Question

the quantitative data set under consideration has roughly a bell - shaped distribution and contains 306 data values. the mean is 515.1 and the standard deviation is 80.2.
a. what can be said about the percentage of data values between 354.7 and 675.5?
approximately % of the data values are between 354.7 and 675.5.
b. what can be said about the number of data values between 354.7 and 675.5?
approximately data values are between 354.7 and 675.5.

Explanation:

Step1: Calculate the number of standard - deviations from the mean

For the lower value $354.7$:
\[z_1=\frac{354.7 - 515.1}{80.2}=\frac{-160.4}{80.2}=- 2\]
For the upper value $675.5$:
\[z_2=\frac{675.5 - 515.1}{80.2}=\frac{160.4}{80.2}=2\]

Step2: Use the empirical rule for a bell - shaped distribution

The empirical rule states that for a bell - shaped (normal) distribution, approximately $95\%$ of the data lies within $z=-2$ and $z = 2$ standard deviations of the mean. So, approximately $95\%$ of the data values are between $354.7$ and $675.5$.

Step3: Calculate the number of data values

The total number of data values is $n = 306$.
The number of data values between $354.7$ and $675.5$ is approximately $0.95\times306=290.7\approx291$.

Answer:

a. $95$
b. $291$