Question

at fast - food restaurants, the lids for drink cups are made with a small amount of flexibility, so they can be stretched across the mouth of the cup and then snugly secured. when lids are too small or too large, customers can get frustrated, especially if they end up spilling their drinks. at one restaurant, large drink cups require lids with a diameter of between 3.95 and 4.05 inches. the restaurants lid supplier claims that the diameter of the large lids follows a normal distribution with mean 3.98 inches and standard deviation 0.02 inch. assume that the suppliers claim is true. the supplier is considering two changes to reduce the percent of its large - cup lids that are too small to 1%: (1) adjusting the mean diameter of its lids, or (2) altering the production process to decrease the standard deviation of the lid diameters. (a) if the mean diameter stays at $mu = 3.98$ inches, what value of the standard deviation will result in only 1% of lids that are too small to fit? standard deviation = inches (round to 4 decimal places.) (b) what effect will the change in part (a) have on the percent of lids that are too large? by the standard deviation the percentage of lids that are too large will .

Explanation:

Step1: Find the z - score for the 1st percentile

The 1st percentile corresponds to a cumulative probability of 0.01. Looking up in the standard - normal distribution table (z - table), the z - score $z$ such that $P(Z

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the original normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know that $x = 3.95$ (the lower - limit of the acceptable diameter), $\mu = 3.98$, and $z=-2.3263$. Rearranging the formula for $\sigma$ gives $\sigma=\frac{x - \mu}{z}$.

Step3: Calculate the standard deviation

Substitute $x = 3.95$, $\mu = 3.98$, and $z=-2.3263$ into the formula: $\sigma=\frac{3.95 - 3.98}{-2.3263}=\frac{-0.03}{-2.3263}\approx0.0129$.

Step4: Analyze the effect on the percentage of lids that are too large

The normal distribution is symmetric. When we decrease the standard deviation while keeping the mean the same, the distribution becomes more concentrated around the mean. Since the distribution is symmetric, if we reduce the standard deviation to make the percentage of lids that are too small equal to 1%, the percentage of lids that are too large will also decrease.

Answer:

(a) 0.0129
(b) decreasing; decrease