Statistical complications please

[Prayerofhope]

A roadway construction process uses a machine that pours concrete onto the roadway and measures the thinckness of the concrete so the roadway will measure up to the required depth in inches. The concrete thickness needs to be consistent across the road, but the machine isn't perfect and it is costly to operate. Since there's a safety hazard if the roadway is thinner than the minimum 23 inches thickness, the company sets the machine to average 26 inches for the batches of concrete. They believe the thickness level of the machine's concrete output can be decribed by a normal model with standard deviation 1.75 inches. [show work]

a) What percent of the concrete roadway is under the minimum depth ?


b) The company's lawyers insist that no more than 3% of the output be under the limit. Because of the expense of operating the machine, they cannot afford to reset the mean to a higher value. Instead they will try to reduce the standard deviation to achieve the "only 3% under" goal. What SD must they attain?


c) Explain what achieving a smaller standard deviation means in this context.

 

[statdad]

a) You have a random variable X that is normally distributed with mean 26 and standard deviation 1.75, and you want $$P(X < 23)$$. If you have a computer program for these use it, otherwise get Z-scores and use the table.

b) Can you find the number from the standard normal distribution that satisfies $$P(Z \le z) = 0.03$$? If so then, with
$$\sigma$$ as the new but unknown standard deviation, solve

$$\frac{23-26}{\sigma} = z$$

for $$\sigma$$.

c) What does standard deviation indicate when you deal with measured quantities?