Question

a restaurant manager suspects that service declines during off - peak hours. to investigate, he selects a random sample of 100 customers who dined in his restaurant during peak hours and a random sample of 70 customers who dined in his restaurant during off - peak hours. each customer rated the service on a scale of 1 to 5, where 1 = highly dissatisfied and 5 = highly satisfied. the results are displayed in the tables.
observed counts:
the conditions for inference are met. what is the value of the chi - square test statistic and p - value of this test?
find the chi - square table here.
○ ( chi^{2}=13.88 ). the p - value is between 0.005 and 0.01.
○ ( chi^{2}=13.88 ). the p - value is between 0.10 and 0.15.
○ ( chi^{2}=15.82 ). the p - value is between 0.0025 and 0.005.
○ ( chi^{2}=15.82 ). the p - value is between 0.05 and 0.10.

Explanation:

Step1: Calculate the degrees of freedom

The formula for degrees of freedom in a chi - square test for independence is \(df=(r - 1)(c - 1)\), where \(r\) is the number of rows and \(c\) is the number of columns. Here, \(r = 5\) (ratings 1 - 5) and \(c=2\) (peak and off - peak). So \(df=(5 - 1)\times(2 - 1)=4\).

Step2: Use the chi - square table

Looking at the chi - square table with \(df = 4\):

• For \(\chi^{2}=13.88\), the critical values: \(\chi_{0.01,4}^{2}=13.28\) and \(\chi_{0.005,4}^{2}=14.86\). So the \(P\) - value for \(\chi^{2}=13.88\) is between \(0.005\) and \(0.01\).
• For \(\chi^{2}=15.82\), the critical values: \(\chi_{0.005,4}^{2}=14.86\) and \(\chi_{0.0025,4}^{2}=16.42\). So the \(P\) - value for \(\chi^{2}=15.82\) is between \(0.0025\) and \(0.005\).

Answer:

\(\chi^{2}=15.82\). The \(P\) - value is between \(0.0025\) and \(0.005\).