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BA 555 Practical Business Analysis
Review of Statistics Confidence Interval Estimation Hypothesis Testing
Linear Regression Analysis Introduction Case Study: Cost of Manufacturing Computers Simple Linear Regression
Agenda
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The Empirical Rule (p.5)
1. Approximately 68% of the observations will fall within 1 standard deviation of the mean. 2. Approximately 95% of the observations will fall within 2 standard deviations of the mean. 3. Approximately 99.7% of the observations will fall within 3 standard deviations of the mean.
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0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15%
68%
95%
99.7%
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0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15%
68%
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99.7%
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Review Example
Suppose that the average hourly earnings of production workers over the past three years were reported to be $12.27, $12.85, and $13.39 with the
standard deviations $0.15, $0.18, and $0.23, respectively. The average hourly earnings of the production
workers in your company also continued to rise over the past three years from $12.72 in 2002, $13.35 in 2003, to $13.95 in 2004.
Assume that the distribution of the hourly earnings for all production workers is mound-shaped.
Do the earnings in your company become less and less competitive? Why or why not.
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Review Example
YearIndustry average
Industry
std.%
increaseCompany average
% increase
Z score
2002 12.27 0.15 12.72 3
2003 12.85 0.18 4.73% 13.35 4.95% 2.77
2004 13.39 0.23 4.20% 13.95 4.50% 2.43
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The Empirical Rule
Generalize the results from the empirical rule. Justify the use of the mound-shaped
distribution.
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0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15%
68%
95%
99.7%
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Sampling Distribution (p.6)
The sampling distribution of a statistic is the probability distribution for all possible values of the statistic that results when random samples of size n are repeatedly drawn from the population.
When the sample size is large, what is the sampling distribution of the sample mean / sample proportion / the difference of two samples means / the difference of two sample proportions? NORMAL !!!
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Central Limit Theorem (CLT) (p.6)
If X ~ N(, 2), then X ~ N( X
,nX
22 )
:::::::
,,, :4 Sample
,,, :3 Sample
,,, :2 Sample
,,, :1 Sample
444241
333231
222221
111211
XXXX
XXXX
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XXXX
n
n
n
n
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Central Limit Theorem (CLT) (p.6)
If X ~ Any distribution with the mean , and variance 2, then X ~ N(
X,
nX
22 ) for large n.
:::::::
,,, :4 Sample
,,, :3 Sample
,,, :2 Sample
,,, :1 Sample
444241
333231
222221
111211
XXXX
XXXX
XXXX
XXXX
n
n
n
n
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Summary: Sampling Distributions
The sampling distribution of a sample mean The sampling distribution of a sample
proportion The sampling distribution of the difference
between two sample means The sampling distribution of the difference
between two sample proportions
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Standard Deviations
Population standard deviation X or simply .
Sample standard deviation Xs or simply s .
Standard deviation of sample means (aka. standard error) X
Standard deviation of sample proportions (aka. standard error) p
Relationships:
o XXXX
Xs
n
s
nor ˆ:
o ppp sn
pp
n
ppˆˆˆ or ˆ:
)ˆ1(ˆ)1(
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Statistical Inference: Estimation
Research Question: What is the parameter value?
Sample of size n
Population
Tools (i.e., formulas):Point EstimatorInterval Estimator
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Confidence Interval Estimation (p.7)
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Example 1: Estimation for the population mean A random sampling of a company’s weekly
operating expenses for a sample of 48 weeks produced a sample mean of $5474 and a standard deviation of $764. Construct a 95% confidence interval for the company’s mean weekly expenses.
Example 2: Estimation for the population proportion
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Statistical Inference: Hypothesis Testing
Research Question: Is the claim supported?
Sample of size n
Population
Tools (i.e., formulas):z or t statistic
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Hypothesis Testing (p.9)
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Example
A bank has set up a customer service goal that the mean waiting time for its customers will be less than 2 minutes. The bank randomly samples 30 customers and finds that the sample mean is 100 seconds. Assuming that the sample is from a normal distribution and the standard deviation is 28 seconds, can the bank safely conclude that the population mean waiting time is less than 2 minutes?
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Setting Up the Rejection RegionType I Error If we reject H0 (accept Ha) when in fact H0 is
true, this is a Type I error. False Alarm.
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The P-Value of a Test (p.11)
The p-value or observed significance level is the smallest value of for which test results are statistically significant.
“the conclusion of rejecting H0 can be reached.”
