXXlll8,<l - }}}+++++++$.n1,X["},,I%I%I%^8X+I%+I%I%:),@5*xXp ) +,0 -)R2 J25*2X5*}>,I%$ }}},,"V}}} -2}}}}}}}}} :  Outline for Confidence Intervals and Tests on One Parameter R.L. Andrews (revised 10/20/2010) If the data are from a process first check to assure that the process is stable. Do not attempt statistical inference for an unstable process. Identify the unknown parameter, proportion or mean. If it is a mean, then determine if the standard deviation came from the phenomenon or from the sample. If it is not the sample standard deviation then one must assume that it is the phenomenon standard deviation. If you try to find a standard deviation and cannot find one, then parameter may be a proportion rather than a mean. I. Confidence Intervals for location parameters with 100(1symbol 97 \f "Symbol")% confidence. General form of a 2-sided interval for a location parameter: (Unbiased Estimator) (Margin of Error) (Sample based Estimate) (Margin of Error) Margin of Error [denoted ME] = (Table Value)(Standard Error of the estimator [denoted SE]) (Sample based Estimate) (Table Value)(Standard Error) The appropriate statistical table and the way the SE value is determined depend on the parameter and information available. The table value can be determined by a statistical table (A-42&43 or A-44) or Excel function (NORMSINV, NORMINV or TINV). For the standard normal, the text table A-42&43 and Excel functions use cumulative probability, hence Z1-a denotes a table value with 1-a area below it and a in the upper tail and Z1-a/2 and 1-a/2 area below and a/2 in the upper tail. For the t distribution, the text table E.2 and the Excel functions use tail probabilities, hence ta denotes a table value with a in the upper tail. ta/2 for 2 tail procedures has symbol 97 \f "Symbol" = area in two tails and symbol 97 \f "Symbol"/2 = the area in one tail. For 1 tail procedures, symbol 97 \f "Symbol" is the area in one tail. For 1-sided intervals only subtract or add the appropriate margin of error to get the appropriate lower or upper limit. The "t" distribution table with ( degrees of freedom is identical to the standard normal distribution C.I. for an unknown phenomenon proportion, p, with n EMBED Equation.3 >10 & n(1- EMBED Equation.3 )>10; where  EMBED Equation.3  is the sample proportion. For table values use the standard normal distribution. Theoretical SE( EMBED Equation.3 ) = embed Equation.2  Sample data based SE( EMBED Equation.3 ) = embed Equation.2  100*(1-a)% 2-sided CI: embed Equation.2  Z1-a/2 = NORMSINV(1-a/2) C.I. for an unknown phenomenon MEAN, symbol 109 \f "Symbol", Theoretical  EMBED Equation.3  Sample data based  EMBED Equation.3  , Use the "t" distribution with n1 degrees of freedom 100*(1-a)% 2-sided CI: embed Equation.2  ta/2 = TINV(a,n-1) You will not be required know 1-sided intervals for MGMT524 1-sided lower bounded CI: embed Equation.2  ta = TINV(2*a,n-1) 1-sided upper bounded CI: embed Equation.2  ta = TINV(2*a,n-1) II. Tests of hypotheses for a parameter with unknown value, significance level = symbol 97 \f "Symbol". Confidence Interval decision rule: If the hypothesized parameter value is outside the 100(1symbol 97 \f "Symbol")% confidence interval, then Reject H0 (conclude Ha), otherwise Fail to Reject H0 (conclude H0). Critical Value decision rule: If the calculated test statistic falls in the rejection region, then Reject H0 (conclude Ha), otherwise Fail to Reject H0 (conclude H0).  EMBED Equation.3  p-value decision rule for all types of hypotheses: If p-value < symbol 97 \f "Symbol", then Reject H0, (conclude Ha). If p-value symbol 179 \f "Symbol" symbol 97 \f "Symbol", then Fail to Reject H0, (conclude H0). p-value = probability of obtaining, by chance, a sample result (test statistic) at least as extreme as the one observed. Probability assumes the null hypothesis is true. Extreme is defined by the alternate hypothesis. (< or > implies only one direction ; ( implies that either < or > are possible directions for extreme.) A. For an unknown phenomenon proportion, p, with null hypothesis H0: p = p0. Condition: n EMBED Equation.3 >10 & n(1- EMBED Equation.3 )>10; where  EMBED Equation.3  is the sample proportion. Use the standard normal distribution {NORMSDIST for p-values & NORMSINV for confidence intervals and critical values}. Note that the SE( EMBED Equation.3 ) for hypothesis testing is different from the SE( EMBED Equation.3 ) for a confidence interval. Theoretical SE( EMBED Equation.3 ) = embed Equation.2  = SD( EMBED Equation.3 ) as denoted in the text Hypothesis based SE( EMBED Equation.3 ) = embed Equation.2  [The text still denotes this as SD( EMBED Equation.3 ).] a. HA: p > p0 pvalue = P(Z > TS) = 1- NORMSDIST(TS) Critical Value = Z1-symbol 97 \f "Symbol" = NORMSINV(1-a). b. HA: p < p0 pvalue = P(Z < TS) = NORMSDIST(TS) Critical Value = Zsymbol 97 \f "Symbol"= NORMSINV(a). c. HA: p symbol 185 \f "Symbol" p0 If TS < 0, pvalue = 2" P(Z < TS) = 2*NORMSDIST(TS). If TS > 0, pvalue = 2" P(Z > TS) = 2*[1-NORMSDIST(TS)]. Critical Values: Zsymbol 97 \f "Symbol"/2 & Z1-symbol 97 \f "Symbol"/2 embed Equation.2  is the C.I. If p0 is outside embed Equation.2 , Reject H0. B. For an unknown phenomenon MEAN with null hypothesis, H0: symbol 109 \f "Symbol" = symbol 109 \f "Symbol"0. embed Equation.2  Theoretical  EMBED Equation.3  [The text denotes this with SD( ).] Sample data based  EMBED Equation.3 , Use the "t" distribution with n1 degrees of freedom a. HA: symbol 109 \f "Symbol" > symbol 109 \f "Symbol"0 pvalue = P(t > TS) = TDIST(TS,df,1) if TS>0 or 1-TDIST(-TS,df,1) if TS<0 Critical Value = tsymbol 97 \f "Symbol", n-1 = TINV(2" a, n-1) b. HA: symbol 109 \f "Symbol" < symbol 109 \f "Symbol"0 pvalue = P(t < TS) = TDIST(-TS,df,1) if TS<0 or 1-TDIST(TS,df,1) if TS>0 Critical Value = -tsymbol 97 \f "Symbol", n-1 = -TINV(2" a, n-1). c. HA: symbol 109 \f "Symbol" eq \O(=,/) symbol 109 \f "Symbol"0 Critical Values: -tsymbol 97 \f "Symbol"/2, n-1 = -TINV(a, n-1) & tsymbol 97 \f "Symbol"/2, n-1 = TINV(a, n-1). If TS < 0, pvalue = 2" P(t < TS) = TDIST(ABS(TS),df,2). If TS > 0, pvalue = 2" P(t > TS) = TDIST(ABS(TS),df,2) embed Equation.2  is the C.I. If symbol 109 \f "Symbol"0 is outside embed Equation.2 , then Reject H0. Hypothesis Testing Methods to test H0: Parameter (is equal to) Hypothesized Value HA: Parameter (differs from) Hypothesized Value Relative to hypothesis testing conclusions based on sample data Rejecting the null hypothesis = strong evidence exists to support that the null is not true. Failing to reject the null (accepting the null) = NO strong evidence exists to support that the null is not true. Rejecting the null provides a strong statement of support for the alternate. Failing to reject the null provides a very weak statement of support for the null. 1. Confidence Interval Method This method uses a confidence interval to create a feasible region for the unknown parameter using sample data. If the confidence interval contains the hypothesized value then this would support a conclusion that H0 is concluded to be true. If the confidence interval does not>?UWXZ[]_ab ! 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Critical Value Method This method creates a feasible region for the test statistic assuming H0 is true. If the feasible region for the test statistic contains the value of the test statistic computed from the sample data then H0 is concluded as being true. The area outside the feasible region for the test statistic is referred to as the rejection region. If the computed value of the test statistic does not lie in the feasible region (It is in the rejection region.) then H0 is rejected as being true. 3. p-value Method This method calculates the probability of obtaining by chance a sample result at least as extreme as the one observed in the actual sample assuming the null hypothesis is true. If this probability is large then the null assumption seems reasonable and H0 is concluded as being true. However, if the calculated probability is small the null assumption would appear to be questionable and H0 is rejected as being true. HA: < This a one-sided test. 1. Confidence Interval Method {You will not be expected to use this method for 1-tailed tests} The feasible region for the unknown parameter opened to the left. embed Excel.Chart.5 \s  (-symbol 165 \f "Symbol" \s 19, upper limit) 2. Critical Value Method The rejection region is on the left of the distribution of the Test Statistic. 3. p-value Method The probability of being extreme is to the left, P(R.V. < TS). HA: > This a one-sided test. 1. Confidence Interval Method {You will not be expected to use this method for 1-tailed tests} embed Excel.Chart.5 \s  The feasible region for the unknown parameter opened to the right. (lower limit, symbol 165 \f "Symbol" \s 19) 2. Critical Value Method The rejection region is on the right of the distribution of the Test Statistic. 3. p-value Method The probability of being extreme is to the right, P(R.V. > TS). HA: symbol 185 \f "Symbol" \s 22 This a two-sided test. 1. Confidence Interval Method The feasible region is bounded below and above. (lower limit, upper limit) embed Excel.Chart.5 \s  2. Critical Value Method The rejection region is on both the right and the left of the distribution of the Test Statistic. 3. p-value Method The probability of being extreme is in two directions. p-value = 2 * one-tail probability.      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