Hypothesis Testing.

[Ethan3141]

For (a) I figured it was matched pairs but to be honest this is kind of a guess and this question might seem stupid but how does one determine whether data is matched pairs of independent groups?

For (b)I am just trying things (not sure how to do it), so far I have:

1. Calculated the differences between platinum plugs and conventional plugs.
2. Calculated the sample mean and standard deviation of these differences
3. Calculated the degrees of freedom via: 8 + 8 - 2 = 14
4. Chose my Hypotheses, [MATH]H_0: \mu_1\geq\mu_2, H_1:\mu_1<\mu_2[/MATH]
5. Looked up the t-value from the table using 0.05 1-tail alpha column and df 14 row, which is [MATH]t_{14,0.05} = 1.7613[/MATH]

Where I am stuck is calculating the t-value from the data, I don't know which formula to use, here is the list I have at my disposal:



Considering I answered for part (a) that there where matched pairs I am thinking I need to use the formula second from the bottom but then I calculated my prior t incorrectly and also I am just guessing. Any guidance would be appreciated which includes spotting mistakes in my previous steps and helping me decide what formula I should be using.

Thanks, Ethan

 

[Ethan3141]

REDRAFT: So I have tried to adapt my answer more towards what I think, e.g the data is matched pairs, here is what I have changed:

1. Same as before
2. Same as before
3. Calculated my degrees of freedom to be [MATH]n-1=8-1=7[/MATH]
4. Chose my hypotheses: [MATH]H_0: \mu_d=0, H_1:\mu_d>0[/MATH]
5. Looked up [MATH]t_{critical} = t_{7,0.05} = 1.8946[/MATH]
6. Calculated t-value: [MATH]t_{calc} = \frac{\bar{x}-\mu_d}{\frac{s_d}{\sqrt{n}}} = \frac{131.25-0}{\frac{96.1305}{\sqrt{8}}} = 3.8617[/MATH]
7. Observed [MATH]t_{calc} > t_{critical}[/MATH]
8. Concluded: There is sufficient evidence to reject [MATH]H_0[/MATH] meaning the data provides evidence that the platinum plug lasts longer.

QUESTIONS:

1. What mistakes have I made? (Just assuming there are some because this is the first question I have attempted on this topic)
2. If my alternative hypothesis was [MATH]H_1:\mu_d<0[/MATH] would my t-value = [MATH]-t_{7,0.05} = -1.8946[/MATH]?
3. What assumptions has this used?