Suppose that a bookseller wants to study the relationship between book preference and residential area. A random sample of readers is chosen for the study, and each participant is asked to choose their favorite genre out of the following choices: mystery, fiction, nonfiction, and self-help. The results are detailed below. Does the evidence gathered show a relationship between book preference and residential area at α=0.1α=0.1? Observed Sample of ReadersMysteryFictionNonfictionSelf-HelpTotalRural

38

72

40

35

185

Urban

56

88

76

94

314

Total

94

160

116

129

499 Calculate the expected value for the number of readers from a rural area who prefer fiction books. Round your answer to six decimal places.

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.

Suppose that a bookseller wants to study the relationship between book preference and residential area. A random sample of readers is chosen for the study, and each participant is asked to choose their favorite genre out of the following choices: mystery, fiction, nonfiction, and self-help. The results are detailed below. Does the evidence gathered show a relationship between book preference and residential area at α=0.1? Observed Sample of ReadersMysteryFictionNonfictionSelf-HelpTotalRural 38 72 40 35 185 Urban 56 88 76 94 314 Total 94 160 116 129 499 Calculate the expected value for the number of readers from a rural area who prefer fiction books. Round your answer to six decimal places.

Learn how to calculate the expected value for the number of rural readers who prefer fiction books in a study examining the relationship between book preference and residential area. This problem involves statistical concepts like contingency tables, chi-square tests, and expected value calculation. Ideal for students in College Level Statistics or AP Statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation