Determine the value for each variable in the two-way table.
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Right & Left & Total \\
\hline Female & 24 & $a$ & $b$ \\
\hline Male & $c$ & 6 & 34 \\
\hline Total & 52 & $d$ & $e$ \\
\hline
\end{tabular}
A group of 65 baseball players were surveyed about which hand they favor for batting. The data from the survey are shown in the table.
$a=\square$
$b=\square$
$c=\square$
$d=\square$
$e=\square$
Answer (1)
Determine the value of c (total players): c = 65 .
Calculate c (Male-Right): c = 34 − 6 = 28 .
Calculate d (Total-Left): d = 65 − 52 = 13 .
Calculate a (Female-Left): a = 13 − 6 = 7 .
Calculate b (Female-Total): b = 24 + 7 = 31 .
The values are: a = 7 , b = 31 , c = 28 , d = 13 .
Explanation
Understanding the Problem We are given a two-way table representing data from a survey of 65 baseball players about their favored batting hand. Our goal is to determine the values of the variables a , b , c , and d in the table. We will use the information provided in the table to find these values.
Finding the value of c (Total) First, we can determine the value of c using the fact that the total number of players surveyed is 65. From the table, we see that the total number of players is also represented by c . Therefore, c = 65 .
Finding the value of c (Male-Right) Next, we can find the value of c (Male-Right) using the information about the number of male players. We know that the total number of male players is 34, and the number of male players who favor the left hand is 6. Therefore, the number of male players who favor the right hand is c = 34 − 6 = 28 .
Finding the value of d Now, we can find the value of d (Total-Left). We know that the total number of players is 65, and the number of players who favor the right hand is 52. Therefore, the number of players who favor the left hand is d = 65 − 52 = 13 .
Finding the value of a We can now find the value of a (Female-Left). We know that the total number of players who favor the left hand is d = 13 , and the number of male players who favor the left hand is 6. Therefore, the number of female players who favor the left hand is a = 13 − 6 = 7 .
Finding the value of b Finally, we can find the value of b (Female-Total). We know that the number of female players who favor the right hand is 24, and the number of female players who favor the left hand is a = 7 . Therefore, the total number of female players is b = 24 + 7 = 31 .
Final Answer In summary, we have found the following values for the variables: a = 7 , b = 31 , c = 28 , and d = 13 .
Examples
Two-way tables are commonly used in surveys and data analysis to organize and understand relationships between different categories. For example, a marketing team might use a two-way table to analyze customer preferences for different product features based on age groups. By organizing the data in this way, they can identify trends and make informed decisions about product development and marketing strategies. This helps in tailoring products to specific customer segments, improving customer satisfaction and increasing sales.
