In: Statistics and Probability
n | mean | standard deviation | |
Shift A | 10 | 3.7 | 1.2 |
Shift B | 10 | 3.3 | 1.4 |
You work at a shipyard where employees work two 12-hour shifts each day-shift A and shift B. You have been assigned to study the number of accidents that occur during each shift, to determine if there are any differences. Using the following information, what is the test statistic, degrees of freedom for this test and the critical value?
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho: ? = ? i.e. there is no significant difference between the number of accidents that occurs in shift A and B.
Ha: ? ? i.e. there is significant difference between the number of accidents that occurs in shift A and B.
This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.
(2) Rejection Region
Based on the information provided, the significance level is ?=0.05, and the degrees of freedom are df=18. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:
Hence, it is found that the critical value for this two-tailed test is tc?=2.101, for ?=0.05 and df=18.
The rejection region for this two-tailed test is R={t:?t?>2.101}.
(3) Test Statistics
Since it is assumed that the population variances are equal, the t-statistic is computed as follows:
??=
= 0.686
(4) Decision about the null hypothesis
Since it is observed that ?t?=0.686?tc?=2.101, it is then concluded that the null hypothesis is not rejected.
Using the P-value approach: The p-value is p=0.5015, and since p=0.5015?0.05, it is concluded that the null hypothesis is not rejected.
(5) Conclusion
It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean ?1? is different than ?2?, at the 0.05 significance level.
i.e. there is no significant difference between the number of accidents that occurs in shift A and B.