Statistics Essay

1778 words - 8 pages

Assignment
Please submit your project to Güzin Akın or İsrafil Boyacı by 14 January, 2013, 10:00 in the morning the latest. Late projects lose 25% of total available points for every late day. Handing in the project between 14 January 10:01 and 15 January 10:00 means losing 25 points out of 100, for example.

* This assignment is to be completed by individual students or by teams of up to three students, depending on the students’ preferences. If you do not know your teammates, please contact Mr. Boyacı.
* Each team will be assigned a number. When answering the questions, put your team number in the places where it says yourteamnumber. If you do not know your team number, please ...view middle of the document...

For example, if you are team 20, then your data are in columns twenty1 and twenty2. Both of these samples are composed of independent random draws from normally distributed populations with unknown means and unknown variances. You can use any method you prefer unless specified.

PART I: The first question is about the sample yourteamnumber1.

1. Test the null hypothesis that the population mean is equal to yourteamnumber (for example, if your team number is 20, then your null hypothesis will be that the population mean is equal to 20) against the two-sided alternative at 1% level of significance. Do not forget to write the hypotheses.

H0:µ=3
H1:µ≠3
α=0.01
X=2,83348548
s=0,973556574079978
n=25
tstat=-0,855186665229796
tα2,n-1=t0,005,24=2,79693949760654
Since tstat<t0,005,24 , do not reject the H0.

PART II: In the following set of questions, compare samples yourteamnumber1 and yourteamnumber2.

2. Test the null hypothesis that the difference between the two population means is 0 at 1% level of significance, assuming that the population variances are equal. Do not forget to write the hypotheses.

H0:µ1-µ2=0 where µ1 and µ2 are the samples'means.
H1:µ1-µ2≠0

α=0.01
n1=n2=25
σ12=σ22
Sp2=24S12+24S2224+24=1,068304976
tstat=X1-X2-(µ1-µ2)Sp2(1n1+1n2)=-2,023592798

tα2,n1-n2-2=t0,005,48=2,68220401815265

Since tstat<t0,005,48 , do not reject the H0.

t-Test: Two-Sample Assuming Equal Variances | | |
| | |
  | Population 1 | Population 2 |
Mean | 2,83348548 | 3,42506856 |
Variance | 0,947812403 | 1,188797549 |
Observations | 25 | 25 |
Pooled Variance | 1,068304976 | |
Hypothesized Mean Difference | 0 | |
df | 48 | |
t Stat | -2,023592798 | |
P(T<=t) one-tail | 0,024299406 | |
t Critical one-tail | 2,406581273 | |
P(T<=t) two-tail | 0,048598811 | |
t Critical two-tail | 2,682204027 |   |

3. Test the null hypothesis that the two population means are equal against the alternative that yourteamnumber1 comes from a population with a larger mean than yourteamnumber2 at 1% level of significance, assuming that the variances of the populations are not equal. Do not forget to write the hypotheses.

H0:µ1≤µ2
H1:µ1>µ2

α=0.01
n1=n2=25
σ12≠σ22
S12=0,947812403
S22=1,188797549
t-Test: Two-Sample Assuming Unequal Variances | | |
| | |
  | Population 1 | Population 2 |
Mean | 2,83348548 | 3,42506856 |
Variance | 0,947812403 | 1,188797549 |
Observations | 25 | 25 |
Hypothesized Mean Difference | 0 | |
df | 47 | |
t Stat | -2,023592798 | |
P(T<=t) one-tail | 0,024359414 | |
t Critical one-tail | 1,677926722 | |
P(T<=t) two-tail | 0,048718828 | |
t Critical two-tail | 2,011740514 |   |
tα,n1-n2-2=t0,01,48=1,67792672164186
tstat=X1-X2-(µ1-µ2)Sp21n1+1n2→
(µ1-µ2)=X1-X2+tstatSp21n1+1n2=-1,18316616<0
So, do not reject the H0.

4. Test the null hypothesis that the two population variances are equal at 1%...

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