Question

Best High Schools. In an issue of Newsweek(Vol. CXLV, No. 20, pp. 48-57), B. Kantrowitz listed "The 100 best high schools in America" according to a ranking devised by J. Mathews. Another characteristic measured from the high school is the percent free lunch, which is the percentage of student body that is eligible for free and reduced-price lunches, an indicator of socioeconomic status. A percentage of 40% or more generally signifies a high concentration of children in poverty. The top 100 schools, grouped according to their percent free lunch, is as follows.

(a) Use the table to design a procedure for obtaining a stratified sample (with proportional allocation) of 25 high schools from the list of the top 100 ranked high schools.

(b) If stratified random sampling with proportional allocation is used to select the sample of 25 high schools, how many would be selected from the stratum with a percent-free-lunch value of 30-under 40?

Short answer

Part (a) Use the procedure of stratified sample.

Part (b) 2 high schools.

Step-by-step solution

Part (a) Step 1. Given information. 

The given statement is:

In an issue of Newsweek, B. Kantrowitz listed "The 100 best high schools in America" according to a ranking devised by J. Mathews.

Another characteristic measured from the high school is the percent free lunch, which is the percentage of the student body that is eligible for free and reduced-price lunches, an indicator of socioeconomic status.

A percentage of 40% or more generally signifies a high concentration of children in poverty. The top 100 schools, are grouped according to their percent free lunch.

The given table is shown below:

Part (a) Step 2. Design a procedure for obtaining a stratified sample (with proportional allocation) of 25 high schools.

The given table containing 100 schools is divided into 5 groups according to bases on the percent free lunch.

A total of 25 samples have to be selected from all the groups.

The stratified sample procedure is going to help us in collecting a total representative sample from all the provided schools.

Part (b) Step 1. Find how many high schools selected from the stratum with a percent-free-lunch value of 30-under 40.

The population contains 100 schools and the sample size is 25.

Therefore,

$\frac{100}{25}=4$

It means we need samples from every 4 schools.

There is a total of 8 schoolings the range of 30-40 in percent free lunch.

Therefore,

role="math" localid="1651566798178" $\frac{8}{4}=2$

In order to stratify random sampling with proportional allocation, we need 2 high schools.