Question

a data set lists weights (lb) of plastic discarded by households. the highest weight is 5.26 lb, the mean of the weights is $\bar{x}=1.899$ lb, and the standard deviation of the weights is $s = 1.026$ lb.
a. what is the difference between the weight of 5.26 lb and the mean of the weights?
b. how many standard deviations is that the difference found in part (a)?
c. convert the weight of 5.26 lb to a z - score.
d. if we consider weights that convert to z - scores between - 2 and 2 to be neither significantly low nor significantly high, is the weight of 5.26 lb significant?
a. the difference is 3.361 lb. (type an integer or a decimal. do not round.)
b. the difference is 3.28 standard deviations. (round to two decimal places as needed.)
c. the z score is $z = 3.28$. (round to two decimal places as needed.)
d. the highest weight is significantly low. significantly high. not significant.

Explanation:

Step1: Calculate difference from mean

Subtract mean from the value. Given value $x = 5.26$ lb and mean $\bar{x}=1.899$ lb. So, $5.26 - 1.899=3.361$ lb.

Step2: Calculate number of standard - deviations

Divide the difference by the standard deviation. Standard deviation $s = 1.026$ lb. So, $\frac{3.361}{1.026}\approx3.28$ (rounded to two decimal places).

Step3: Calculate z - score

The formula for z - score is $z=\frac{x-\bar{x}}{s}$. Substituting $x = 5.26$, $\bar{x}=1.899$ and $s = 1.026$, we get $z=\frac{5.26 - 1.899}{1.026}\approx3.28$ (rounded to two decimal places).

Step4: Determine significance

Since the z - score $z = 3.28>2$, the weight is significantly high.

Answer:

a. $3.361$
b. $3.28$
c. $3.28$
d. significantly high