Outline for Confidence Intervals and Tests on One Parameter
Prepared by Robert L. Andrews (revised 4/9/2011)
If the data are from a process (data observed over time, referred to as time-series data in the text), first check to assure that the process has stable characteristics. Do not attempt statistical inference for a process with unstable characteristics. Identify the unknown parameter (either proportion or mean). If the parameter is a mean, then determine the sample mean and standard deviation. If you try to find a standard deviation and cannot find one, then parameter may be a proportion rather than a mean. If the parameter is a proportion then determine the sample proportion. For Excel, the DIST functions find probabilities for p-values and the INV functions find critical values used for confidence intervals or hypothesis testing with the critical value method. Functions having a period (.) in the name are functions only found in Excel 2010, while those without any period work in
2010 and previous versions of Excel.
I. Confidence Intervals for location parameters with 100(1-

)% confidence (95% confidence means α = .05).
General form of a 2-sided interval for a location parameter: ( Unbiased Estimator
) ± (
M argin of E rror)
( Sample based Estimate
) ± (
M argin of E rror)
M argin of E rror [denoted ME ] = (Critical Value)
•
( S tandard E rror of the estimator [denoted SE ])
( Sample based Estimate ) ± (Critical Value)•( S tandard E rror)
The appropriate statistical table and the way the SE value is determined depend on the parameter and information available. The critical value can be determined by a statistical table (Table Z or Table T) or Excel
(2007 functions: NORMSINV or TINV) or (2010 functions: NORM.S.INV, T.INV or T.INV.2T). For the standard normal, Table Z in the text and Excel normal distribution functions use cumulative probability, hence
Z
1-
 denotes a table value with 1-

area below it and

in the upper tail and Z
1-

/2
denotes the value with 1-
 area below and

in the upper tail. For the t distribution, Table T in the text and all of the 2007 Excel functions use tail probabilities, hence t  denotes a table value with

in the upper tail. t 
/2 for 2 tail procedures has
 = area in two tails and

/2 = the area in one tail. For 1-tail procedures,

is the area in one tail. For onesided intervals only subtract or add the appropriate margin of error to get the appropriate lower or upper limit.
The "t" distribution table (Table T) with

degrees of freedom is identical to the standard normal distribution.
A.
C.I. for an unknown phenomenon PROPORTION , p, with n
• p >10 & n
•(1p )>10; where p is the sample proportion. For table values use the standard normal distribution.
Theoretical SE( p ) = p

1
 p
 n Sample data based SE( p ) = p

1
 p

100*(1-

)% 2-sided CI:
 z
1
 
/ 2


1

 n n
Z
1-

/2
= NORMSINV(1-

/2) or NORM.S.INV(1-

/2)
B.
C.I. for an unknown phenomenon MEAN,

,
Theoretical SE ( y )
 
/ n
Sample data based SE ( y )
 s / n , Use the "t" distribution with n-1 degrees of freedom
100*(1-

)% 2-sided CI : y
 t

/ 2 , n

1
 t 
/2
= TINV(

,n-1) = T.INV.2T(

,n-1)
You will not be required know 1-sided intervals for MGMT524
1-sided lower bounded CI:
 y
 t

, n

1
 s
 
,

 t  = TINV(2*

,n-1) = T.INV (1-

,n-1) or
1-sided upper bounded CI:



, y
 t

, n

1
 s
   t  = TINV(2*

,n-1) = T.INV(1-

,n-1)

II.
Tests of hypotheses for a parameter with unknown value, significance level =

.
Confidence Interval decision rule: If the hypothesized parameter value is outside the 100(1-

)% 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). p-value decision rule for all types of hypotheses: If p-value <

, then Reject H0 , (conclude Ha).
If p-value
 
, 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 H
0
: p = p
0
.
Condition: n• p >10 & n•(1p )>10; where p is the sample proportion. Use the standard normal distribution {NORMSDIST for p-values & NORMSINV for confidence intervals and critical values}.
Note that the SE( p
Theoretical SE( p
) for hypothesis testing is different from the SE( p
) = p

1
 p
 n = SD( p ) as denoted in the text
Hypothesis based SE( p ) = p
0

1
 p
0
 n = SD( p
) for a confidence interval.
) as denoted in the text
HA: p < p
0
p-value = P(Z < TS) & Critical Value = Z 
If p
0
is outside (0, p
 z
1
 
 p
0

1
 p
0
 n
= NORMSINV(

) or NORM.S.INV(

).
), then Reject H0.
HA: p > p
0
p-value = P(Z > TS) & Critical Value = Z1
If p
0
is outside ( p
 z
1
 
 p
0

1
 p
0
 n
= NORMSINV(1-
,1), then Reject H0.

