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Determine the critical value.

A local retailer currently schedules employees based on the assumption that they serve customers uniformly throughout the week (the same number each day). Management is starting to question this assumption and decides to collect data on the number of customers served each day of the week to perform a Chi-Square goodness-of-fit test at a [tex]5\%[/tex] significance level.

\begin{tabular}{|l|c|c|c|c|c|c|}
\hline & Monday & Tuesday & Wednesday & Thursday & Friday & Total \\
\hline & & & & & & \\
\begin{tabular}{l}
Number \\
Served
\end{tabular} & 40 & 33 & 35 & 32 & 60 & 200 \\
\hline
\end{tabular}

Provided the assumptions of the test are satisfied, determine the critical value for the test.

Sagot :

GinnyAnswer
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]
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