We need to calculate the expected value for the number of readers from a rural area who prefer fiction books, assuming the book preference and residential area are independent. This is done using the formula for the expected frequency in a contingency table:

$E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}}$

Where:

• $E_{ij}$ is the expected value for the cell in row $i$ and column $j$,
• $\text{Row Total}_i$ is the total for the $i$-th row (in this case, the total number of rural readers),
• $\text{Column Total}_j$ is the total for the $j$-th column (in this case, the total number of fiction book preferences),
• $\text{Grand Total}$ is the overall total number of readers.

Step-by-Step Calculation

1. Row total for rural area:
   The total number of readers from the rural area is 185.

2. Column total for fiction books:
   The total number of readers who prefer fiction books is 160.

3. Grand total:
   The overall total number of readers is 499.

Now, using the formula for expected frequency:

$E_{\text{rural, fiction}} = \frac{( \text{Rural Total} \times \text{Fiction Total} )}{\text{Grand Total}} = \frac{( 185 \times 160 )}{499}$

Let me calculate this.The expected value for the number of readers from a rural area who prefer fiction books is approximately 59.318637.

Would you like to explore more details or have any other questions?

Here are 5 related questions you might consider:

1. How can we test if book preference is independent of residential area using a chi-square test?
2. What is the expected value for the number of urban readers who prefer nonfiction books?
3. How do we calculate the chi-square test statistic using the observed and expected values?
4. What is the critical value for the chi-square test at a significance level of 0.1?
5. How do you interpret the result of the chi-square test in this context?

Tip: The larger the difference between observed and expected values, the more evidence there is of a relationship between the two variables.