Question

the data on the major and home state for a randomly chosen group of students at a private college are: ( a_{11}=68 ), ( a_{12}=99 ), ( a_{13}=64 ), ( a_{14}=145 ), ( a_{21}=41 ), ( a_{22}=217 ), ( a_{23}=22 ), ( a_{24}=93 ), ( a_{31}=75 ), ( a_{32}=219 ), ( a_{33}=72 ), ( a_{34}=285 ), ( a_{41}=128 ), ( a_{42}=144 ), ( a_{43}=52 ), ( a_{44}=58 ) the following contingency table was created based on the above data, where you will need to calculate the grand total, the total for each row, and the total for each column on your own before answering the question that appears after the table. calculate the probability of a student whose home state is pennsylvania, given that the student is majoring in biology (for answers up to four decimal places, enter the number in the box below without rounding it. for answers that have more than four decimal values, round your answer to four decimal places).

Explanation:

Step1: Calculate row totals

• Pennsylvania: \(68 + 99+64 + 145=376\)
• New Jersey: \(41+217 + 22+93 = 373\)
• New York: \(75+219+72 + 285=651\)
• Maryland: \(128+144+52 + 58=382\)

Step2: Calculate column totals

• Biology: \(68+41 + 75+128=312\)
• Accounting: \(99+217+219+144 = 679\)
• History: \(64+22+72+52=210\)
• Education: \(145+93+285+58 = 581\)

Step3: Calculate grand - total

\(376+373+651+382=1782\) (or \(312+679+210+581 = 1782\))

Step4: Use the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the context of contingency tables, if \(A\) is the event “home - state is Pennsylvania” and \(B\) is the event “majoring in Biology”, then \(P(A\cap B)=\frac{n(A\cap B)}{n(\text{grand - total})}\) and \(P(B)=\frac{n(B)}{n(\text{grand - total})}\), so \(P(A|B)=\frac{n(A\cap B)}{n(B)}\)

Here, \(n(A\cap B) = 68\) (the number of students from Pennsylvania majoring in Biology) and \(n(B)=312\) (the total number of students majoring in Biology)

\(P=\frac{68}{312}\approx0.2180\)

Answer:

\(0.2180\)