Question

2) The surface finish of aluminum which is suspected to be a normally dissboeoan ds as follows measured in micro-inches. The roughness for 12 such instruments (Use the 1% significance level) Perform, Manual Analysis variable SURFACE ROUGHNESS 25.1 28.9 22.I 20.9 23.5 27.9 38.0 26.0 28.7 20.9 26.8 38.5 a) Find the mean and standard deviation of the roughness data above. b) Find the 75 percentile for the data, i e. the ON+1)th value for the distribution (Hint: arrange the numbers from the lowest to the highest and find the upper quartile) c) If you wish to test, the hypothesis that the mean surface roughness of the aluminum exceeds 25.0 micro-inches, what statistical test would you employ, i.e. state and perform a manual analysis of the data. d) Setup the appropriate hypothesis for investigating this question. Test the hypothesis you have ormulated in part (c) at the 1% level of significance or the 99% level of confidence. e) What are your conclusions ? Use α 0.01

Answer 1

a) n= 12

Sample mean: $\bar{X}=\sum X_i/n= 27.275$

Sample standard deviation: $S=\sqrt{\sum (X_i-\bar{X})/n-1}= 5.838606$

b) 75% percentile: Arrange data set in increasing order.

3* (n+1)/4 = 3* 13/4 = 9.75 th observation= 9th observation+ 0.75(10th - 9th observation)

=> 28.7+0.75(28.9-28.7)= 28.85

c) Populaiton mean= 25

One sample t-test:

$H_0:\mu =25$

$H_a:\mu >25$

df= n-1= 12-1 =11

test statistic:

$t= \frac{\bar{X}-\mu }{S/\sqrt{n}}=\frac{27.275-25}{5.838606/\sqrt{12}}=1.34978$

d) Critical value: $t_c=t_{\alpha ,n-1}=t_{0.01,11}=2.7181$

e) P-value: 0.102

At 0.01 significant level, failed to reject H0 and the test is not significant. There is insufficient evidence to support the claim that the roughness of aluminium exceeds 25 micro inches.