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Regression Analysis
A technique to examine the relationship between an outcome variable (dependent variable, Y) and a group of explanatory variables (independent variables, X1, X2, … Xk).
The model allows us to understand (quantify) the effect of each X on Y.
It also allows us to predict Y based on X1, X2, …. Xk.
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Types of Relationship
Linear Relationship Simple Linear Relationship
Y = 0 + 1 X + Multiple Linear Relationship
Y = 0 + 1 X1 + 2 X2 + … + k Xk +
Nonlinear Relationship Y = 0 exp(1X+ Y = 0 + 1 X1 + 2 X1
2 + … etc.
Will focus only on linear relationship.
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Simple Linear Regression Model
XY 10
population
sample
XY 10ˆˆˆ
True effect of X on Y
Estimated effect of X on Y
Key questions:1. Does X have any effect on Y?2. If yes, how large is the effect?3. Given X, what is the estimated Y?
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Least Squares Method
Least squares line: It is a statistical procedure for finding the “best-
fitting” straight line. It minimizes the sum of squares of the deviations of
the observed values of Y from those predicted Y
XY 10ˆˆˆ
Deviations are minimized. Bad fit.
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Case: Cost of Manufacturing Computers (pp.13 – 45) A manufacturer produces computers. The goal is to
quantify cost drivers and to understand the variation in production costs from week to week.
The following production variables were recorded: COST: the total weekly production cost (in $millions) UNITS: the total number of units (in 000s) produced
during the week. LABOR: the total weekly direct labor cost (in $10K). SWITCH: the total number of times that the production
process was re-configured for different types of computers
FACTA: = 1 if the observation is from factory A; = 0 if from factory B.
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Raw Data (p. 14)
Case FactA Units Switch Labor Cost(1000) (10,000) (million)
1 1 1.104 8 5.591181 1.1554562 0 1.044 12 6.836490 1.1441983 1 1.020 12 5.906357 1.1414904 1 0.986 6 5.050069 1.1196565 1 0.972 13 4.790412 1.1248156 0 1.005 11 5.474329 1.1373397 0 0.953 10 5.614134 1.1212758 0 1.083 9 6.002122 1.1532249 1 0.978 9 5.971627 1.119525
10 0 0.993 12 5.679461 1.13463511 1 0.958 12 4.320123 1.11938612 1 0.945 13 5.884950 1.11354313 1 1.012 7 4.593554 1.13212414 0 0.974 10 4.915151 1.13123815 1 0.910 11 4.969754 1.10497616 0 1.086 7 5.722599 1.15154717 0 0.962 11 6.109507 1.12747818 1 0.941 10 5.006398 1.11405819 0 1.046 9 6.141096 1.14087220 1 0.955 11 5.019560 1.11129021 1 1.096 12 5.741166 1.15904422 1 1.004 9 4.990734 1.12780523 0 0.997 8 4.662818 1.13066124 1 0.967 13 6.150249 1.12707325 1 1.068 6 6.038454 1.14104126 1 1.041 11 4.988593 1.14031927 1 0.989 16 6.104960 1.13017228 0 1.001 10 4.605764 1.13511829 1 1.008 9 5.529746 1.12132630 1 1.001 7 4.941728 1.12428431 1 0.984 10 6.456427 1.11501632 1 0.981 12 7.058013 1.12435333 1 0.944 11 4.626091 1.11631834 0 0.967 10 4.054482 1.12851735 0 1.018 9 5.820684 1.15023836 1 0.902 9 4.932339 1.09406137 0 1.049 11 5.798058 1.14379338 0 1.024 11 5.528302 1.14513539 1 1.044 7 6.635490 1.14215640 0 1.018 9 5.617445 1.14028541 0 0.937 11 5.275923 1.11441842 0 0.942 9 2.927715 1.11577443 0 1.061 11 6.750682 1.15406944 0 0.901 7 5.029670 1.10533545 0 1.078 9 7.005407 1.15336746 0 1.030 10 4.885713 1.14693447 0 0.981 8 6.362366 1.13042348 1 1.011 10 6.261692 1.13092949 1 1.016 9 5.677634 1.13634950 0 1.008 9 6.630767 1.14061651 0 1.059 11 6.930117 1.154121
