You have conducted a study to determine whether your products are meeting the cu

Question

You have conducted a study to determine whether your products are meeting the customer requirements. The customer's design states that parts must have a mean length of 220mm. You've collected 8 samples and the mean length of the samples (x) is 218mm. The value of o for part length is 1.1mm. (a) Conduct a hypothesis test with alpha = 0.05 to see if the design requirements are being met. Yout work should show all 7 steps of the hypothesis testing procedure. On the cover sheet, just write your conclusion, (i. E., "There is/is not sufficient evidence at the alpha = 0.05 level of significance to conclude that mean length....") (b) For the results of your hypothesis test in part (a), what is the p-value for this test? (c) What is the probability of making a Type I error? (d) What would it mean in the context of this situation (i. E., in terms of mean length) if you made a Type I error? (e) If a Type I error occurs, who is penallized-your company or your customer who receives the parts? (Examples of "penalties" could be your company discarding parts that aren't really detective or the customer receiving parts that don't meet their specifications.) Briefly explain your answer. (f) If you? Want to detect a shift of mean length from 220mm to 219mm, what it the probability of making an error and not detecting this shift? (g) Based on your answer to part (f), what is the power of this test to detect a shift of a mean length of 219mm? (h) Who is penallzed if the error in part (f) occurs, your company or your customer who receives the parts? Briefly explain your answer.

Explanation / Answer

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   220  
Ha:    u   =/   220  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical t,              
df = n - 1 =    7          
tcrit =    +/-   2.364624252      
              
Getting the test statistic, as              
              
X = sample mean =    218          
uo = hypothesized mean =    220          
n = sample size =    8          
s = standard deviation =    1.1          
              
Thus, t = (X - uo) * sqrt(n) / s =    -5.142594772          

As |t| > 2.635, we REJECT THE NULL HYPOTHESIS.

There is significant evidence at 0.05 level that the true mean length of the parts is not 220 mm. [CONCLUSION]

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b)
              
Also, the p value is, as this is two tailed,              
              
p =    0.001334898   [ANSWER]

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c)

The probability of making a type I error is alpha, so

P(type I) = alpha = 0.05 [ANSWER]

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d)

A type I error is incorrectly rejecting a true Ho.

Hence, it is incorrectly saying that the true mean length of the parts is not 220 mm, when in fact, it is 220 mm. [ANSWER]
              
Comparing t and tcrit (or, p and significance level), we   REJECT THE NULL HYPOTHESIS.      

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