If the heights of male teenagers are normally distributed with a mean of 60 inches and standard deviation of 10, And the sample size was 25, what percentage of boy’s heights in inches would be:
Between 57 and 63
Lass than 58
61 or larger
7.2a
60 0
HeightZ
57 63
%
Z = x - s / n
57 - 60
10 / 25
63 - 60
10 / 25
Z= -1.5 = .4332
Z= 1.5 = .4332
.8664 = 86.8%
Hypothesis Testing
Hypothesis: A statement of belief…
Null Hypothesis, H0: …there is no difference between the population mean and the hypothesized value 0.
Alternative Hypothesis, Ha: …reject the null hypothesis and accept that there is a difference between the population mean and the hypothesized value 0.
Probabilities of Type I and Type II errors
H0 True H0 False
Accept H0
Reject H0
Type I Error
Type IIError
Correctresults
Correctresults
Truth
Testresult
1 -
1 -
H0 True = statistically insignificantH0 False = statistically significantAccept H0 = statistically insignificantReject H0 = statistically significant
Differences
a b
c d
http://en.wikipedia.org/wiki/False_positive
-3 -3 -2 -2 -1-1 00 11 22 33
SE
Probability Distribution for a two-tailed test
SEMagnitude of (XE – XC)
1.96 SE
XE < XC XE > XC
= 0.05
0.0250.025
-3 -3 -2 -2 -1-1 00 11 22 33
SE
Probability Distribution for a one-tailed test
SEMagnitude of (XE – XC)
1.645 SE
XE < XC XE > XC
= 0.05= 0.05
t-Tests
• Students t-test is used if:– two samples come from two different
groups.– e.g. A group of students and a group of
professors
• Paired t-test is used if:– two samples from the sample group.– e.g. a pre and post test on the same
group of subjects.
One-Tailed vs. Two Tailed Tests
• The Key Question: “Am I interested in the deviation from the mean of the sample from the mean of the population in one or both directions.”
• If you want to determine whether one mean is significantly from the other, perform a two-tailed test.
• If you want to determine whether one mean is significantly larger, or significantly smaller, perform a one-tailed test.
t-Test(Two Tailed)
Independent Sample means
x A - xB - 0
t =
Sp [ ( 1/NA ) + ( 1/NB) ]
d f = N A + N B - 2
Independent Sample Means
Sample A (A – Mean)2
26 34.3424 14.9018 4.5817 9.8618 4.5820 .0218 4.58Mean = 20.14A2 = 2913N = 7(A – Mean)2 = 72.86Var = 12.14s = 3.48
Sample B (B – Mean)2
38 113.8526 1.7724 11.0924 11.0930 7.1322 28.41
Mean = 27.33B2 = 4656N = 6(B – Mean)2 = 173.34Var = 34.67s = 5.89
Standard error of the difference between the means (SED)
SED of E - C =
s A 2Estimate of the s B
2
N AN B
+SED of x E - x C =
A 2 B
2
N AN B
+Theoretical
Theoretical
Population
Sample
Pooled estimate of the SED (SEDp)
1Estimate of the 1
N AN B
+SEDp of x A - x B = Sp
s2(nA-1) + s2 (nB – 1)Sp =
n A + n B - 2
12.14 (6) + 34.67 (5)Sp =
7 + 6 - 2= 22.38 = 4.73
t-Test(Two Tailed)
d f = N E + N C - 2 = 11
x A - xB - 0
t =
Sp [ ( 1/NA ) + ( 1/NB) ]
20.14 - 27.33 - 0 =
4.73 ( 1/7 ) + ( 1/6)= -2.73
Critical Value 95% = 2.201
One-tailed and two-tailed t-tests
• A two-tailed test is generally recommended because differences in either direction need to be known.
Paired t-test
t paired = t p = d - 0
Standard error of d
= -------------d - 0
S d 2
N
df = N - 1
d = D/N
d 2 = D 2 – ( D) 2 / N
S d2 = d 2 / N - 1
Pre/post attitude assessment
Student Before After Difference D squared
1 25 28 3 9
2 23 19 -4 16
3 30 34 4 16
4 7 10 3 9
5 3 6 3 9
6 22 26 4 16
7 12 13 1 1
8 30 47 17 289
9 5 16 11 121
10 14 9 -5 25
Total 171 208 D = 37 D2 = 511
Pre/post attitude assessment
Student Before After Difference D squaredTotal 171 208 37 511
t paired = t p = d - 0
Standard error of d
= -------------d - 0
S d 2
N
d = D/N
N = 10
d 2 = D 2 – ( D) 2 / N
S d2 = d 2 / N - 1
= 37/10 = 3.7
= 511 - 1369/10 = 374.1
= 374.1 / 10 – 1 = 41.5667
= 3.7 / 2.0387
= 1.815
= 3.7 / 41.5667 / 10
= 3.7 / 4. 15667
df = N – 1 = 90.05 > 1.833
Probabilities of Type I and Type II errors
H0 True H0 False
Accept H0
Reject H0
Type I Error
Type IIError
Correctresults
Correctresults
Truth
Testresult
1 -
1 -
H0 True = statistically insignificantH0 False = statistically significantAccept H0 = statistically insignificantReject H0 = statistically significant
Differences
Standard 2 X 2 table
a = subjects with both the risk factor and the diseaseb = subjects with the risk factor but not the diseasec = subjects with the disease but not the risk factord = subjects with neither the risk factor nor the diseasea + b = all subjects with the risk factorc + d = all subjects without the risk factora + c = all subjects with the diseaseb + d = all subjects without the diseasea + b + c + d = all study subjects
Present Absent
Present
Absent
Disease status
Risk FactorStatus
aa bb
cc dd
a + ba + b
c + dc + d
a + ca + c b + db + d a+b+c+da+b+c+dTotal
Total
Standard 2 X 2 table
Sensitivity = a/a+cSpecificity = d/b+d
Present Absent
Present
Absent
Disease status
Risk FactorStatus
aa bb
cc dd
a + ba + b
c + dc + d
a + ca + c b + db + d a+b+c+da+b+c+dTotal
Total
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