The design tolerance thickness for an automotive part is 55mm to 65mm. If the process mean thickness is 61mm with a standard deviation of 2.3mm, then what percentage of the sheets will be acceptable?(What percentage will fall within the design tolerence?) Assume that the distribution of these parts is normally distributed

55mm is 61-2.61σ

65mm is 61+1.54σ
Consult your Z table to see that
.4955+.4382 = .9337 = 93.37% will pass

how u came up with 2.61 and 1.54

How did you come up with 2.61 and 1.54 when SD is 2.3?

To find the percentage of sheets that will be acceptable, we need to calculate the proportion of the distribution that falls within the design tolerance range.

Step 1: Calculate the Z-scores
Z-score is a measure of how many standard deviations an individual data point is from the mean. We need to calculate the Z-scores for the lower and upper limits of the design tolerance range.

Lower Z-score = (Lower Limit - Mean) / Standard Deviation
Lower Z-score = (55 - 61) / 2.3

Upper Z-score = (Upper Limit - Mean) / Standard Deviation
Upper Z-score = (65 - 61) / 2.3

Step 2: Look up the Z-score in the Z-table
Using a standard normal distribution table (also known as Z-table), we can find the area under the curve associated with each Z-score. This represents the percentage of values that fall below or above those Z-scores.

Step 3: Calculate the percentage within the design tolerance
To find the percentage within the design tolerance, we subtract the percentage from the upper Z-score from the percentage from the lower Z-score, and multiply by 100.

Percentage within design tolerance = (Percentage from Upper Z-score - Percentage from Lower Z-score) * 100

Let's calculate the values:

Lower Z-score = (55 - 61) / 2.3 ≈ -2.61
Upper Z-score = (65 - 61) / 2.3 ≈ 1.74

From the Z-table, we find:
Percentage from Lower Z-score ≈ 0.0043 (or approximately 0.43%)
Percentage from Upper Z-score ≈ 0.9599 (or approximately 95.99%)

Percentage within design tolerance = (0.9599 - 0.0043) * 100
Percentage within design tolerance ≈ 95.99%

Therefore, approximately 95.99% of the sheets will be acceptable (i.e., fall within the design tolerances).