Question

Put a lid on it! At some fast-food restaurants, customers who want a lid for their drinks get them from a large stack left near straws, napkins, and condiments. The lids are made with a small amount of flexibility so they can be stretched across the mouth of the cup and then snuggly secured. When lids are too small or too large, customers can get very frustrated, especially if they end up spilling their drinks. At one particular restaurant, large drink cups require lids with a “diameter” of between $3.95$and $4.05$inches. The restaurant’s lid supplier claims that the mean diameter of their large lids is $3.98$inches with a standard deviation of $0.02$inches. Assume that the supplier’s claim is true.

(a) What percent of large lids are too small to fit? Show your method.

(b) What percent of large lids are too big to fit? Show your method.

(c) Compare your answers to (a) and (b). Does it make sense for the lid manufacturer to try to make one of these values larger than the other? Why or why not?

Short answer

a) Approximately $6.7\%$of lids are too small

b) The proportion of observation less than $3.5$is approximately $1$so the proportion of observation greater than $3.5$is approximately $0$,it is extremely rare for a lid to be too long.

c) it makes more sense to have a larger proportion of lids too small rather than too large.

Step-by-step solution

Part (a) Step-1 Given Information

Given in the question that The restaurant’s lid supplier claims that the mean diameter of their large lids is $3.98$inches with a standard deviation of 0. $0.02$inches we have to find out that at What percent of large lids are too small to fit.

Part (a) Step-2 Explanation

If a lid too small to fit, that means that its diameter is less than$3.95$.let x be the width of lid, we wish to find the proportion of lids for which $x<3.95$

The graph below shows the area below$3.95$

we first find the z value corresponding to $x=3.95$as follows:

localid="1649997527082" $$\begin{gathered} z=\frac{3.95-3.98}{0.02} \\ =-1.5 \end{gathered}$$

from standard normal tables,the proportion of observation less than$-1.5is0.0668$so we can say that approximately$6.7\%$of lids are too small.

Part (b) Step-1: Given Information

Given in the question that The restaurant’s lid supplier claims that the mean diameter of their large lids is $3.98$inches with a standard deviation of $0.02$inches we have to find out that at What percent of large lids are too big to fit.

Part (b) Step-2: Explanation

If a lid is too large to fit, that means that its diameter is more than $4.05$. Let x be the width of lid. We wish to find the proportion of lids for which $x>4.05$

The graph below shows the area above $4.05$:

We first find the z value corresponding to $4.05$as follows:

localid="1649997541648" $$\begin{gathered} z=\frac{4.05-3.98}{0.02} \\ =3.5 \end{gathered}$$

From the standard normal tables, the proportion of observations less than $3.5$is approximately $1$, so the proportion of observations greater than $3.5$is approximately $0$.

Hence, it is extremely rare for a lid to be too big.

Part (c) Step-1: Given Information

We have to find out that to compare (a) and (b) in order to know Does it make sense for the lid manufacturer to try to make one of these values larger than the other .

Part (c) Step-2: Explanation

It makes more sense to have a larger proportion of lids too small rather than too large. The company wants to make sure that the fit is snug. If more lids are too large, there will be more spills. If lids are too small, customers will just try another lid. But if lids are too large, the customer may not notice and then spill the drink.