1 1 1.104 8 5.591181 1.1554562 0 1.044 12 6.836490 1.1441983 1 1.020 12 5.906357 1.1414904 1 0.986 6 5.050069 1.1196565 1 0.972 13 4.790412 1.1248156 0 1.005 11 5.474329 1.1373397 0 0.953 10 5.614134 1.1212758 0 1.083 9 6.002122 1.1532249 1 0.978 9 5.971627 1.119525
10 0 0.993 12 5.679461 1.13463511 1 0.958 12 4.320123 1.11938612 1 0.945 13 5.884950 1.11354313 1 1.012 7 4.593554 1.13212414 0 0.974 10 4.915151 1.13123815 1 0.910 11 4.969754 1.10497616 0 1.086 7 5.722599 1.15154717 0 0.962 11 6.109507 1.12747818 1 0.941 10 5.006398 1.11405819 0 1.046 9 6.141096 1.14087220 1 0.955 11 5.019560 1.11129021 1 1.096 12 5.741166 1.15904422 1 1.004 9 4.990734 1.12780523 0 0.997 8 4.662818 1.13066124 1 0.967 13 6.150249 1.12707325 1 1.068 6 6.038454 1.14104126 1 1.041 11 4.988593 1.14031927 1 0.989 16 6.104960 1.13017228 0 1.001 10 4.605764 1.13511829 1 1.008 9 5.529746 1.12132630 1 1.001 7 4.941728 1.12428431 1 0.984 10 6.456427 1.11501632 1 0.981 12 7.058013 1.12435333 1 0.944 11 4.626091 1.11631834 0 0.967 10 4.054482 1.12851735 0 1.018 9 5.820684 1.15023836 1 0.902 9 4.932339 1.09406137 0 1.049 11 5.798058 1.14379338 0 1.024 11 5.528302 1.14513539 1 1.044 7 6.635490 1.14215640 0 1.018 9 5.617445 1.14028541 0 0.937 11 5.275923 1.11441842 0 0.942 9 2.927715 1.11577443 0 1.061 11 6.750682 1.15406944 0 0.901 7 5.029670 1.10533545 0 1.078 9 7.005407 1.15336746 0 1.030 10 4.885713 1.14693447 0 0.981 8 6.362366 1.13042348 1 1.011 10 6.261692 1.13092949 1 1.016 9 5.677634 1.13634950 0 1.008 9 6.630767 1.14061651 0 1.059 11 6.930117 1.15412152 0 1.019 13 6.415978 1.142435
How many possible regression models can we build?
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Simple Linear Regression Model (pp. 17 – 26) Question1: Is Labor a significant cost driver?
This question leads us to think about the following model: Cost = f(Labor) + . Specifically, Cost = 0 + 1 Labor +
Question 2: How well does this model perform? (How accurate can Labor predict Cost?) This question leads us to try other regression
models and make comparison.
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Initial Analysis (pp. 15 – 16)
Summary statistics + Plots (e.g., histograms + scatter plots) + Correlations
Things to look for Features of Data (e.g., data range, outliers)
do not want to extrapolate outside data range because the relationship is unknown (or un-established).
Summary statistics and graphs. Is the assumption of linearity appropriate? Inter-dependence among variables? Any potential
problem? Scatter plots and correlations.
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Correlation (p. 15)
(rho): Population correlation (its value most likely is unknown.) r: Sample correlation (its value can be calculated from the
sample.) Correlation is a measure of the strength of linear relationship. Correlation falls between –1 and 1. No linear relationship if correlation is close to 0. But, ….
= –1 –1 < < 0 = 0 0 < < 1 = 1r = –1 –1 < r < 0 r = 0 0 < r < 1 r = 1
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Correlation (p. 15)
Cost Units Switch Labor Cost 0.9297 -0.0232 0.4520 ( 52) ( 52) ( 52) 0.0000 0.8706 0.0008 Units 0.9297 -0.1658 0.4603 ( 52) ( 52) ( 52) 0.0000 0.2402 0.0006 Switch -0.0232 -0.1658 0.1554 ( 52) ( 52) ( 52) 0.8706 0.2402 0.2714 Labor 0.4520 0.4603 0.1554 ( 52) ( 52) ( 52) 0.0008 0.0006 0.2714
Is 0.9297 a or r?
