How to calculate expected values in chi-square test

A table represents the number of students who passed or failed an aptitude test at two different campuses.East Campus West CampusPassed 48 37Failed 52 63In order to determine if there is a significant difference between campuses and pass rate, the chi-square test for association and independence should be performed.What is the expected frequency of West Campus and failed?50 students63 students50.7 students57.5 students

Based on the below table, answer the following question:xP(x)30.0540.4050.3060.1570.10What is the expected value?Question 1Answera.4.85b.2.48c.0.3

To calculate the chi-squared statistic, we need to follow these steps: ### Step-by-Step Calculation: #### iii) Calculate the Expected FrequenciesThe expected frequency for each cell in a contingency table is calculated using the formula: \[ E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}} \] Let's calculate the expected frequencies for each cell: 1. **Facebook:** - Female: \( E_{11} = \frac{(117 \times 64)}{249} \approx 30.07 \) - Male: \( E_{12} = \frac{(132 \times 64)}{249} \approx 33.93 \) 2. **Instagram:** - Female: \( E_{21} = \frac{(117 \times 64)}{249} \approx 30.07 \) - Male: \( E_{22} = \frac{(132 \times 64)}{249} \approx 33.93 \) 3. **Snapchat:** - Female: \( E_{31} = \frac{(117 \times 58)}{249} \approx 27.25 \) - Male: \( E_{32} = \frac{(132 \times 58)}{249} \approx 30.75 \) 4. **Twitter:** - Female: \( E_{41} = \frac{(117 \times 63)}{249} \approx 29.6 \) - Male: \( E_{42} = \frac{(132 \times 63)}{249} \approx 33.4 \) The expected frequencies are: | | Facebook | Instagram | Snapchat | Twitter | Row Total | |------------|----------|-----------|----------|---------|-----------| | **Female** | 30.07 | 30.07 | 27.25 | 29.6 | 117 | | **Male** | 33.93 | 33.93 | 30.75 | 33.4 | 132 | | **Column Total** | 64 | 64 | 58 | 63 | 249 | #### iv) Calculate the Chi-Squared StatisticThe chi-squared statistic is calculated using the formula: \[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \] Where \( O_{ij} \) is the observed frequency and \( E_{ij} \) is the expected frequency. Let's calculate the chi-squared statistic step by step: 1. **Facebook:** - Female: \( \frac{(33 - 30.07)^2}{30.07} \approx 0.29 \) - Male: \( \frac{(31 - 33.93)^2}{33.93} \approx 0.25 \) 2. **Instagram:** - Female: \( \frac{(30 - 30.07)^2}{30.07} \approx 0.00 \) - Male: \( \frac{(34 - 33.93)^2}{33.93} \approx 0.00 \) 3. **Snapchat:** - Female: \( \frac{(26 - 27.25)^2}{27.25} \approx 0.06 \) - Male: \( \frac{(32 - 30.75)^2}{30.75} \approx 0.05 \) 4. **Twitter:** - Female: \( \frac{(28 - 29.6)^2}{29.6} \approx 0.09 \) - Male: \( \frac{(35 - 33.4)^2}{33.4} \approx 0.08 \) Summing these values: \[ \chi^2 = 0.29 + 0.25 + 0.00 + 0.00 + 0.06 + 0.05 + 0.09 + 0.08 = 0.82 \] So, the chi-squared statistic is: \[ \chi^2 \approx 0.82 \] This value should be entered in the box for the chi-squared statistic.

Suppose that χ2 follows a chi-square distribution with 28 degrees of freedom.Use the ALEKS calculator to answer the following.(a) Compute P≤χ216. Round your answer to at least three decimal places.=P≤χ216(b) Find k such that =P≥χ2k0.05. Round your answer to at least two decimal places.=k

Suppose that χ2 follows a chi-square distribution with 12 degrees of freedom.Use the ALEKS calculator to answer the following.(a) Compute P≤χ217. Round your answer to at least three decimal places.=P≤χ217(b) Find k such that =P>χ2k0.025. Round your answer to at least two decimal places.=k