) or NORM.S.INV(1-

)
HA: p ≠ p
0
[TS < 0, p-value = 2
If p
0
is outside (
•
P(Z < TS)] or [TS > 0, p-value = 2
•
P(Z > TS)] Critical Values = Z  /2 & Z1 /2
 z
1
 
/ 2
 p
0

1
 p
0
 n ,
 z
1
 
/ 2
 p
0

1
 p
0
 n ), then Reject H0.
P(Z < TS) = NORMSDIST(TS) or NORM.S.DIST(TS)
P(Z > TS) = 1-NORMSDIST(TS) or 1- NORM.S.DIST(TS)
B. For an unknown phenomenon MEAN, µ, with null hypothesis, H0:

=


 
 
0
 
(
)
 0
.
Theoretical SE ( y )
 
/ n
Sample data based SE ( y )
 s / n , Use the "t" distribution with n-1 degrees of freedom
HA:

<

0
p-value = P(t < TS)
&
Critical Value = -t
 , n-1 = -TINV(2•

, n-1) or T.INV(

, n-1).
P(t < TS) = T.DIST(TS,df,TRUE) or if TS<0 TDIST(ABS(TS),df,1), or if TS>0 1-TDIST(TS,df,1)
If µ
0
is outside ( - ∞ , y
 t

, n

1

), then Reject H0.
HA:

>

0
p-value = P(t > TS) Critical Value = t  , n-1 = TINV(2•

, n-1) or T.INV(1-

, n-1)
P(t > TS) = 1-T.DIST(TS,df,TRUE) or if TS<0 = 1-TDIST(ABS(TS),df,1), or if TS>0 = TDIST(TS,df,1)
If µ
0
is outside ( y
 t

, n

1

,
∞
), then Reject H0.
HA:

=,/

0 p-value = 2-tail area = TDIST(ABS(TS),df,2) or T.DIST.2T(ABS(TS),df,2);
Critical Values = -t
 /2,n-1 & t  /2,n-1
If µ
0
is outside ( y
 t

/ 2 , n

1
 s
 
, y
 t

/ 2 , n

1

), then Reject H0.

Hypothesis Testing Methods to test
H
0
: Parameter (is equal to) Hypothesized Value
HA: Parameter (differs from) Hypothesized Value
When testing hypotheses all conclusions must be based on statistics calculated from the sample data.
Rejecting the null hypothesis implies strong evidence exists to support the alternate, hence the null does not appear to be true. Failing to reject the null (accepting the null) implies that the evidence is NOT strong enough to support concluding 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 H
0 appears to be true. If the confidence interval does not contain the hypothesized value then H
0 is rejected as being true and H A
appears to be true.
2. Critical Value Method
This method creates a feasible region for the test statistic assuming H
0
is true. If the feasible region for the test statistic contains the value of the test statistic computed from the sample data then H
0 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 H
0 is rejected as being true because the data indicate that H
A
appears to be 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 calculated probability is large then the null assumption seems reasonable and H
0 is concluded as being true. However, if the calculated probability is small, the null assumption would appear to be questionable and H
0 is rejected as being true.
1. Confidence Interval Method
{You will not be expected to use this method for 1-tailed tests}
The feasible region for the unknown parameter only has an upper limit.
For Means, (-∞, upper limit) & for Proportions, (0, upper limit).
2. Critical Value Method 0.000873
The rejection region is on the
0.002029
0.004432
0.5
left side of the distribution of the
Test Statistic (see diagram →).
0.009094
0.017528
0.4
0.3
0.03174
3. p-value Method
0.053991
0.2
0.1
The probability of being extreme 0.086277
0 is the area to the left of the
0.129518
Reject
Null observed Test Statistic, P(R.V. < TS).

1. Confidence Interval Method
{You will not be expected to use this method for 1-tailed tests}
The feasible region for the unknown parameter only has a lower limit.
For Means, (lower limit, ∞) & for Proportions: (lower limit, 1)
2. Critical Value Method 0.4
0.3
The rejection region is on the right side of the distribution of the Test Statistic
(see diagram →).
0.2
0.1
0
Reject
Null
3. p-value Method
The probability of being extreme is the area to the right of the observed Test
Statistic, P(R.V. > TS).

1. Confidence Interval Method
The feasible region is bounded below and above.
(lower limit, upper limit)
2. Critical Value Method
The rejection region is on both
0.4
0.3
the left and the right sides of the distribution of the Test Statistic
(see diagram →).
0.2
0.1
0
Reject
Null
Reject
Null
3. p-value Method
The probability of being extreme includes being either in the lower or upper tail of the distribution. p-value = 2 * (area in the one-tail where the observed test statistic lies) p-value = 2 * (observed one-tail area).
In Excel, the TDIST and T.DIST.2T functions can calculate the 2-tail area directly, which eliminates doubling the observed 1-tail area.
Prepared by Robert L. Andrews (revised 4/9/2011)