In: Statistics and Probability
A Problem with a phone line that prevents a customer from receiving or making calls is upsetting to both the customer and the telecommunications company. Samples of 20 problems reported to two different offices of a telecommunications company and the time to clear those problems (in minutes) from the customers’ lines.
Data summary table:
Problem 1 data |
|
Mean 1 |
2.21 |
Mean 2 |
2.01 |
Std. Dev. 1 |
1.72 |
Std. Dev. 2 |
1.89 |
Using the output below answer the answer the flowing questions:
Pooled-Variance t Test for the Difference Between Two Means |
||
(assumes equal population variances) |
||
Data |
||
Hypothesized Difference |
0 |
|
Level of Significance |
0.05 |
|
Population 1 Sample |
||
Sample Size |
20 |
|
Sample Mean |
2.21 |
|
Sample Standard Deviation |
1.72 |
|
Population 2 Sample |
||
Sample Size |
20 |
|
Sample Mean |
2.01 |
|
Sample Standard Deviation |
1.89 |
|
Intermediate Calculations |
||
Population 1 Sample Degrees of Freedom |
19 |
|
Population 2 Sample Degrees of Freedom |
19 |
|
Total Degrees of Freedom |
38 |
|
Pooled Variance |
3.2653 |
|
Standard Error |
0.5714 |
|
Difference in Sample Means |
0.2000 |
|
t Test Statistic |
0.3500 |
|
Two-Tail Test |
||
Lower Critical Value |
-2.0244 |
|
Upper Critical Value |
2.0244 |
|
p-Value |
0.7283 |
Assuming the population variances are unknown but equal, to test if (at level of significance a = .05) there is any significant difference between the mean waiting time between the two offices?,
a) What is the null hypothesis?
b) What is the correct t-statistic?
c) What is the correct decision rule?
d) What is the correct conclusion?
e) Using only the p-value in the Excel output for problem 1, can it be concluded that there is any significant difference between the mean waiting time of the 2 groups at level of significance a=.1?
f) If in fact the true means were significantly different, based on the correct conclusion for Problem 1, would an error be made?
Solution :
We have been given the following summary of two samples:
Sample 1 | Sample 2 | |
Mean | 2.21 | 2.01 |
Standard deviation | 1.72 | 1.89 |
Sample size | 20 | 20 |
Assuming the population variances are unknown and equal we need to test if there is any significant difference between the mean waiting time between the two offices.
To test the claim we use pooled variance t-test as follows;
Hypothesized Difference |
0 |
|
Level of Significance |
0.05 |
|
Intermediate Calculations |
||
Total Degrees of Freedom |
38 |
|
Pooled Variance |
3.2653 |
|
Standard Error |
0.5714 |
|
Difference in Sample Means |
0.2000 |
|
t Test Statistic |
0.3500 |
|
Two-Tail Test |
||
Lower Critical Value |
-2.0244 |
|
Upper Critical Value |
2.0244 |
|
p-Value |
0.7283 |
The null and alternative hypothesis are given by;
Vs
The test Statistic used in this case is ;
Where,
Following these formulas we get, t statistic = 0.3500
And p-value = 0.7283, so the decision rule based on this p-value is given by;
If p-value <= significance level then we reject H0 otherwise we do not reject H0.
Here significance level = 0.05.
Since p-value = 0.7283 > significance level = 0.05 we do not reject H0.
Conclusion :
There is not a significant difference between the mean waiting time between the two offices.
And hence for significance level 0.1 also we do not reject H0 since p-value > 0.1.
This gives us the conclusion, there is not a significant difference between the mean waiting time between the two offices.
If in fact the true means were significantly different (that is, H1 is true). Based on the conclusion stated above we accept H0. And accepting H0 when actually it was not true is an Type II error.
So in such situation we are committing Type II error.