Blocks of wood used for flooring are cut by machine. Their lengths are normally distributed with mean 230mm & std deviation 2mm, while their widths are normally distributed with mean 80mm and std deviation 1.5mm; the 2 measurements are independent.
The setting on the machine which cuts the blocks to length is to be changed so that while the std deviation remains unchanged, 95% of the blocks will be no longer than 232.7mm. Calculate the new mean length. Ans: 229.41.
After this resetting a block is produced that is only 224.6mm long. Does this suggest that the machine is not correctly set? Ans: z=-2.405, Yes, Not correctly set
The setting on the machine which cuts the blocks to length is to be changed so that while the std deviation remains unchanged, 95% of the blocks will be no longer than 232.7mm. Calculate the new mean length. Ans: 229.41.
After this resetting a block is produced that is only 224.6mm long. Does this suggest that the machine is not correctly set? Ans: z=-2.405, Yes, Not correctly set
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Polly -
Since the standard deviation of 2 mm did not change, we can simply calculate the "number of standard deviations" that is equivalent to an area of 95% or 0.95:
Look up 0.95 in a Standard Normal table. You should find 0.9495 and 0.9505. Well, 0.95 would be right in the middle, so z = 1.645. Find it? Great!
Now, 1.645 is simply the "number of standard deviations" so let's simply multiply it times a standard deviation of 2mm:
1.65 x 2mm = 3.29 mm. So, this is what was added to the new mean:
mean + 3.29 mm = 232.7 mm, solve for mean
mean = 229.41mm (that wasn't too hard, was it!)
B) Given the new mean of 229.41 mm, use it to find z at 224.6:
z = (224.6 - 229.41) / 2 = -2.405
Now, if you look again in the Standard Normal table, P(Z < -2.405) = 0.0081. Now this value of probability is really really low (0.8%), so the machine is NOT CORRECTLY SET.
Hope that helps
Since the standard deviation of 2 mm did not change, we can simply calculate the "number of standard deviations" that is equivalent to an area of 95% or 0.95:
Look up 0.95 in a Standard Normal table. You should find 0.9495 and 0.9505. Well, 0.95 would be right in the middle, so z = 1.645. Find it? Great!
Now, 1.645 is simply the "number of standard deviations" so let's simply multiply it times a standard deviation of 2mm:
1.65 x 2mm = 3.29 mm. So, this is what was added to the new mean:
mean + 3.29 mm = 232.7 mm, solve for mean
mean = 229.41mm (that wasn't too hard, was it!)
B) Given the new mean of 229.41 mm, use it to find z at 224.6:
z = (224.6 - 229.41) / 2 = -2.405
Now, if you look again in the Standard Normal table, P(Z < -2.405) = 0.0081. Now this value of probability is really really low (0.8%), so the machine is NOT CORRECTLY SET.
Hope that helps
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z = (X-Mean)/Standard deviation
i) 95% = 0.9500
The z-value which divides the area under the standard normal curve into two parts of 95% (0.9500) and 5% (0.0500) is + 1.645
This z value is to be located from the area under the standard normal curve table or by using a scientific calculator.
If the table is consulted,
0.9500 area is the total of 0.5000 area on the left side of the mean and 0.4500 area on the right side of the mean.
Therefore z-value corresponding to 0.4500 area is to be located. The process is reverse to that of determining the area for a certain z value.
The z value is 1.645 and it is +ve because it lies on the right side.
1.645 = (232.7-Mean)/2
3.29 = 232.7-Mean
Mean = 232.7-3.29 = 229.41
New mean = 229.41mm
ii) z value corresponding to X=224.6 is
z = (224.6-229.41)/2 = - 2.405
The calculations are tallied BUT how the decision is taken ???
It seems that z = - 2.405 is less than z = + 1.645
Then the answer should be NO and the machine is correctly set.
Verify the conclusion drawn with others.
i) 95% = 0.9500
The z-value which divides the area under the standard normal curve into two parts of 95% (0.9500) and 5% (0.0500) is + 1.645
This z value is to be located from the area under the standard normal curve table or by using a scientific calculator.
If the table is consulted,
0.9500 area is the total of 0.5000 area on the left side of the mean and 0.4500 area on the right side of the mean.
Therefore z-value corresponding to 0.4500 area is to be located. The process is reverse to that of determining the area for a certain z value.
The z value is 1.645 and it is +ve because it lies on the right side.
1.645 = (232.7-Mean)/2
3.29 = 232.7-Mean
Mean = 232.7-3.29 = 229.41
New mean = 229.41mm
ii) z value corresponding to X=224.6 is
z = (224.6-229.41)/2 = - 2.405
The calculations are tallied BUT how the decision is taken ???
It seems that z = - 2.405 is less than z = + 1.645
Then the answer should be NO and the machine is correctly set.
Verify the conclusion drawn with others.