Sample size
P-value for
H0: = 0Ha: ≠ 0
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Regression Analysis - Linear model: Y = a + b*X Dependent variable: Cost Independent variable: Labor Standard T Parameter Estimate Error Statistic P-Value
Intercept 1.08673 0.0127489 85.2409 0.0000 Slope 0.00810182 0.00226123 3.58293 0.0008 Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value Model 0.00231465 1 0.00231465 12.84 0.0008 Residual 0.00901526 50 0.000180305 Total (Corr.) 0.0113299 51
Fitted Model (Least Squares Line) (p.18)
Error Standard
0-Estimate Statistic T
H0: 1 = 0Ha: 1 ≠ 0
1 or b1?0 or b0?
Sb1
Sb0b1b0
Degrees of freedom = n – k – 1, where n = sample size, k = # of Xs.
** Divide the p-value by 2for one-sided test. Makesure there is at least weakevidence for doing this step.
XY 0081.008673.1ˆ
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Regression Analysis - Linear model: Y = a + b*X Dependent variable: Cost Independent variable: Labor Standard T Parameter Estimate Error Statistic P-Value
Intercept 1.08673 0.0127489 85.2409 0.0000 Slope 0.00810182 0.00226123 3.58293 0.0008 Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value Model 0.00231465 1 0.00231465 12.84 0.0008 Residual 0.00901526 50 0.000180305 Total (Corr.) 0.0113299 51
Hypothesis Testing and Confidence Interval Estimation for (pp. 19 – 20)
Sb1 Sb0b1b0
Degrees of freedom = n – k – 1k = # of independent variables
Q1: Does Labor have any impact on Cost → Hypothesis TestingQ2: If so, how large is the impact? → Confidence Interval Estimation
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Analysis of Variance (p. 21) Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 0.00231465 1 0.00231465 12.84 0.0008 Residual 0.00901526 50 0.000180305 Total (Corr.) 0.0113299 51
Syy = SS Total =
2
1
)( yyn
ii 0.0113299.
SSR =SS of Regression Model =
2
1
)ˆ( yyn
ii 0.00231465.
SSE = SS of Error =
2
1
)ˆ( i
n
ii yy 0.00901526.
SS Total = SS Model + SS Error.
- Not very useful in simple regression.- Useful in multiple regression.
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Sum of Squares (p.22)
Syy = Total variation in YSSE = remaining variation that cannot be explained by the model.
SSR = Syy – SSE = variation in Y that has been explained by the model.
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Fit Statistics (pp. 23 – 24)
Total
sidual
Total
Model
SS
SS
SS
SS Re1
Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 0.00231465 1 0.002315 12.84 0.0008 Residual 0.00901526 50 0.000180 Total (Corr.) 0.0113299 51
Correlation Coefficient = 0.45199 R-squared = 20.4295 percent R-squared (adjusted for d.f.) = 18.8381 percent Standard Error of Est. = 0.0134278
sidualMSkn Re
Residual
1
SS
0.45199 x 0.45199 = 0.204295
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Prediction (pp. 25 – 26)
What is the predicted production cost of a given week, say, Week 21 of the year that Labor = 5 (i.e., $50,000)? Point estimate: predicted cost = b0 + b1 (5) = 1.0867 +
0.0081 (5) = 1.12724 (million dollars). Margin of error? → Prediction Interval
What is the average production cost of a typical week that Labor = 5? Point estimate: estimated cost = b0 + b1 (5) = 1.0867 +
0.0081 (5) = 1.12724 (million dollars). Margin of error? → Confidence Interval
100(1-)% prediction interval:
X of variance1
)(11 Est.of Error Standardˆ
2
2/
n
xx
nty g ,
100(1-)% confidence interval:
X of variance1
)(1 Est.of Error Standardˆ
2
2/
n
xx
nty g ,
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95.00% 95.00% Predicted Prediction Limits Confidence Limits X Y Lower Upper Lower Upper 3.0 1.11103 1.08139 1.14067 1.09874 1.12332 4.0 1.11913 1.09098 1.14729 1.11105 1.12722 5.0 1.12724 1.09988 1.15459 1.12267 1.1318 6.0 1.13534 1.10804 1.16263 1.13113 1.13954
95% Prediction and Confidence Intervals for Cost
Labor ($10,000)
Co
st (
$ m
illio
n)
2.9 3.9 4.9 5.9 6.9 7.91.09
1.11
1.13
1.15
1.17
Prediction vs. Confidence Intervals (pp. 25 – 26)
☺
☺☺☺
☺
☻☻☻ ☻☻☻
☺
Variation (margin of error) on both ends seems larger. Implication?
