One hundred fifty measurements are taken of the thickness of a coil of rolled st

Question

One hundred fifty measurements are taken of the thickness of a coil of rolled steel sheet measured at approximately equidistant points along the center line of its length. Measurements have a mean value of 11.291 mm and a standard deviation of 0.263 mm. The smallest and largest measurements in the sample are 10.73 and 11.89 mm. Measurements are divided into eight data bins with boundaries at 10.695, 10.845 10.995, 11.145, 11.295, 11.445, 11595, 11.745, and 11.895. The first bin, containing measurements between 10,695 and 10.845, has eight measurements in it, and the count of measurements in the following successive bins is 12, 21, 34, 31, 25, 14, and 5. Apply the 2 test to see whether the measurements fit a Gaussian distribution to a 95% confidence level.

Explanation / Answer

Here there are total 150 measurements

Mean value = 11.291 mm

Standard deviaton = 0.263 mm

Smallest measurement = 10.695 ; Largest measurement = 11.89

Here by using excel we will calculate expected probability and expected frequency of the given range

Here as for column three we use formula = NORMDIST (upper limit,11.291,0.263, TRUE)

for last value it is ranomly attributed one

for column 3, we deduct the probability of upper cell from lower cell.

Now as we get the chi- square value from the table ;

X2 = 1.8561

here dF = n - k - 1 = 10 - 2 -1 = 7 and alpha = 0.05

so X2criticial = 14.067

so here X2 < X2criticial   so we fail to reject the null hypothesis and can conluce that the measurements follow normal distribution.

data bins Frequency Cumu. Probability Respective probability Expected Chi value < 10.845 8 0.0450 0.0450 6.74 0.2339 10.845 - 10.995 12 0.1302 0.0852 12.79 0.0482 10.995 - 11.145 21 0.2517 0.1215 18.22 0.4230 11.295 - 11.445 34 0.5061 0.2544 38.16 0.4530 11.445 - 11.595 31 0.7209 0.2148 32.23 0.0467 11.595 - 11.745 25 0.8761 0.1552 23.28 0.1265 11.745 - 11.895 14 0.9578 0.0817 12.26 0.2480 > 11.895 5 1 0.0422 6.32 0.2768 Sum 150 1 150 1.8561

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