Question 1200718: A bank has taken a random sample of 4 customers for each set of various criteria to determine their level of customer satisfaction on a scale of 0 to 10. The data appear in the table below.
Time of day Automated Bank Representative Row means
Morning 6, 5, 8, 4 8, 7, 9, 9 Row 1
𝑥 = 5.75 𝑥 = 8.25 𝑥 = 7.00
Afternoon 3, 5, 6, 5 9, 10, 6, 8 Row 2
𝑥 = 4.75 𝑥 = 8.25 𝑥 = 6.50
Evening 5, 5, 7, 5 9, 10, 10, 9 Row 3
𝑥 = 5.50 𝑥 = 9.50 𝑥 =7.50
Column means Column 1 𝑥 = 5.33 Column 2 𝑥 = 8.67 Total 𝑥 =7.00
For the two measurement problem, use an a = 0.05 level of significance. Conduct appropriate hypothesis tests and conclude whether to reject or not reject the claim that:
a. there is no difference in population mean satisfaction depending on time of contact,
b. there is no difference in population mean satisfaction depending on type of customer contact and
c. there is no interaction between type of contact and time of contact.
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! Certainly, let's analyze the customer satisfaction data using a two-way ANOVA.
**a. Test for Difference in Mean Satisfaction Depending on Time of Contact**
* **Null Hypothesis (H0):** There is no difference in population mean satisfaction among the three time periods (Morning, Afternoon, Evening).
* **Alternative Hypothesis (Ha):** There is a difference in population mean satisfaction among the three time periods.
**b. Test for Difference in Mean Satisfaction Depending on Type of Contact**
* **Null Hypothesis (H0):** There is no difference in population mean satisfaction between Automated and Bank Representative contact.
* **Alternative Hypothesis (Ha):** There is a difference in population mean satisfaction between Automated and Bank Representative contact.
**c. Test for Interaction Between Time of Contact and Type of Contact**
* **Null Hypothesis (H0):** There is no interaction between time of contact and type of contact on customer satisfaction.
* **Alternative Hypothesis (Ha):** There is an interaction between time of contact and type of contact on customer satisfaction.
**Two-Way ANOVA Procedure**
1. **Calculate Sum of Squares:**
* **Total Sum of Squares (SST):** Calculate the total variation in the data.
* **Sum of Squares Between Rows (SSR):** Calculate the variation between the row means (time of contact).
* **Sum of Squares Between Columns (SSC):** Calculate the variation between the column means (type of contact).
* **Sum of Squares of Interaction (SS(RC)):** Calculate the variation due to the interaction between rows and columns.
* **Sum of Squares Within Cells (SSE):** Calculate the variation within each cell (combination of time and contact).
2. **Calculate Degrees of Freedom:**
* **Rows (dfR):** Number of rows - 1 = 3 - 1 = 2
* **Columns (dfC):** Number of columns - 1 = 2 - 1 = 1
* **Interaction (dfRC):** dfR * dfC = 2 * 1 = 2
* **Error (dfE):** Total number of observations - (number of rows * number of columns) = 12 - (3 * 2) = 6
* **Total (dfT):** Total number of observations - 1 = 12 - 1 = 11
3. **Calculate Mean Squares:**
* MSR = SSR / dfR
* MSC = SSC / dfC
* MS(RC) = SS(RC) / dfRC
* MSE = SSE / dfE
4. **Calculate F-statistics:**
* F-statistic for Rows: F_R = MSR / MSE
* F-statistic for Columns: F_C = MSC / MSE
* F-statistic for Interaction: F_RC = MS(RC) / MSE
5. **Determine Critical Values:**
* Use an F-distribution table to find the critical F-values for each test at the 0.05 significance level.
6. **Compare F-statistics to Critical Values:**
* If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.
**Interpretation:**
* **If the null hypothesis for rows is rejected:** There is evidence to suggest that the mean satisfaction scores differ significantly across the different times of contact (Morning, Afternoon, Evening).
* **If the null hypothesis for columns is rejected:** There is evidence to suggest that the mean satisfaction scores differ significantly between Automated and Bank Representative contact.
* **If the null hypothesis for interaction is rejected:** There is evidence to suggest that the effect of time of contact on satisfaction depends on the type of contact (and vice versa).
**Note:**
* This is a general outline of the two-way ANOVA procedure.
* The actual calculations can be quite complex and are typically performed using statistical software (such as R, Python, SPSS, or Excel).
**Disclaimer:**
* This information is for general knowledge and educational purposes only and does not constitute financial, investment, or professional advice.
* The analysis assumes that the data meets the assumptions of ANOVA (normality, homogeneity of variances).
I hope this explanation is helpful! Let me know if you have any further questions.
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