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Another Simple Regression Model: Cost = 0 + 1 Units + (p. 27)
Regression Analysis - Linear model: Y = a + b*X Standard T Parameter Estimate Error Statistic P-Value Intercept 0.849536 0.0158346 53.6506 0.0000 Slope 0.281984 0.0157938 17.8541 0.0000 Analysis of Variance Source Sum of Squares Df Mean Square F-Ratio P-Value Model 0.00979373 1 0.00979373 318.77 0.0000 Residual 0.00153618 50 0.0000307235 Total (Corr.) 0.0113299 51 Correlation Coefficient = 0.929739 R-squared = 86.4414 percent R-squared (adjusted for d.f.) = 86.1702 percent Standard Error of Est. = 0.00554288
95% Prediction and Confidence Intervals for Cost
Units (1000)
Co
st (
$ m
illio
n)
0.9 0.94 0.98 1.02 1.06 1.1 1.14 1.181.09
1.11
1.13
1.15
1.17
A better model?Why?
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Statgraphics
Simple Regression Analysis Relate / Simple Regression X = Independent variable, Y = dependent variable For prediction, click on the Tabular option icon and
check Forecasts. Right click to change X values. Multiple Regression Analysis
Relate / Multiple Regression For prediction, enter values of Xs in the Data Window
and leave the corresponding Y blank. Click on the Tabular option icon and check Reports.
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Normal Probabilitiesz .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .03590.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .07530.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .11410.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .15170.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .18790.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .22240.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .25490.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .28520.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .31330.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .33891.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .36211.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .38301.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .40151.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .41771.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .43191.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .44411.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .45451.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .46331.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .47061.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .47672.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .48172.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .48572.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .48902.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .49162.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .49362.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .49522.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .49642.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .49742.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .49812.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .49863.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
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Critical Values of t
DEGREES OF FREEDOM 100.t 050.t 025.t 010.t 005.t DEGREES OF
FREEDOM 100.t 050.t 025.t 010.t 005.t
1 3.078 6.314 12.706 31.821 63.657 24 1.318 1.711 2.064 2.492 2.797 2 1.886 2.920 4.303 6.965 9.925 25 1.316 1.708 2.060 2.485 2.787 3 1.638 2.353 3.182 4.541 5.841 26 1.315 1.706 2.056 2.479 2.779 4 1.533 2.132 2.776 3.747 4.604 27 1.314 1.703 2.052 2.473 2.771 5 1.476 2.015 2.571 3.365 4.032 28 1.313 1.701 2.048 2.467 2.763 6 1.440 1.943 2.447 3.143 3.707 29 1.311 1.699 2.045 2.462 2.756 7 1.415 1.895 2.365 2.998 3.499 30 1.310 1.697 2.042 2.457 2.750 8 1.397 1.860 2.306 2.896 3.355 35 1.306 1.690 2.030 2.438 2.724 9 1.383 1.833 2.262 2.821 3.250 40 1.303 1.684 2.021 2.423 2.704
10 1.372 1.812 2.228 2.764 3.169 45 1.301 1.679 2.014 2.412 2.690 11 1.363 1.796 2.201 2.718 3.106 50 1.299 1.676 2.009 2.403 2.678 12 1.356 1.782 2.179 2.681 3.055 60 1.296 1.671 2.000 2.390 2.660 13 1.350 1.771 2.160 2.650 3.012 70 1.294 1.667 1.994 2.381 2.648 14 1.345 1.761 2.145 2.624 2.977 80 1.292 1.664 1.990 2.374 2.639 15 1.341 1.753 2.131 2.602 2.947 90 1.291 1.662 1.987 2.368 2.632 16 1.337 1.746 2.120 2.583 2.921 100 1.290 1.660 1.984 2.364 2.626 17 1.333 1.740 2.110 2.567 2.898 120 1.289 1.658 1.980 2.358 2.617 18 1.330 1.734 2.101 2.552 2.878 140 1.288 1.656 1.977 2.353 2.611 19 1.328 1.729 2.093 2.539 2.861 160 1.287 1.654 1.975 2.350 2.607 20 1.325 1.725 2.086 2.528 2.845 180 1.286 1.653 1.973 2.347 2.603 21 1.323 1.721 2.080 2.518 2.831 200 1.286 1.653 1.972 2.345 2.601 22 1.321 1.717 2.074 2.508 2.819 ? 1.282 1.645 1.960 2.326 2.576 23 1.319 1.714 2.069 2.500 2.807