Let's walk through the solution step-by-step to determine the critical value for the Chi-Square goodness-of-fit test.
### Step 1: Determine the Observed Frequencies
The observed frequencies are given in the problem:
- Monday: 40
- Tuesday: 33
- Wednesday: 35
- Thursday: 32
- Friday: 60
### Step 2: Calculate the Total Number of Observations
The total number of customers served throughout the week is:
[tex]\[ N = 40 + 33 + 35 + 32 + 60 = 200 \][/tex]
### Step 3: Determine the Number of Categories
Each day of the week represents a category. Since we are considering five days (Monday to Friday), the number of categories is:
[tex]\[ \text{Number of categories} = 5 \][/tex]
### Step 4: Calculate the Degrees of Freedom
The degrees of freedom for a Chi-Square test is determined by the number of categories minus one:
[tex]\[ \text{Degrees of freedom} = \text{Number of categories} - 1 = 5 - 1 = 4 \][/tex]
### Step 5: Determine the Significance Level
The significance level ([tex]\(\alpha\)[/tex]) given in the problem is 0.05.
### Step 6: Lookup the Critical Value
Using a Chi-Square distribution table, we find the critical value corresponding to [tex]\( \alpha = 0.05 \)[/tex] and [tex]\( \text{degrees of freedom} = 4 \)[/tex].
Based on these parameters, the critical value from the Chi-Square distribution is:
[tex]\[ \text{Critical value} = 9.487729036781154 \][/tex]
### Final Answer
The critical value for the Chi-Square goodness-of-fit test at a 5% significance level with 4 degrees of freedom is:
[tex]\[ 9.487729036781154 \][/tex]
