# Mathematical models to design effective hypothesis testing during experiments

Trying to use mathematical models to design effective hypothesis testing during experiments. In hypothesis analysis the null hypothesis and significance level of the test must be set prior to data collection. When the consequences of false judgment are severe, one needs to fine tune test parameters in order to control the magnitude of Type I and Type II errors.

Drug A is administered to reduce migraine headaches. The question is whether the drug causes significant increase in body weight or not. Due to long term side effects, we need to be able to stop drug A's usage if indeed it causes obesity.

Drug A's effect on body weight was tested on a sample of N volunteers. The change in body weight of the sample in a 6 month period was recorded. The average change in body weight was a > 0. First assume that the population variance of the change in body weight is s2 (s squared).

1. Determine the significance level (in terms of other parameters) that makes the test as powerful as it is confident in case that the population average change in body weight turns out to be b > a > 0.

Assume that N = 20, a =1, b = 4, s2 = 4.

2. Determine the significance level that makes the test as powerful as it is confident. Determine the significance level that makes Type II error twice the significance level.

3. Repeat question 2 with b = 6. Discuss the changes in all the test parameters and their statistical meaning.

4. Repeat question 2 assuming that population variance is not known but the sample variance is 4.

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#### Solution Preview

1. Determine the significance level (in terms of other parameters) that makes the test as powerful as it is confident in case that the population average change in body weight turns out to be b > a > 0. Assume that N = 20, a =1, b = 4, s2 = 4.

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<br>I doubt if there're mistakes in the your raw data, because the two values of a and b are too different to make significant test. (which means that b>a is almost for certain) This means that we should find the significance level that makes the probability of type I and type II error equal. Because population variance is known, we can use z distribution.

<br>Then the standard error is SE= SQRT(S2/N)= SQRT(4/20)=0.447

<br>z = ...

#### Solution Summary

The solution addresses - Trying to use mathematical models to design effective hypothesis testing during experiments. In hypothesis analysis the null hypothesis and significance level of the test must be set prior to data collection. When the consequences of false judgment are severe, one needs to fine tune test parameters in order to control the magnitude of Type I and Type II errors.

Drug A is administered to reduce migraine headaches. The question is whether the drug causes significant increase in body weight or not. Due to long term side effects, we need to be able to stop drug A's usage if indeed it causes obesity.

Drug A's effect on body weight was tested on a sample of N volunteers. The change in body weight of the sample in a 6 month period was recorded. The average change in body weight was a > 0. First assume that the population variance of the change in body weight is s2 (s squared).

1. Determine the significance level (in terms of other parameters) that makes the test as powerful as it is confident in case that the population average change in body weight turns out to be b > a > 0.

Assume that N = 20, a =1, b = 4, s2 = 4.

2. Determine the significance level that makes the test as powerful as it is confident. Determine the significance level that makes Type II error twice the significance level.

3. Repeat question 2 with b = 6. Discuss the changes in all the test parameters and their statistical meaning.

4. Repeat question 2 assuming that population variance is not known but the sample variance is 4.