Question

In Exercises 9 –12, consider a value to be significantly low if its z score is less than or equal to -2 or consider the value to be significantly high if its z score is greater than or equal to 2.

Quarters Data Set 29 “Coin Weights” lists weights (grams) of quarters manufactured after 1964. Those weights have a mean of 5.63930 g and a standard deviation of 0.06194 g. Identify the weights that are significantly low or significantly high.

Short answer

The values of weights equal to or greater than 5.76318 grams are significantly high, and the values of weights equal to or less than 5.51542 grams are significantly low.

Step-by-step solution

Given information

The sample mean for the weights is given to be equal to 5.63930 grams.

The sample standard deviation for the weights is given to be equal to 0.06194 grams.

Formula of z-score

The given formula is used to calculate thez-scoreof a value. The z-score value gives the number of standard deviations an observation is above or below the mean value.

$Z=\frac{x-\overline{x}}{s}$

An observation that has a z-score equal to or less than -2 is a significantly low value.

An observation that has a z-score equal to or larger than 2 is a significantly high value.

Identify the weights that are significantly low or high

The significantly low weights are calculated as shown below:

$$\begin{gathered} Z⩽-2 \\ \frac{x-\overline{x}}{s}⩽-2 \\ \frac{x-5.63930}{0.06194}⩽-2 \\ x⩽5.51542 \end{gathered}$$

Hence, the weights equal to or less than 5.51542 grams aresignificantly low.

The significantly high weights are calculated as shown below:

$$\begin{gathered} Z⩾2 \\ \frac{x-\overline{x}}{s}⩾2 \\ \frac{x-5.63930}{0.06194}⩾2 \\ x⩾5.76318 \end{gathered}$$

Hence, the weights equal to or larger than 5.76318 grams aresignificantly high.