You have sample data and you are asked to assess the credibility of a statement about population using sample data.
In other words, we use a random sample of data taken from a population to describe and make inferences about the population. For example, Indian Government wants to know about response of its citizens on a new policy. It is not possible to reach out each citizen to collect feedback as it’s a very expensive and time-consuming process. Instead they reach out to a sample of it from each district or state and make judgement whether people are happy with the policy or not.
Statistical significance evaluates the likelihood that an observed (actual) difference is due to chance. It deals with the following question :
If we selected many samples from the same population, would we still find the same relationship between these two variables in every sample? Or is our finding due only to random chance?
Independent T-Test
The independent t test evaluates whether the means for two independent groups are significantly different from each other. It is used for just 2 groups of samples. If you have more than 2 groups of samples, you should use ANOVA.
Assumptions
- Each score is sampled independently and randomly.
- The scores are normally distributed within each of the two groups.
- The variance in each of the groups is equal.
Case StudyHow powerful are rumors? Frequently, students ask friends and/or look at instructor evaluations to decide if a class is worth taking. Kelley (1950) found that instructor reputation has a profound impact on actual teaching ratings. [Source : Journal of Personality, 18, 431-439]Experimental Design Before viewing the lecture, students were given a summary of the instructors prior teaching evaluations. There were two types of instructors : Charismatic instructor and Punitive instructor.Null HypothesisIt is a statement that you want to test. It usually states that there is no relationship between the two variables. In this case, the null hypothesis states that there is no difference between the mean ratings of the charismatic-teacher-reputation condition and the punitive-teacher-reputation condition.Alternate HypothesisIt is contrary to the null hypothesis. It usually states that there is a relationship between the two variables. In this case, the alternate hypothesis states that there is a difference between the mean ratings of the charismatic-teacher-reputation condition and the punitive-teacher-reputation condition.
What is p-value in simple terms?
P-value evaluate how well the sample data support that the null hypothesis is true. A low P value means that your sample provides enough evidence that you can reject the null hypothesis for the entire population. In technical language, it means lowest level of significance at which you can reject the null hypothesis.Type I and II Errors Examples1. Let’s say you are testing a new drug for some disease.
Null Hypothesis : New Drug has no effect on disease. In a test of its effectiveness, a type I error would be to say it has an effect when it does not (False Positive) ; a type II error would be to say it has no effect when it does (False Negative).
2. Null Hypothesis : Person is innocent
If an innocent person is convicted, it is a type I error (False Positive). If court lets guilty person go free, it is type II error (False Negative). Description of type I and II error is shown in the image below –
InterpretationAn independent-samples t-test was used to test the difference between the mean ratings of the charismatic-teacher-reputation condition and the punitive-teacher-reputation condition. The output from SPSS is shown below
Assumption Check
The columns labeled “Levene’s Test for Equality of Variances” tell us whether an assumption of the t-test has been met. The t-test assumes that the variance in each of the groups is approximately equal.
Look at the column labeled “Sig.” under the heading “Levene’s Test for Equality of Variances”. In this example, the significance (p value) of Levene’s test is .880. If this value is less than or equal to 5% level of significance (.05), then you can reject the null hypothesis that the variability of the two groups is equal, implying that the variances are unequal.
If the significance (p value) of Levene’s test is less than or equal to 5% level of significance (.05), then you should use the bottom row of the output (the row labeled “Equal variances not assumed”)
If the significance (p value) of Levene’s test is greater than 5% level of significance (.05), then you should use the middle row of the output (the row labeled “Equal variances assumed”)
In this example, .880 is larger than 0.05, so we will assume that the variances are equal and we will use the middle row of the output.ConclusionThe column labeled “Sig. (2-tailed)” gives the two-tailed p value associated with the test. In this example, the p value is .018. Since p-value .018 is less than .05, so we reject null hypothesis. That implies that there is a significant difference between the mean ratings of the charismatic-teacher-reputation condition and the punitive-teacher-reputation condition.