5. A fastener company is supplying washers to a manufacturer of kitchen appliances who has specified that the outside diameter be 0.500 ± 0.025 inch. Measurements made on 250 washers show that the process is in good statistical control, the diameters are normally distributed, and the mean and standard deviation for the process are 0.505 and 0.0065 inch, respectively. What percent of the washers made by this process are within the specifications? Sketch the appropriate graph to represent the distribution of the process along with the specifications.

Question

5. A fastener company is supplying washers to a manufacturer of kitchen appliances who has specified that the outside diameter be 0.500 ± 0.025 inch. Measurements made on 250 washers show that the process is in good statistical control, the diameters are normally distributed, and the mean and standard deviation for the process are 0.505 and 0.0065 inch, respectively. What percent of the washers made by this process are within the specifications? Sketch the appropriate graph to represent the distribution of the process along with the specifications.

==this is what i have so far P(0.475<X<0.525)=P((0.475-0.505)/0.0065<Z<(0.525-0.505)/0.0065)=P(-4.6154<Z<3.0769)=0.998954-0.0000=0.998952
=== is this correct

Answer 1

Your calculations are close, but there is a small mistake that needs to be corrected. Let's go through the calculations step by step.

First, we need to standardize the upper and lower limits of the specifications using the mean and standard deviation of the process.

Lower limit = 0.500 - 0.025 = 0.475 inch
Upper limit = 0.500 + 0.025 = 0.525 inch

Next, we calculate the z-scores for the lower and upper limits using the formula:

z = (x - μ) / σ

where x is the limit, μ is the mean of the process, and σ is the standard deviation of the process.

For the lower limit:
z_lower = (0.475 - 0.505) / 0.0065 ≈ -4.6154

For the upper limit:
z_upper = (0.525 - 0.505) / 0.0065 ≈ 3.0769

Now, we need to calculate the area under the normal distribution curve between these z-scores. We can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.

P(z_lower < Z < z_upper) = P(-4.6154 < Z < 3.0769)

Looking up the probabilities or using a calculator, we find that P(-4.6154 < Z < 3.0769) is approximately 0.999819.

So, the percent of washers made by this process that are within the specifications is 99.9819% (or rounded to 99.98%).

To sketch the appropriate graph, we can plot a normal distribution curve with the mean (μ) at 0.505 inch and the standard deviation (σ) at 0.0065 inch. Then, shade the area between the z-scores -4.6154 and 3.0769 to represent the washers within the specifications.