At the time in the year that we begin formal testing, students should be well immersed in their questions of interest. Having looked at survey or experimental data, hopefully students have recognized trends, and possibly generated more questions. The question that students will want to know is: Can I trust my results? How do my results apply to the world at large? These are the questions we can finally emphasize within inference.
Students begin inference by drawing confidence intervals. These begin the construction of what is reasonable to expect based on sampling variation. It bears repeating again and again that sampling distributions rely on the assumption and condition that the data are not biased. Rather than this being a purely theoretical consideration, it should be a habit to question sources of bias in data at all times.
Moving into hypothesis testing, the components of a successful test can be broken down into four basic procedures.
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1) Formal statement of the question or hypothesis (symbolically and verbally)
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2) Examination of the conditions that exist, and any assumptions.
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3) Selection of an appropriate test and the calculations
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4) Interpretation of results and statement of the conclusion.
With so much familiarity with their own questions of interest, we can segue into a hypothesis, by forcing the question into a narrower frame: H
0
and H
a
. This hypothesis should arise in a natural fashion from the previous data collection and analysis around the students' questions of interest. The previous Question Formation Techniques can be used to form a hypothesis. This time we practice with the formal structure. We first use language to state the null and alternate hypothesis, and later state it symbolically.
If the student has pursued the question of gender difference and memory tasks, and collected data around that question, we can formalize the hypothesis in words. The null hypothesis assumes that there is no difference, no association, or no change from the status quo. Therefore in this instance the null hypothesis should be that there is no difference in performance between girls and boys. Simply put, the proportion of girls who succeed equals the proportion of boys that succeed; symbolically: (H
0
p
g
=p
b
) .
The alternative hypothesis is the one that arises through consideration of any ideas based on exploratory data analysis. Perhaps our data shows that girls have a slightly higher proportion of successes at the task than boys. The question is not whether our sample favored girls, but whether this edge experienced by the girls is significantly better, or simply due to sampling variation. We state that our alternative hypothesis is that girls have more success at memory tasks than boys. Simply put the proportion of girls who succeed is greater than the proportion of boys that succeed; symbolically: (H
0
p
g
>p
b
).
While some students will not maintain a single question throughout the year, the plan is to emphasize working through each level of statistics content while referring back to the students' initial questions of interest. Students have been encouraged to change and refine their questions, focusing on how we measured outcomes. Inference computes the probability of students' findings if the hypothesis that we are focused on is true. The calculation of that probability gives us a measure of the certainty that our results match or do not match the hypothesis.
Hypothesis Formation Worksheet
State your question of interest:
Ex. I want to know if girls are faster than boys in solving math computations.
What did you measure?
Ex. I measured the time it took girls and boys to finish several computations
Was your measurement Quantitative (Can be averaged) or Categorical (counted)?
Ex. Quantitative
H
0
(null hypothesis) is a statement that there is nothing different, no association or nothing changing in the subject of your interest
In words:
Ex. Girls and boys will have the same speed when solving computations
In symbols: H
0
p
1
=p
2
for categorical data H
0
μ
1
= μ
2
for quantitative data
Ex.
H
0
μ
g
= μ
b
H
a
(alternate hypothesis) is a statement that there is a measurable difference, an association or a change the subject of your interest
In words:
Ex.
I think that girls and boys will not have equal speeds when solving computations