Alpha Significance: Understanding & Analyzing Data Set

In summary, the conversation discusses the process of calculating a test statistic and its relationship to the normal curve. The conclusion drawn is that, based on the calculated test statistic being greater than a certain threshold, the null hypothesis can be rejected with a small risk. This indicates that the samples did not come from a population with the stated mean and standard deviation.
  • #1
gummz
32
2

Homework Statement


What conclusions can be drawn from this data set? What assumptions are you making?

Homework Equations


http://i.imgur.com/M9YQGAF.png
I hope this is legible.

The Attempt at a Solution


The solution is what I'm having trouble with.
I just don't get how that test statistic has anything to do with whether that hypothesis is rejected or not, and what meaning it has for the test statistic to be larger than that number.
 
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  • #2
This is an interesting question that might require a lengthy discussion.
In general, the process of calculating a test statistic (Z value) is designed so that you can relate your data set to the normal curve. Remember that most normally distributed stuff will be observed in the biggest part of the curve, and it is less common to see something in the tails (left or right extremes) of the curve.
When you find ##|Z_{0.025}| = 1.96##, that is giving you a value which says that 2.5% of observations from a population that has mean of zero and standard deviation 1 (the normal curve standard) will be greater that 1.96 and 2.5% will be less that -1.96. Those two tails account for 5% of the population.
Your calculated test statistic was greater than 1.96, which indicates that it would be an uncommon observation if your null hypothesis is true.
Your conclusion, then is that, accepting the small risk that your observation was a random chance (alpha), you can reject the null hypothesis. That is that these samples did not come from a population with the stated mean and standard deviation.
 

1. What is alpha significance?

Alpha significance refers to the threshold or level of probability at which a researcher determines that the results of a statistical test are considered statistically significant. It is commonly set at 0.05, meaning that there is a 5% chance that the results occurred by chance and not due to the variables being studied.

2. How is alpha significance determined?

Alpha significance is typically determined by the researcher prior to conducting the statistical test. It is chosen based on the level of confidence the researcher wants in their results, as well as the potential consequences of making a Type I error (rejecting the null hypothesis when it is actually true). It is commonly set at 0.05 or 0.01, but can vary depending on the study.

3. What is the relationship between alpha and p-value?

Alpha and p-value are closely related in that they both represent the probability of obtaining the observed results by chance. Alpha is set by the researcher and serves as the threshold for determining whether the p-value (calculated from the data) is considered statistically significant. If the p-value is lower than the alpha level, the results are considered statistically significant.

4. How does alpha significance affect the interpretation of data?

Alpha significance plays a crucial role in the interpretation of data, as it helps determine whether the results of a statistical test are considered significant or not. If the p-value is lower than the set alpha level, the researcher can reject the null hypothesis and conclude that there is a significant relationship between the variables being studied. If the p-value is higher than the alpha level, the null hypothesis cannot be rejected and there is no statistical significance found in the data.

5. Can alpha significance be changed after conducting a study?

It is generally not recommended to change the alpha significance level after conducting a study, as this can lead to biased results and undermine the validity of the study. However, if there are valid reasons for changing the alpha level (such as adjusting for multiple comparisons), it must be clearly stated and justified in the study's methods and results.

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