Q. 3.141 - Elementary Statistics

Question

Student-to-Faculty Ratio. A sample of fifth-grade classes was studied in the journal article "Predicting Feelings of School Safety for Lower, Middle, and Upper School Students"  by R. Bachman et al. One of the variables collected was the class size in terms of student-to-faculty ratio. The student-to-faculty ratios of the $81$fifth grade classes sampled have a mean of $15.83$ and a standard deviation of $1.74$

a. Construct a graph
b. Apply Property $1$ of the empirical rule to make pertinent statements about the observations in the sample.
c. Repeat part (b) for Property $2$ of the empirical rule.
d. Repeat part (b) for Property $3$ of the empirical rule.

Step-by-Step Solution

Answer

Part(a) Required graph is given below.

Part(b) By property $1$ of empirical rule $68\%$ of sample to make statement about observation in sample.

Part(c)  By property $2$ of empirical rule $95\%$ of sample to make statement about observation in sample.

Part(d)  By property $3$ of empirical rule $99.7\%$ of sample to make statement about observation in sample.

Part(a) Step 1 : Given information

We are given that the student-to-faculty ratios of the $81$ fifth grade classes sampled have a mean of $15.83$ and a standard deviation of $1.74$

Part(a) Step 2 : Simplify

For drawing the graph, first we need to find standard deviations to either side of mean from empirical rule, which means we need to find $\overline{x}-3s, \overline{x}-2s, \overline{x}-s, \overline{x}, \overline{x}+s, \overline{x}+2s, \overline{x}+3s$ which is discussed in further steps.

Required graph is

Part(b) Step 1 : Given information

We are given that the student-to-faculty ratios of the $81$ fifth grade classes sampled have a mean of $15.83$ and a standard deviation of $1.74$

Part(b) Step 2 : Simplify

Student to faculty ratio is given in roughly bell shaped . Therefore, by Property I of the empirical rule, we can say that nearly $68\%$ of the ratio in the sample have within one standard deviation to either side of the mean.

Therefore, 

$81\times \frac{68}{100}=55.08\approx 55$

 One standard deviation to either side of the mean is from 

$$\begin{gathered} \overline{x}-s=15.83-1.74=14.09 \\ \overline{x}+s=15.83+1.74=17.57 \end{gathered}$$

From above calculations , we can say that,

 Approximately $55$ of the $81$  in the sample have ratio between $14.09$ and $17.57$

Part(c) Step 1 : Given information

We are given that the student-to-faculty ratios of the $81$ fifth grade classes sampled have a mean of $15.83$ and a standard deviation of $1.74$

Part(c) Step 2 : Simplify

Student to faculty ratio is given in roughly bell shaped . Therefore, by Property 2 of the empirical rule, nearly$95\%$ of the classes  in the sample have ratio within two standard deviations to either side of the mean.

Now, 

$81\times \frac{95}{100}=76.95\approx 77$

Two standard deviations to either side of the mean is from
$$\begin{gathered} \overline{x}-2s=15.83-2(1.74)=12.35 \\ \overline{x}+2s=15.83+2(1.74)=19.31 \end{gathered}$$

From above calculations we can say that,

 nearly,$77$ of the $81$ classes in the sample have ratio between $12.35$ and $19.31$

Part(d) Step 1 : Given information

We are given that the student-to-faculty ratios of the $81$ fifth grade classes sampled have a mean of $15.83$ and a standard deviation of $1.74$

Part(d) Step 2 : Simplify

Student to faculty ratio is given in roughly bell shaped . Therefore, by Property 3 of the empirical rule, nearly$99.7\%$ of the classes in the sample have ratio within three standard deviations to either side of the mean.

Now, 

$81\times \frac{99.7}{100}=80.76\approx 81$

Three standard deviations to either side of the mean is from

$$\begin{gathered} \overline{x}-3s=15.83-3(1.74)=10.61 \\ \overline{x}+3s=15.83+3(1.74)=21.05 \end{gathered}$$
From above calculation , we can say that, 

nearly$81$, that is, all of the $81$ fifth-grade classes in the sample have ratio between $10.61$ and $21.05$