Question

determine the expected count under the null hypothesis.

is achieving a basic skill level in a subject independent to the location of the school?

a random sample of students categorized by the location of school and the number achieving basic skills in three subjects is shown in the table.

two - way table: location vs subject

location of school	reading	math	science	totals
suburban	63	66	65	194
totals	106	108	103	317

under the null hypothesis for a chi - square test of independence, what would be the expected number of students achieving basic math skills in the urban schools?

• 52.8
• 54.0
• 41.9
• 41.0

Explanation:

Step1: Recall Expected Count Formula

For a Chi - square test of independence, the expected count for a cell in a two - way table is given by the formula: \(E=\frac{\text{Row Total}\times\text{Column Total}}{\text{Grand Total}}\)

Step2: Identify Row, Column, and Grand Totals

• The row corresponding to "Urban" schools has a total of \(123\) (Row Total).
• The column corresponding to "Math" subject has a total of \(108\) (Column Total).
• The grand total of all students is \(317\) (Grand Total).

Step3: Calculate Expected Count

Substitute the values into the formula: \(E = \frac{123\times108}{317}\)

First, calculate \(123\times108=123\times(100 + 8)=123\times100+123\times8 = 12300+984 = 13284\)

Then, divide by the grand total: \(E=\frac{13284}{317}\approx41.9\) (Wait, no, let's recalculate: \(123\times108 = 123\times108\). Let's do \(123\times108=(120 + 3)\times108=120\times108+3\times108 = 12960+324 = 13284\). Then \(13284\div317\approx41.9\)? Wait, no, maybe I made a mistake. Wait, row total for urban is 123, column total for math is 108, grand total is 317. Wait, \(123\times108 = 13284\), \(13284\div317\approx41.9\)? Wait, but let's check again. Wait, maybe the row total is urban (123), column total is math (108). Wait, \(123\times108 = 13284\), \(13284\div317\approx41.9\). Wait, but let's check the options. Wait, maybe I mixed up row and column? Wait, no, the formula is (row total * column total)/grand total. Wait, urban row total is 123, math column total is 108, grand total 317. So \(123\times108 = 13284\), \(13284\div317\approx41.9\). But wait, let's check the calculation again: \(317\times41 = 317\times40+317\times1 = 12680 + 317 = 12997\), \(317\times42=12997 + 317 = 13314\). \(13284\) is between \(317\times41\) and \(317\times42\). \(13284-12997 = 287\), so \(41+\frac{287}{317}\approx41 + 0.905\approx41.9\)

Answer:

41.9 (The option corresponding to 41.9)